Monoidal Categories and Monoids

A monoid is a concept in mathematics similar to that of a group (see Groups), except that every element need not have an inverse. Therefore, a monoid is a set, equipped with a law of composition which is associative, and an identity element. An example of a monoid is the natural numbers (including zero) with the law of composition given by addition.

In this post, we will introduce certain concepts in category theory (see Category Theory) that are abstractions of the classical idea of a monoid.

A monoidal category is given by a category \mathbf{C}, a bifunctor \Box: \mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}, an object I of \mathbf{C}, and three natural isomorphisms \alpha (also known as the associator), \lambda (also known as the left unitor), and \rho (also known as the right unitor), with components

\displaystyle \alpha_{A,B,C}:A\Box (B\Box C)\cong (A\Box B)\Box C

\displaystyle \lambda_{A}:I\Box A\cong A

\displaystyle \rho_{A}:A\Box I\cong A

satisfying the conditions

\displaystyle 1_{A}\Box\alpha_{A,B,C}\circ\alpha_{A,B\Box C,D}\circ\alpha_{A,B,C}\Box 1_{D}=\alpha_{A,B,C\Box D}\circ\alpha_{A\Box B,C,D}

for any four objects A, B, C, and D of \mathbf{C}, and

\displaystyle \alpha_{A,I,B}\circ 1_{A}\Box \lambda_{B}=\rho_{A}\Box 1_{B}

for any two objects A and B in \mathbf{C}.

The following “commutative diagrams” courtesy of user IkamusumeFan of Wikipedia may help express these conditions better (the symbol \otimes is used here instead of \Box to denote the bifunctor; this is very common notation, but we use \Box following the book Categories for the Working Mathematician by Saunders Mac Lane in order to differentiate it from the tensor product, which is just one specific example of the bifunctor in question; I hope this will not cause any confusion):



If the natural isomorphisms \alpha, \lambda, and \rho are identities, then we have a strict monoidal category.

A monoid object, or monoid in a monoidal category (\mathbf{C},\Box,I) is an object M of \mathbf{C} together with two morphisms \mu:M\Box M\rightarrow M and \eta:I\rightarrow M satisfying the conditions

\displaystyle \mu\circ 1\Box\mu\circ\alpha=\mu\circ\mu\Box 1

\displaystyle \mu\circ \eta\Box 1=\lambda

\displaystyle \mu\circ 1\Box\eta=\rho

Again we can use the following commutative diagrams made by User IkamusumeFan of Wikipedia to help express these conditions:



As examples of monoidal categories, we have the following:

\displaystyle (\mathbf{Set},\times,1)

\displaystyle (\mathbf{Ab},\otimes,\mathbb{Z})

\displaystyle (K\mathbf{-Mod},\otimes_{K},K)


(\mathbf{C}^{\mathbf{C}},\circ,\text{Id})    (\mathbf{C}^{\mathbf{C}} denotes the category of functors from \mathbf{C} to itself)

The monoids in these monoidal categories are given respectively by the following:

Ordinary monoids



Strict monoidal categories

Monads (see Adjoint Functors and Monads)

Among the important kinds of monoidal categories with extra structure are braided monoidal categories and symmetric monoidal categories. A braided monoidal category \mathbf{C} is a monoidal category equipped with a natural isomorphism \gamma (also known as a commutativity constraint) with components \gamma_{A,B}:A\Box B\cong B\Box A satisfying the following coherence conditions

\displaystyle \alpha_{B,C,A}\circ\gamma_{A,B\Box C}\circ\alpha_{A,B,C}=1_{B}\Box\gamma_{A,C}\circ\alpha_{B,A,C}\circ \gamma_{A,B}\Box 1_{C}

\displaystyle \alpha_{C,A,B}^{-1}\circ\gamma_{A\Box B,C}\circ\alpha_{A,B,C}^{-1}=\gamma_{A,C}\Box 1_{B}\circ\alpha_{A,C,B}^{-1}\circ 1\Box\gamma_{A}\gamma_{A,B}

which can be expressed in the following commutative diagrams (once again credit goes to User IkamusumeFan of Wikipedia):



The category \mathbf{C} is a symmetric monoidal category if the isomorphisms \gamma_{A,B} satisfy the condition \gamma_{B,A}\circ\gamma_{A,B}=1_{A\Box B}. We have already encountered an example of this category in The Theory of Motives in the form of tensor categories, defined as a symmetric monoidal categories whose Hom-sets (the sets of morphisms from a fixed object A to another object B) form a vector space (the term “tensor category” is sometimes used to refer to other concepts in mathematics though, including symmetric monoidal categories themselves).

Another important kind of monoidal category is a closed monoidal category. A closed monoidal category is a monoidal category where the functor -\Box B has a right adjoint (see Adjoint Functors and Monads) also known as the “internal Hom functor”, which is like a Hom functor that takes values in the category itself instead of in sets, and is denoted by (\ )^{B}. We have already seen an example of a closed monoidal category in Adjoint Functors and Monads, given by the category of R-modules for a fixed commutative ring R. There A^{B} was given by \text{Hom}(A,B) (this is the set of R-linear transformations from A to B, which itself is an R-module).

We see therefore that the concepts of monoidal categories and monoids can be found everywhere in mathematics. Studying these structures are not only interesting for their own sake, but can also help us find or construct other useful new concepts in mathematics.


Monoidal Category on Wikipedia

Monoid on Wikipedia

Braided Monoidal Category on Wikipedia

Symmetric Monoidal Category on Wikipedia

Closed Monoidal Category on Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Categories for the Working Mathematician by Saunders Mac Lane


The Yoneda Lemma

Update: Some time after I published this post, I came across the following post on another blog that makes for a really nice intuitive introduction to the ideas expressed by the Yoneda lemma:

The Most Obvious Secret in Mathematics at Math3ma

I must admit that my own post might not offer much in the way of intuition (and really could have been written better), so I highly recommend reading the above link in conjunction with this one.

In Algebraic Spaces and Stacks we introduced the notion of a representable functor, and we made use of it to “transfer” the properties of schemes to functors and categories over some fixed category. In this short post we discuss an important related concept, one of the most important concepts in category theory, called the Yoneda lemma.

Let \mathbf{C} be a category, and let A be any object of \mathbf{C}. The Yoneda lemma states that the set of natural transformations from the functor \text{Hom}(-,A) to any contravariant functor G from \mathbf{C} to the category of sets is in bijection with the set G(A).

In the case that G is the contravariant functor \text{Hom}(-,B), where B is an element of \mathbf{C}, the Yoneda lemma says that the set of natural transformations from \text{Hom}(-,A) to \text{Hom}(-,B) is in bijection with the set \text{Hom}(A,B).

We can treat the functor \text{Hom}(-,-) as a covariant functor from the category \mathbf{C} to the category \textbf{Sets}^{\mathbf{C}^{\text{op}}} of contravariant functors from \mathbf{C} to the category of sets, which sends an object A of \mathbf{C} to the contravariant functor \text{Hom}(-,A), and a morphism f:A\rightarrow B of \mathbf{C} to the natural transformation \text{Hom}(-,f):\text{Hom}(-,A)\rightarrow\text{Hom}(-,B). Then the Yoneda lemma, via the result given in the preceding paragraph, says that the functor \text{Hom}(-,-): \mathbf{C}\rightarrow\mathbf{Sets}^{\mathbf{C}^{\text{op}}} is fully faithful. We also say that this functor is an embedding; in particular, it is called the Yoneda embedding. It embeds the category \mathbf{C} into the category \mathbf{Sets}^{\mathbf{C}^{\text{op}}}.

The Yoneda lemma is an important ingredient of the functor of points approach to the theory of schemes. Furthermore, the Yoneda lemma tells us that the category of schemes is embedded as a subcategory of the category of contravariant functors from the category of schemes to the category of sets, so we can also try looking at a bigger subcategory of the latter category, and see if we can come up with interesting objects to study – this actually leads us to the theory of algebraic spaces.


Yoneda Lemma on Wikipedia

Yoneda Lemma on nLab

Representable Functors on Rigorous Trivialities

The Most Obvious Secret in Mathematics at Math3ma

Localization and Gromov-Witten Invariants by Kai Behrend

Categories for the Working Mathematician by Saunders Mac Lane

Adjoint Functors and Monads

In Category Theory we introduced the language of categories, and in many posts in this blog we have seen how useful it is in describing concepts in modern mathematics, for example in the two most recent posts, The Theory of Motives and Algebraic Spaces and Stacks. In this post, we introduce another important concept in category theory, that of adjoint functors, as well as the closely related notion of monads. Manifestations of these ideas are quite ubiquitous in modern mathematics, and we enumerate a few examples in this post.

An adjunction between two categories \mathbf{C} and \mathbf{D} is a pair of functors, F:\mathbf{C}\rightarrow \mathbf{D}, and G:\mathbf{D}\rightarrow \mathbf{C}, such that there exists a bijection

\displaystyle \text{Hom}_{\mathbf{D}}(F(X),Y)\cong\text{Hom}_{\mathbf{C}}(X,G(Y))

for all objects X of \mathbf{C} and all objects Y of \mathbf{D}. We say that F is left-adjoint to G, and that G is right-adjoint to F. We may also write F\dashv G.

An adjunction determines two natural transformations \eta: 1_{\mathbf{C}}\rightarrow G\circ F and \epsilon:F\circ G\rightarrow 1_{\mathbf{D}}, called the unit and counit, respectively. Conversely, the functors F and G, together with the natural transformations \eta and \epsilon, are enough to determine the adjunction, therefore we can also denote the adjunction by (F,G,\eta,\epsilon).

We give an example of an adjunction. Let K be a fixed field, and consider the functors


\displaystyle G:\textbf{Vect}_{K}\rightarrow\textbf{Sets}

where F is the functor which assigns to a set X the vector space F(X) made up of formal linear combinations of elements of X with coefficients in K; in other words, an element of F(X) can be written as \sum_{i}a_{i}x_{i}, where a_{i}\in K and x_{i}\in X, and G is the forgetful functor, which assigns to a vector space V the set G(V) of elements (vectors) of V; in other words it simply “forgets” the vector space structure on V.

For every function g:X\rightarrow G(V) in \textbf{Sets} we have a linear transformation f:F(X)\rightarrow V in \textbf{Vect}_{K} given by f(\sum_{i}a_{i}x_{i})=\sum_{i}a_{i}g(x_{i}). The correspondence \psi:g\rightarrow f has an inverse \varphi, given by restricting f to X (so that our only linear transformations are of the form f(x_{i}), and we can obtain set-theoretic functions corresponding to these linear transformations). Hence we have a bijection

\displaystyle \text{Hom}_{\textbf{Vect}_{K}}(F(X),V)\cong\text{Hom}_{\textbf{Sets}}(X,G(V)).

We therefore see that the two functors F and G form an adjunction; the functor F (sometimes called the free functor) is left-adjoint to the forgetful functor G, and G is right-adjoint to F.

As another example, consider now the category of modules over a commutative ring R, and the functors -\otimes_{R}B and \text{Hom}_{R}(B,-) (see The Hom and Tensor Functors). For every morphism g:A\otimes_{R}B\rightarrow C we have another morphism f: A\rightarrow\text{Hom}_{R}(B,C) given by [f(a)](b)=g(a,b). We actually have a bijection

\displaystyle \text{Hom}(A\otimes_{R}B,C)\cong\text{Hom}(A,\text{Hom}_{R}(B,C)).

This is called the Tensor-Hom adjunction.

Closely related to the concept of an adjunction is the concept of a monad. A monad is a triple (T,\eta,\mu) where T is a functor from \mathbf{C} to itself, \eta is a natural transformation from 1_{\mathbf{C}} to T, and \mu is a natural transformation from \mu:T^{2}\rightarrow T, satisfying the following properties:

\displaystyle \mu\circ\mu_{T}=\mu\circ T\mu

\displaystyle \mu\circ\eta_{T}=\mu\circ T\eta=1

Dual to the concept of a monad is the concept of a comonad. A comonad on a category \mathbf{C} may be thought of as a monad on the opposite category \mathbf{C}^{\text{op}}.

As an example of a monad, we can consider the action of a fixed group G on a set (such as the symmetric group permuting the elements of the set, for example). In this case, our category will be \mathbf{Sets}, and T, \eta, and \mu are given by

\displaystyle T(X)=G\times X

\displaystyle \eta:X\rightarrow G\times X given by x\rightarrow\langle g,x\rangle

\displaystyle \mu:G\times (G\times X)\rightarrow G\times X given by \langle g_{1},\langle g_{2},x\rangle\rangle\rightarrow \langle g_{1}g_{2},x\rangle

Adjunctions and monads are related in the following way. Let F:\mathbf{C}\rightarrow\mathbf{D} and G:\mathbf{D}\rightarrow\mathbf{C} be a pair of adjoint functors with unit \eta and counit \epsilon. Then we have a monad on \mathbf{C} given by (G\circ F,\eta,G\epsilon_{F}). We can also obtain a comonad given by (F\circ G,\epsilon,F\eta_{G}).

Conversely, if we have a monad (T,\eta,\mu) on the category \mathbf{C}, we can obtain a pair of adjoint functors F:\mathbf{C}\rightarrow\mathbf{C}^{T} and G:\mathbf{C}^{T}\rightarrow\mathbf{C}, where \mathbf{C}^{T} is the Eilenberg-Moore category, whose objects (called T-algebras) are pairs (A,\alpha), where A is an object of \mathbf{C}, and \alpha is a morphism T(A)\rightarrow A satisfying

\displaystyle \alpha\circ \eta_{A}=1_{A}

\displaystyle \alpha\circ \mu_{A}=\alpha\circ T(\alpha),

and whose morphisms h:(A,\alpha)\rightarrow (B,\beta) are morphisms h:A\rightarrow B in \mathbf{C} such that

\displaystyle h\circ\alpha=\beta\circ T(h).

In the example we gave above in the discussion on monads, the T-algebras are exactly the sets with the action of the group G. If X is such a set, then the corresponding T-algebra is the pair (X,h), where the function h:G\times X\rightarrow X satisfies

\displaystyle h(g_{1},h(g_{2},x))=h(g_{1}g_{2},x)

\displaystyle h(e,x)=x.

For comonads, we have a dual notion of coalgebras. These “dual” ideas are important objects of study in themselves, for example in topos theory. Another reason to consider comonads and coalgebras is that in mathematics there often arises a situation where we have three functors

\displaystyle L:\mathbf{D}\rightarrow\mathbf{C}

\displaystyle F:\mathbf{C}\rightarrow\mathbf{D}

\displaystyle R:\mathbf{D}\rightarrow\mathbf{C}

where L is left-adjoint to F, and R is right-adjoint to F (a so-called adjoint triple). As an example, consider the forgetful functor F:\textbf{Top}\rightarrow\textbf{Sets} which assigns to a topological space its underlying set. It has both a left-adjoint L:\textbf{Sets}\rightarrow\textbf{Top} which assigns to a set X the trivial topology (where the only open sets are the empty set and X itself), and a right-adjoint R:\textbf{Sets}\rightarrow\textbf{Top} which assigns to the set X the discrete topology (where every subset of X is an open set). Therefore we have a monad and a comonad on \textbf{Sets} given by F\circ L and F\circ R respectively.

Many more examples of adjoint functors and monads can be found in pretty much all areas of mathematics. And according to a principle attributed to the mathematician Saunders Mac Lane (one of the founders of category theory, along with Samuel Eilenberg), such a structure that occurs widely enough in mathematics deserves to be studied for its own sake.


Adjoint Functors on Wikipedia

Monad on Wikipedia

Adjoint Functor on nLab

Adjunction on nLab

Monad on nLab

Adjunctions by TheCatsters on YouTube

Monads by TheCatsters on YouTube

Categories for the Working Mathematician by Saunders Mac Lane

Category Theory by Steve Awodey

Algebraic Spaces and Stacks

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category \mathcal{C}, we say that some other category \mathcal{S} is a category over \mathcal{C} if there is a functor p from \mathcal{S} to \mathcal{C} (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If \mathcal{S} is a category over some other category \mathcal{C}, we say that it is a fibered category (over \mathcal{C}) if for every object U=p(x) and morphism f: V\rightarrow U in \mathcal{C}, there is a strongly cartesian morphism \phi: f^{*}x\rightarrow x in \mathcal{S} with f=p(\phi).

This means that any other morphism \psi: z\rightarrow x whose image p(\psi) under the functor p factors as p(\psi)=p(\phi)\circ h must also factor as \psi=\phi\circ \theta under some unique morphism \theta: z\rightarrow f^{*}x whose image under the functor p is h. We refer to f^{*}x as the pullback of x along f.

Under the functor p, the objects of \mathcal{S} which get sent to U in \mathcal{C} and the morphisms of \mathcal{S} which get sent to the identity morphism i_{U} in \mathcal{C} form a subcategory of \mathcal{S} called the fiber over U. We will also write it as \mathcal{S}_{U}.

An important example of a fibered category is given by an ordinary presheaf on a category \mathcal{C}, i.e. a functor F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set}); we can consider it as a category fibered in sets \mathcal{S}_{F}\rightarrow\mathcal{C}.

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let \mathcal{U}=\{f_{i}:U_{i}\rightarrow U\} be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object U of \mathcal{C}. An object with descent data is a collection of objects X_{i} in \mathcal{S}_{U} together with transition isomorphisms \varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j} in \mathcal{S}_{U_{i}\times_{U}U_{j}}, satisfying the cocycle condition

\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}

The morphisms \text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i} and the \text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j} are the projection morphisms. The notations \text{pr}_{0}^{*}X_{i} and \text{pr}_{1}^{*}X_{j} means that we are “pulling back” X_{i} and X_{j} from \mathcal{S}_{U_{i}} and \mathcal{S}_{U_{j}}, respectively, to \mathcal{S}_{U_{i}\times_{U}U_{j}}.

A morphism between two objects with descent data is a a collection of morphisms \psi_{i}:X_{i}\rightarrow X'_{i} in \mathcal{S}_{U_{i}} such that \varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}. Therefore we obtain a category, the category of objects with descent data, denoted \mathcal{DD}(\mathcal{U}).

We can define a functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) by assigning to each object X of \mathcal{S}_{U} the object with descent data given by the pullback f_{i}^{*}X and the canonical isomorphism \text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X. An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor F:\mathcal{A}\rightarrow\mathcal{B} is faithful if the induced map \text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y)) is injective for any two objects x and y of \mathcal{A}.

A functor F:\mathcal{A}\rightarrow\mathcal{B} is full if the induced map \text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y)) is surjective for any two objects x and y of \mathcal{A}.

A functor F:\mathcal{A}\rightarrow\mathcal{B} is essentially surjective if any object y of \mathcal{B} is isomorphic to the image F(x) of some object x in \mathcal{A} under F.

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) we have defined earlier.

If the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) is fully faithful, then the fibered category \mathcal{S}\rightarrow\mathcal{C} is a prestack.

If the functor \mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U}) is an equivalence of categories, then the fibered category \mathcal{S}\rightarrow\mathcal{C} is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) F is a prestack if and only if it is a separated functor,

(ii) F is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from \mathcal{C} to \textbf{Sets} which is naturally isomorphic to the functor which assigns to any object X the set of morphisms \text{Hom}(X,U), for some fixed object U of \mathcal{C}.

A representable presheaf is a contravariant functor from \mathcal{C} to \textbf{Sets} which is naturally isomorphic to the functor which assigns to any object X the set of morphisms \text{Hom}(U,X), for some fixed object U of \mathcal{C}. If \mathcal{C} is the category of schemes, the latter functor is also called the functor of points of the object U.

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points h_{U} of a scheme U may be identified with U itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets} be functors, and let a:F\rightarrow G and b:H\rightarrow G be natural transformations between these functors. Then the fiber product F\times_{a,G,b}H is a functor from \mathcal{C}^{\text{op}} to \textbf{Sets}, and is given by the formula

\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)

for any object X of \mathcal{C}.

Let F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets} be functors. We say that a natural transformation a:F\rightarrow G is representable, or that F is relatively representable over G if for every U\in\text{Ob}(\mathcal{C}) and any \xi\in G(U) the functor h_{U}\times_{G}F is representable.

We now let (\text{Sch}/S)_{\text{fppf}} be the site (a category with a Grothendieck topology –  see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of S-schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme S is a presheaf

\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}

with the following properties

(1) The presheaf F is a sheaf.

(2) The diagonal morphism F\rightarrow F\times F is representable.

(3) There exists a scheme U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and a map h_{U}\rightarrow F which is surjective, and etale (This is often written simply as U\rightarrow F). The scheme U is also called an atlas.

The diagonal morphism being representable implies that the natural transformation h_{U}\rightarrow F is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have U is the disjoint union of affine schemes U_{i} and where the map h_{U}\rightarrow F is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let \mathcal{C} be a category. A category fibered in groupoids p:\mathcal{S}\rightarrow\mathcal{C} is called representable if there exists an object X of \mathcal{C} and an equivalence j:\mathcal{S}\rightarrow \mathcal{C}/X (The notation \mathcal{C}/X signifies a slice category, whose objects are morphisms f:U\rightarrow X in \mathcal{C}, and whose morphisms are morphisms h:U\rightarrow V in \mathcal{C} such that f=g\circ h, where g:U\rightarrow X).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let \mathcal{X} be a category fibered in groupoids over (\text{Sch}/S)_{\text{fppf}}. Then \mathcal{X} is representable by a scheme if there exists a scheme U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and an equivalence j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}} of categories over (\text{Sch}/S)_{\text{fppf}}.

A category fibered in groupoids p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}} is representable by an algebraic space over S if there exists an algebraic space F over S and an equivalence j:\mathcal{X}\rightarrow \mathcal{S}_{F} of categories over (\text{Sch}/S)_{\text{fppf}}.

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let F:\mathcal{X}\rightarrow\mathcal{S} and G:\mathcal{Y}\rightarrow\mathcal{S} be morphisms of categories over \mathcal{C}. The fiber product \mathcal{X}\times_{\mathcal{S}}\mathcal{Y} is given by the following description:

(1) an object of \mathcal{X}\times_{\mathcal{S}}\mathcal{Y} is a quadruple (U,x,y,f), where U\in\text{Ob}(\mathcal{C}), x\in\text{Ob}(\mathcal{X}_{U}), y\in\text{Ob}(\mathcal{Y}_{U}), and f : F(x)\rightarrow G(y) is an isomorphism in \mathcal{S}_{U},

(2) a morphism (U,x,y,f) \rightarrow (U',x',y',f') is given by a pair (a,b), where a:x\rightarrow x' is a morphism in X, and b:y\rightarrow y' is a morphism in Y such that a and b induce the same morphism U\rightarrow U', and f'\circ F(a)=G(b)\circ f.

Let S be a scheme. A morphism f:\mathcal{X}\rightarrow \mathcal{Y} of categories fibered in groupoids over (\text{Sch}/S)_{\text{fppf}} is called representable by algebraic spaces if for any U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}}) and any y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y} the category fibered in groupoids

\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}

over (\text{Sch}/U)_{\text{fppf}} is representable by an algebraic space over U.

An algebraic stack (or Artin stack) over a scheme S is a category

\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}

with the following properties:

(1) The category \mathcal{X} is a stack in groupoids over (\text{Sch}/S)_{\text{fppf}} .

(2) The diagonal \Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X} is representable by algebraic spaces.

(3) There exists a scheme U\in\text{Ob}((\text{Sch/S})_{\text{fppf}}) and a morphism (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} which is surjective and smooth (This is often written simply as U\rightarrow\mathcal{X}). Again, the scheme U is called an atlas.

If the morphism (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor (\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X} is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by \mathcal{M}_{1,1} – the reason for this notation will become clear later). A family of elliptic curves over some “base space” B is a fibration \pi:X\rightarrow B with a section O:B\rightarrow X such that the fiber \pi^{-1}(b) over any point b of B is an elliptic curve with origin O(b).

Ideally what we want is to be able to obtain every family X\rightarrow B by pulling back a “universal object” E\rightarrow\mathcal{M}_{1,1} via the map B\rightarrow\mathcal{M}_{1,1}. This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by M_{1,1}). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus g with \nu marked points, denoted \mathcal{M}_{g,\nu}. The moduli stack of elliptic curves is simply the special case \mathcal{M}_{1,1}. Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.


Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

The Theory of Motives

The theory of motives originated from the observation, sometime in the 1960’s, that in algebraic geometry there were several different cohomology theories (see Homology and Cohomology and Cohomology in Algebraic Geometry), such as Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The search for a “universal cohomology theory”, such that all these other cohomology theories could be obtained from such a universal cohomology theory is what led to the theory of motives.

The four cohomology theories enumerated above are examples of what is called a Weil cohomology theory. A Weil cohomology theory, denoted H^{*}, is a functor (see Category Theory) from the category \mathcal{V}(k) of smooth projective varieties over some field k to the category \textbf{GrAlg}(K) of graded K-algebras, for some other field K which must be of characteristic zero, satisfying the following axioms:

(1) (Finite-dimensionality) The homogeneous components H^{i}(X) of H^{*}(X) are finite dimensional for all i, and H^{i}(X)=0 whenever i<0 or i>2n, where n is the dimension of the smooth projective variety X.

(2) (Poincare duality) There is an orientation isomorphism H^{2n}\cong K, and a nondegenerate bilinear pairing H^{i}(X)\times H^{2n-i}(X)\rightarrow H^{2n}\cong K.

(3) (Kunneth formula) There is an isomorphism

\displaystyle H^{*}(X\times Y)\cong H^{*}(X)\otimes H^{*}(Y).

(4) (Cycle map) There is a mapping \gamma_{X}^{i} from C^{i}(X), the abelian group of algebraic cycles of codimension i on X (see Algebraic Cycles and Intersection Theory), to H^{i}(X), which is functorial with respect to pullbacks and pushforwards, has the multiplicative property \gamma_{X\times Y}^{i+j}(Z\times W)=\gamma_{X}^{i}(Z)\otimes \gamma_{Y}^{j}(W), and such that \gamma_{\text{pt}}^{i} is the inclusion \mathbb{Z}\hookrightarrow K.

(5) (Weak Lefschetz axiom) If W is a smooth hyperplane section of X, and j:W\rightarrow X is the inclusion, the induced map j^{*}:H^{i}(X)\rightarrow H^{i}(W) is an isomorphism for i\leq n-2, and a monomorphism for i\leq n-1.

(6) (Hard Lefschetz axiom) The Lefschetz operator

\displaystyle \mathcal{L}:H^{i}(X)\rightarrow H^{i+2}(X)

given by

\displaystyle \mathcal{L}(x)=x\cdot\gamma_{X}^{1}(W)

for some smooth hyperplane section W of X, with the product \cdot provided by the graded K-algebra structure of H^{*}(X), induces an isomorphism

\displaystyle \mathcal{L}^{i}:H^{n-i}(X)\rightarrow H^{n+i}(X).

The idea behind the theory of motives is that all Weil cohomology theories should factor through a “category of motives”, i.e. any Weil cohomology theory

\displaystyle H^{*}: \mathcal{V}(k)\rightarrow \textbf{GrAlg}(K)

can be expressed as the following composition of functors:

\displaystyle H^{*}: \mathcal{V}(k)\xrightarrow{h} \mathcal{M}(k)\rightarrow\textbf{GrAlg}(K)

where \mathcal{M}(k) is the category of motives. We can get different Weil cohomology theories, such as Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology, via different functors (called realization functors) from the category of motives to a category of graded algebras over some field K. This explains the term “motive”, which actually comes from the French word “motif”, which itself is already used in music and visual arts, among other things, as some kind of common underlying “theme” with different possible manifestations.

Let us now try to construct this category of motives. This category is often referred to in the literature as a “linearization” of the category of smooth projective varieties. This means that we obtain it from some sense starting with the category of smooth projective varieties, but we also want to modify it so that it we can do linear algebra, or more properly homological algebra, in some sense. In other words, we want it to behave like the category of modules over some ring. With this in mind, we want the category to be an abelian category, so that we can make sense of notions such as kernels, cokernels, and exact sequences.

An abelian category is a category that satisfies the following properties:

(1) The morphisms form an abelian group.

(2) There is a zero object.

(3) There are finite products and coproducts.

(4) Every morphism f:X\rightarrow Y has a kernel and cokernel, and satisfies a decomposition

\displaystyle K\xrightarrow{k} X\xrightarrow{i} I\xrightarrow{j} Y\xrightarrow{c} K'

where K is the kernel of f, K' is the cokernel of f, and I is the kernel of c and the cokernel of k (not to be confused with our notation for fields).

In order to proceed with our construction of the category of motives, which we now know we want to be an abelian category, we discuss the notion of correspondences.

The group of correspondences of degree r from a smooth projective variety X to another smooth projective variety Y, written \text{Corr}^{r}(X,Y), is defined to be the group of algebraic cycles of X\times Y of codimension n+r, where n is the dimension of X, i.e.

\text{Corr}^{r}(X,Y)=C^{n+r}(X\times Y)

A morphism (of varieties, in the usual sense) f:Y\rightarrow X determines a correspondence from X to Y of degree 0 given by the transpose of the graph of f in X\times Y. Therefore we may think of correspondences as generalizations of the usual concept of morphisms of varieties.

As we have learned in Algebraic Cycles and Intersection Theory, whenever we are dealing with algebraic cycles, it is often useful to consider them only up to some equivalence relation. In the aforementioned post we introduced the notion of rational equivalence. This time we consider also homological equivalence and numerical equivalence between algebraic cycles.

We say that two algebraic cycles Z_{1} and Z_{2} are homologically equivalent if they have the same image under the cycle map, and we say that they are numerically equivalent if the intersection numbers Z_{1}\cdot Z and Z_{2}\cdot Z are equal for all Z of complementary dimension. There are other such equivalence relations on algebraic cycles, but in this post we will only mostly be using rational equivalence, homological equivalence, and numerical equivalence.

Since correspondences are algebraic cycles, we often consider them only up to these equivalence relations, and denote the quotient group we obtain by \text{Corr}_{\sim}^{r}(X,Y), where \sim is the equivalence relation imposed, for example, for numerical equivalence we write \text{Corr}_{\text{num}}^{r}(X,Y).

Taking the tensor product of the abelian group \text{Corr}_{\sim}^{r}(X,Y) with the rational numbers \mathbb{Q}, we obtain the vector space

\displaystyle \text{Corr}_{\sim}^{r}(X,Y)_{\mathbb{Q}}=\text{Corr}_{\sim}^{r}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q}

To obtain something closer to an abelian category (more precisely, we will obtain what is known as a pseudo-abelian category, but in the case where the equivalence relation is numerical equivalence, we will actually obtain an abelian category), we need to consider “projectors”, correspondences p of degree 0 from a variety X to itself such that p^{2}=p. So now we form a category, whose objects are h(X,p) for a variety X and projector p, and whose morphisms are given by

\displaystyle \text{Hom}(h(X,p),h(Y,q))=q\circ\text{Corr}_{\sim}^{0}(X,Y)_{\mathbb{Q}}\circ p.

We call this category the category of pure effective motives, and denote it by \mathcal{M}_{\sim}^{\text{eff}}(k). The process described above is also known as passing to the pseudo-abelian (or Karoubian) envelope.

We write h^{i}(X,p) for the objects of \mathcal{M}_{\sim}^{\text{eff}}(k) that map to H^{i}(X). In the case that X is the projective line \mathbb{P}^{1}, and p is the diagonal \Delta_{\mathbb{P}^{1}}, we find that

h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=h^{0}\mathbb{P}^{1}\oplus h^{2}\mathbb{P}^{1}

which can be rewritten also as

\displaystyle h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=\mathbb{I}\oplus\mathbb{L}

where \mathbb{I} is the image of a point in the category of pure effective motives, and \mathbb{L} is known as the Lefschetz motive. It is also denoted by \mathbb{Q}(-1). The above decomposition corresponds to the projective line \mathbb{P}^{1} being a union of the affine line \mathbb{A}^{1} and a “point at infinity”, which we may denote by \mathbb{A}^{0}:

\displaystyle \mathbb{P}^{1}=\mathbb{A}^{0}\cup\mathbb{A}^{1}

More generally, we have

\displaystyle h(\mathbb{P}^{n},\Delta_{\mathbb{P}^{n}})=\mathbb{I}\oplus\mathbb{L}\oplus...\oplus\mathbb{L}^{n}

corresponding to

\displaystyle \mathbb{P}^{n}=\mathbb{A}^{0}\cup\mathbb{A}^{1}\cup...\cup\mathbb{A}^{n}.

The category of effective pure motives is an example of a tensor category. This means it has a bifunctor \otimes: \mathcal{M}_{\sim}^{\text{eff}}\times\mathcal{M}_{\sim}^{\text{eff}}\rightarrow\mathcal{M}_{\sim}^{\text{eff}} which generalizes the usual notion of a tensor product, and in this particular case it is given by taking the product of two varieties. We can ask for more, however, and construct a category of motives which is not just a tensor category but a rigid tensor category, which provides us with a notion of duals.

By formally inverting the Lefschetz motive (the formal inverse of the Lefschetz motive is then known as the Tate motive, and is denoted by \mathbb{Q}(1)), we can obtain this rigid tensor category, whose objects are triples h(X,p,m), where X is a variety, e is a projector, and m is an integer. The morphisms of this category are given by

\displaystyle \text{Hom}(h(X,p,n),h(Y,q,m))=q\circ\text{Corr}_{\sim}^{n-m}(X,Y)_{\mathbb{Q}}\circ p.

This category is called the category of pure motives, and is denoted by \mathcal{M}_{\sim}(k). The category \mathcal{M}_{\text{rat}}(k) is called the category of Chow motives, while the category \mathcal{M}_{\text{num}}(k) is called the category of Grothendieck (or numerical) motives.

The category of Chow motives has the advantage that it is known to be “universal”, in the sense that every Weil cohomology theory factors through it, as discussed earlier; however, in general it is not even abelian, which is a desirable property we would like our category of motives to have. Meanwhile, the category of Grothendieck motives is known to be abelian, but it is not yet known if it is universal. If the so-called “standard conjectures on algebraic cycles“, which we will enumerate below, are proved, then the category of Grothendieck motives will be known to be universal.

We have seen that the category of pure motives forms a rigid tensor category. Closely related to this concept, and of interest to us, is the notion of a Tannakian category. More precisely, a Tannakian category is a k-linear rigid tensor category with an exact faithful functor (called a fiber functor) to the category of finite-dimensional vector spaces over some field extension K of k.

One of the things that makes Tannakian categories interesting is that there is an equivalence of categories between a Tannakian category \mathcal{C} and the category \text{Rep}_{G} of finite-dimensional linear representations of the group of automorphisms of its fiber functor, which is also known as the Tannakian Galois group, or, if the Tannakian category is a “category of motives” of some sort, the motivic Galois group. This aspect of Tannakian categories may be thought of as a higher-dimensional analogue of the classical theory of Galois groups, which can be stated as an equivalence of categories between the category of finite separable field extensions of a field k and the category of finite sets equipped with an action of the Galois group \text{Gal}(\bar{k}/k), where \bar{k} is the algebraic closure of k.

So we see that being a Tannakian category is yet another desirable property that we would like our category of motives to have. For this not only do we have to tweak the tensor product structure of our category, we also need certain conjectural properties to hold. These are the same conjectures we have hinted at earlier, called the standard conjectures on algebraic cycles, formulated by Alexander Grothendieck at around the same time he initially developed the theory of motives.

These conjectures have some very important consequences in algebraic geometry, and while they remain unproved to this day, the search for their proof (or disproof) is an important part of modern mathematical research on the theory of motives. They are the following:

(A) (Standard conjecture of Lefschetz type) For i\leq n, the operator \Lambda defined by

\displaystyle \Lambda=(\mathcal{L}^{n-i+2})^{-1}\circ\mathcal{L}\circ (\mathcal{L}^{n-i}):H^{i}\rightarrow H^{i-2}

\displaystyle \Lambda=(\mathcal{L}^{n-i})\circ\mathcal{L}\circ (\mathcal{L}^{n-i+2})^{-1}:H^{2n-i+2}\rightarrow H^{2n-i}

is induced by algebraic cycles.

(B) (Standard conjecture of Hodge type) For all i\leq n/2, the pairing

\displaystyle x,y\mapsto (-1)^{i}(\mathcal{L}x\cdot y)

is positive definite.

(C) (Standard conjecture of Kunneth type) The projectors H^{*}(X)\rightarrow H^{i}(X) are induced by algebraic cycles in X\times X with rational coefficients. This implies the following decomposition of the diagonal:

\displaystyle \Delta_{X}=\pi_{0}+...+\pi_{2n}

which in turn implies the decomposition

\displaystyle h(X,\Delta_{X},0)=h(X,\pi_{0},0)\oplus...\oplus h(X,\pi_{2n},0)

which, writing h(X,\Delta_{X},0) as hX and h(X,\pi_{i},0) as h^{i}(X), we can also compactly and suggestively write as

\displaystyle hX=h^{0}X\oplus...\oplus h^{2n}X.

In other words, every object hX=h(X,\Delta_{X},0) of our “category of motives” decomposes into graded “pieces” h^{i}(X)=h(X,\pi_{i},0) of pure “weighti. We have already seen earlier that this is indeed the case when X=\mathbb{P}^{n}. We will need this conjecture to hold if we want our category to be a Tannakian category.

(D) (Standard conjecture on numerical equivalence and homological equivalence) If an algebraic cycle is numerically equivalent to zero, then its cohomology class is zero. If the category of Grothendieck motives is to be “universal”, so that every Weil cohomology theory factors through it, this conjecture must be satisfied.

In Algebraic Cycles and Intersection Theory and Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry, we have made mention of the two famous conjectures in algebraic geometry known as the Hodge conjecture and the Tate conjecture. In fact, these two closely related conjectures can be phrased in the language of motives as the conjectures stating that the realization functors from the category of motives to the category of pure Hodge structures and continuous l-adic representations of \text{Gal}(\bar{k}/k), respectively, be fully faithful. These conjectures are closely related to the standard conjectures on algebraic cycles as well.

We have now constructed the category of pure motives, for smooth projective varieties. For more general varieties and schemes, there is an analogous idea of “mixed motives“, which at the moment remain conjectural, although there exist several related constructions which are the closest thing we currently have to such a theory of mixed motives.

If we want to construct a theory of mixed motives, instead of Weil cohomology theories we must instead consider what are known as “mixed Weil cohomology theories“, which are expected to have the following properties:

(1) (Homotopy invariance) The projection \pi:X\rightarrow\mathbb{A}^{1} induces an isomorphism

\displaystyle \pi^{*}:H^{*}(X)\xrightarrow{\cong}H^{*}(X\times\mathbb{A}^{1})

(2) (Mayer-Vietoris sequence) If U and V are open coverings of X, then there is a long exact sequence

\displaystyle ...\rightarrow H^{i}(U\cap V)\rightarrow H^{i}(X)\rightarrow H^{i}(U)\oplus H^{i}(V)\rightarrow H^{i}(U\cap V)\rightarrow...

(3) (Duality) There is a duality between cohomology H^{*} and cohomology with compact support H_{c}^{*}.

(4) (Kunneth formula) This is the same axiom as the one in the case of pure motives.

We would like a category of mixed motives, which serves as an analogue to the category of pure motives in that all mixed Weil cohomology theories factor through it, but as mentioned earlier, no such category exists at the moment. However, the mathematicians Annette Huber-Klawitter, Masaki Hanamura, Marc Levine, and Vladimir Voevodsky have constructed different versions of a triangulated category of mixed motives, denoted \mathcal{DM}(k).

A triangulated category \mathcal{T} is an additive category with an automorphism T: \mathcal{T}\rightarrow\mathcal{T} called the “shift functor” (we will also denote T(X) by X[1], and T^{n}(X) by X[n], for n\in\mathbb{Z}) and a family of “distinguished triangles

\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]

 which satisfies the following axioms:

(1) For any object X of \mathcal{T}, the triangle X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow X[1] is a distinguished triangle.

(2) For any morphism u:X\rightarrow Y of \mathcal{T}, there is an object Z of \mathcal{T} such that X\xrightarrow{u}Y\rightarrow Z\rightarrow X[1] is a distinguished triangle.

(3) Any triangle isomorphic to a distinguished triangle is a distinguished triangle.

(4) If X\rightarrow Y\rightarrow Z\rightarrow X[1] is a distinguished triangle, then the two “rotations” Y\rightarrow Z\rightarrow Z[1]\rightarrow Y[1] and Z[-1]\rightarrow X\rightarrow Y\rightarrow Z are also distinguished triangles.

(5) Given two distinguished triangles X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1] and X'\xrightarrow{u'}Y'\xrightarrow{v'}Z'\xrightarrow{w'}X'[1] and morphisms f:X\rightarrow X' an g:Y\rightarrow Y' such that the square “commutes”, i.e. u'\circ f=g\circ u, there exists a morphisms h:Z\rightarrow Z such that all other squares commute.

(6) Given three distinguished triangles X\xrightarrow{u}Y\xrightarrow{j}Z'\xrightarrow{k}X[1]Y\xrightarrow{v}Z\xrightarrow{l}X'\xrightarrow{i}Y[1], and X\xrightarrow{v\circ u}Z\xrightarrow{m}Y'\xrightarrow{n}X[1], there exists a distinguished triangle Z'\xrightarrow{f}Y'\xrightarrow{g}X'\xrightarrow{h}Z'[1] such that “everything commutes”.

A t-structure on a triangulated category \mathcal{T} is made up of two full subcategories \mathcal{T}^{\geq 0} and \mathcal{T}^{\leq 0} satisfying the following properties (writing \mathcal{T}^{\leq n} and \mathcal{T}^{\leq n} to denote \mathcal{T}^{\leq 0}[-n] and \mathcal{T}^{\geq 0}[-n] respectively):

(1) \mathcal{T}^{\leq -1}\subset \mathcal{T}^{\leq 0} and \mathcal{T}^{\geq 1}\subset \mathcal{T}^{\geq 0}

(2) \displaystyle \text{Hom}(X,Y)=0 for any object X of \mathcal{T}^{\leq 0} and any object Y of \mathcal{T}^{\geq 1}

(3) for any object Y of \mathcal{T} we have a distinguished triangle

\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]

where X is an object of \mathcal{T}^{\leq 0} and Z is an object of \mathcal{T}^{\geq 1}.

The full subcategory \mathcal{T}^{0}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0} is called the heart of the t-structure, and it is an abelian category.

It is conjectured that the category of mixed motives \mathcal{MM}(k) is the heart of the t-structure of the triangulated category of mixed motives \mathcal{DM}(k).

Voevodsky’s construction proceeds in a manner somewhat analogous to the construction of the category of pure motives as above, starting with schemes (say, over a field k, although a more general scheme may be used) as objects and correspondences as morphisms, but then makes use of concepts from abstract homotopy theory, such as taking the bounded homotopy category of bounded complexes, and localization with respect to a certain subcategory, before passing to the pseudo-abelian envelope and then formally inverting the Tate object \mathbb{Z}(1). The triangulated category obtained is called the category of geometric motives, and is denoted by \mathcal{DM}_{\text{gm}}(k). The schemes and correspondences involved in the construction of \mathcal{DM}_{\text{gm}}(k) are required to satisfy certain properties which eliminates the need to consider the equivalence relations which form a large part of the study of the category of pure motives.

Closely related to the triangulated category of mixed motives is motivic cohomology, which is defined in terms of the former as

\displaystyle H^{i}(X,\mathbb{Z}(m))=\text{Hom}_{\mathcal{DM}(k)}(X,\mathbb{Z}(m)[i])

where \mathbb{Z}(m) is the tensor product of m copies of the Tate object \mathbb{Z}(1), and the notation \mathbb{Z}(m)[i] tells us that the shift functor of the triangulated category is applied to the object \mathbb{Z}(m) i times.

Motivic cohomology is related to the Chow group, which we have introduced in Algebraic Cycles and Intersection Theory, and also to algebraic K-theory, which is another way by which the ideas of homotopy theory are applied to more general areas of abstract algebra and linear algebra. These ideas were used by Voevodsky to prove several related theorems, from the Milnor conjecture to its generalization, the Bloch-Kato conjecture (also known as the norm residue isomorphism theorem).

Historically, one of the motivations for Grothendieck’s attempt to obtain a universal cohomology theory was to prove the Weil conjectures, which is a higher-dimensional analogue of the Riemann hypothesis for curves over finite fields first proved by Andre Weil himself (see The Riemann Hypothesis for Curves over Finite Fields). In fact, if the standard conjectures on algebraic cycles are proved, then a proof of the Weil conjectures would follow via an approach that closely mirrors Weil’s original proof (since cohomology provides a Lefschetz fixed-point formula –  we have mentioned in The Riemann Hypothesis for Curves over Finite Fields that the study of fixed points is an important part of Weil’s proof). The last of the Weil conjectures were eventually proved by Grothendieck’s student Pierre Deligne, but via a different approach that bypassed the standard conjectures. A proof of the standard conjectures, which would lead to a perhaps more elegant proof of the Weil conjectures, is still being pursued to this day.

The theory of motives is not only related to analogues of the Riemann hypothesis, which concerns the location of zeroes of L-functions, but to L-functions in general. For instance, it is also related to the Langlands program, which concerns another aspect of L-functions, namely their analytic continuation and functional equation, and to the Birch and Swinnerton-Dyer conjecture, which concerns their values at special points.

We recall in The Riemann Hypothesis for Curves over Finite Fields that the Frobenius morphism played an important part in counting the points of a curve over a finite field, which in turn we needed to define the zeta function (of which the L-function can be thought of as a generalization) of the curve. The Frobenius morphism is an element of the Galois group, and we recall that a category of motives which is a Tannakian category is equivalent to the category of representations of its motivic Galois group. Therefore we can see how we can define “motivic L-functions” using the theory of motives.

As the L-functions occupy a central place in many areas of modern mathematics, the theory of motives promises much to be gained from its study, if only we could make progress in deciphering the many mysteries that surround it, of which we have only scratched the surface in this post. The applications of motives are not limited to L-functions either – the study of periods, which relate Betti cohomology and de Rham cohomology, and lead to transcendental numbers which can be defined using only algebraic concepts, is also strongly connected to the theory of motives. Recent work by the mathematicians Alain Connes and Matilde Marcolli has also suggested applications to physics, particularly in relation to Feynman diagrams in quantum field theory. There is also another generalization of the theory of motives, developed by Maxim Kontsevich, in the context of noncommutative geometry.


Weil Cohomology Theory on Wikipedia

Motive on Wikipedia

Standard Conjectures on Algebraic Cycles on Wikipedia

Motive on nLab

Pure Motive on nLab

Mixed Motive on nLab

The Tate Conjecture over Finite Fields on Hard Arithmetic

What is…a Motive? by Barry Mazur

Motives – Grothendieck’s Dream by James S. Milne

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Algebraic Cycles and the Weil Conjectures by Steven L. Kleiman

The Standard Conjectures by Steven L. Kleiman

Feynman Motives by Matilde Marcolli

Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Periodes) by Yves Andre

An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry

String theory is by far the most popular of the current proposals to unify the as of now still incompatible theories of quantum mechanics and general relativity. In this post we will give a short overview of the concepts involved in string theory, but not with the goal of discussing the theory itself in depth (hopefully there will be more posts in the future working towards this task). Instead, we will focus on introducing a very interesting and very beautiful branch of mathematics that arose out of string theory called mirror symmetry. In particular, we will focus on a version of it originally formulated by the mathematician Maxim Kontsevich in 1994 called homological mirror symmetry.

We will start with string theory. String theory started out as a theory of the nuclear forces that held together the protons and electrons in the nucleus of an atom. It was abandoned later on, due to a more successful theory called quantum chromodynamics taking its place. However, it was soon found out that string theory could model the elusive graviton, a particle “carrier” of gravity in the same way that a photon is a particle “carrier” of electromagnetism (the photon is more popularly referred to as a particle of light, but because light itself is an electromagnetic wave, it is also a manifestation of an electromagnetic field), and since then physicists have started developing string theory, no longer in the sole context of nuclear forces, but as a possible candidate for a working theory of quantum gravity.

The incompatibility of quantum mechanics and general relativity (which is currently our accepted theory of gravity) arises from the nonrenormalizability of gravity. In calculations in quantum field theory (see Some Basics of Relativistic Quantum Field Theory and Some Basics of (Quantum) Electrodynamics), there appear certain “nonsensical” quantities which are made sense of via a “corrective” procedure called renormalization (not to be confused with some other procedures called “normalization”). While the way that renormalization works is not really completely understood at the moment, it is known that this procedure at least “works” – this means that it produces the correct values of quantities, as can be checked via experiment.

Renormalization, while it works for the other forces, however fails for gravity. Roughly this is sometimes described as gravity “wildly fluctuating” at the smallest scales. What we know is that this signals, for us, a lack of knowledge of  what physics is like at these extremely small scales (much smaller than the current scale of quantum mechanics).

String theory attempts to solve this conundrum by proposing that particles, at the very smallest scales, are not “particles” at all, but “strings”. This takes care of the problem of fluctuations at the smallest scales, since there is a limit to how small the scale can be, set by the length of the strings. It is perhaps worth noting at this point that the next most popular contender to string theory, loop quantum gravity, tackles this problem by postulating that space itself is not continuous, but “discretized” into units of a certain length. For both theories, this length is predicted to be around 10^{-35} meters, a constant quantity which is known as the Planck length.

Over time, as string theory was developed, it became more ambitious, aiming to provide not only the unification of quantum mechanics and general relativity, but also the unification of the four fundamental forces – electromagnetism, the weak nuclear force, the strong nuclear force, and gravity, under one “theory of everything“. At the same time, it needed more ingredients – to be able to account for bosons, the particles carrying “forces”, such as photons and gravitons, and the fermions, particles that make up matter, such as electrons, protons, and neutrons, a new ingredient had to be added, called supersymmetry. In addition, it worked not in the four dimensions of spacetime that we are used to, but instead required ten dimensions (for the “bosonic” string theory, before supersymmetry, the number of dimensions required was a staggering twenty-six)!

How do we explain spacetime having ten dimensions, when we experience only four? It turns out, even before string theory, the idea of extra dimensions was already explored by the physicists Theodor Kaluza and Oskar Klein. They proposed a theory unifying electromagnetism and gravity by postulating an “extra” dimension which was “curled up” into a loop so small we could never notice it. The usual analogy is that of an ant crossing a wire – when the radius of the wire is big, the ant realizes that it can go sideways along the wire, but when the radius of the wire is small, it is as if there is only one dimension that the ant can move along.

So we now have this idea of six curled up dimensions of spacetime, in addition to the usual four. It turns out that there are so many ways that these dimensions can be curled up. This phenomenon is called the string theory landscape, and it is one of the biggest problems facing string theory today. What could be the specific “shape” in which these dimensions are curled up, and why are they not curled up in some other way? Some string theorists answer this by resorting to the controversial idea of a multiverse, so that there are actually several existing universes, each with its own way of how the extra six dimensions are curled up, and we just happen to be in this one because, perhaps, this is the only one where the laws of physics (determined by the way the dimensions are curled up) are able to support life. This kind of reasoning is called the anthropic principle.

In addition to the string theory landscape, there was also the problem of having several different versions of string theory. These problems were perhaps alleviated by the discovery of mysterious dualities. For example, there is the so-called T-duality, where a compactification (a “curling up”) with a bigger radius gives the same laws of physics as a compactification with a smaller, “reciprocal” radius. Not only do the concept of dualities connect the different ways in which the extra dimensions are curled up, they also connect the several different versions of string theory! In 1995, the physicist Edward Witten conjectured that this is perhaps because all these different versions of string theory come from a single “mother theory”, which he called “M-theory“.

In 1991, physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes used these dualities to solve a mathematical problem that had occupied mathematicians for decades, that of counting curves on a certain manifold (a manifold is a shape without sharp corners or edges) known as a Calabi-Yau manifold. In the context of Calabi-Yau manifolds, which are some of the shapes in which the extra dimensions of spacetime are postulated to be curled up, these dualities are known as mirror symmetry. With the success of Candelas, de la Ossa, Green, and Parkes, mathematicians would take notice of mirror symmetry and begin to study it as a subject of its own.

Calabi-Yau manifolds are but special cases of Kahler manifolds, which themselves are very interesting mathematical objects because they can be studied using three aspects of differential geometry – Riemannian geometry, symplectic geometry, and complex geometry.

We have already encountered examples of Kahler manifolds on this blog – they are the elliptic curves (see Elliptic Curves and The Moduli Space of Elliptic Curves). In fact elliptic curves are not only Kahler manifolds but also Calabi-Yau manifolds, and they are the only two-dimensional Calabi-Yau manifolds (we sometimes refer to them as “one-dimensional” when we are considering “complex dimensions”, as is common practice in algebraic geometry – this apparent “discrepancy” in counting dimensions arises because we need two real numbers to specify a complex number). In string theory of course we consider six-dimensional (three-dimensional when considering complex dimensions) Calabi-Yau manifolds, since there are six extra curled up dimensions of spacetime, but often it is also fruitful to study also the other cases, especially the simpler ones, since they can serve as our guide for the study of the more complicated cases.

Riemannian geometry studies Riemannian manifolds, which are manifolds equipped with a metric tensor, which intuitively corresponds to an “infinitesimal distance formula” dependent on where we are on the manifold. We have already encountered Riemannian geometry before in Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry. There we have seen that Riemannian geometry is very important in the mathematical formulation of general relativity, since in this theory gravity is just the curvature of spacetime, and the metric tensor expresses this curvature by showing how the formula for the infinitesimal distance between two points (actually the infinitesimal spacetime interval between two events) changes as we move around the manifold.

Symplectic geometry, meanwhile, studies symplectic manifolds. If Riemannian manifolds are equipped with a metric tensor that measures “distances”, symplectic manifolds are equipped with a symplectic form that measures “areas”. The origins of symplectic geometry are actually related to William Rowan Hamilton’s formulation of classical mechanics (see Lagrangians and Hamiltonians), as developed later on by Henri Poincare. There the object of study is phase space, which gives the state of a system based on the position and momentum of the objects that comprise it. It is this phase space that is expressed as a symplectic manifold.

Complex geometry, following our pattern, studies complex manifolds. These are manifolds which locally look like \mathbb{C}^{n}, in the same way that ordinary differentiable manifolds locally look like \mathbb{R}^{n}. Just as Riemannian geometry has metric tensors and symplectic geometry has symplectic forms, complex geometry has complex structures, mappings of tangent spaces with the property that applying them twice is the same as multiplication by -1, mimicking the usual multiplication by the imaginary unit i on the complex plane.

Complex manifolds are not only part of differential geometry, they are also often studied using the methods of algebraic geometry! We recall (see Basics of Algebraic Geometry) that algebraic geometry studies varieties and schemes, which are shapes such as lines, conic sections (parabolas, hyperbolas, ellipses, and circles), and elliptic curves, that can be described by polynomials (their modern definitions are generalizations of this concept). In fact, all Calabi-Yau manifolds can be described by polynomials, such as the following example, due to user Andrew J. Hanson of Wikipedia:


This is a visualization (actually a sort of “cross section”, since we can only display two dimensions and this object is actually six-dimensional) of the Calabi-Yau manifold described by the following polynomial equation:

\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0

This polynomial equation (known as the Fermat quintic) actually describes the Calabi-Yau manifold  in projective space using homogeneous coordinates. This means that we are using the concepts of projective geometry (see Projective Geometry) to include “points at infinity“.

We note at this point that Kahler manifolds and Calabi-Yau manifolds are interesting in their own right, even outside of the context of string theory. For instance, we have briefly mentioned in Algebraic Cycles and Intersection Theory the Hodge conjecture, one of seven “Millenium Problems” for which the Clay Mathematics Institute is currently offering a million-dollar prize, and it concerns Kahler manifolds. Perhaps most importantly, it “unifies” several different branches of mathematics; as we have already seen, the study of Kahler manifolds and Calabi-Yau manifolds involves Riemannian geometry, symplectic geometry, complex geometry, and algebraic geometry. The more recent version of mirror symmetry called homological mirror symmetry further adds category theory and homological algebra to the mix.

Now what mirror symmetry more specifically states is that a version of string theory called Type IIA string theory, on a spacetime with extra dimensions compactified onto a certain Calabi-Yau manifold V, is the same as another version of string theory, called Type IIB string theory, on a spacetime with extra dimensions compactified onto another Calabi-Yau manifold W, which is “mirror” to the Calabi-Yau manifold V.

The statement of homological mirror symmetry (which is still conjectural, but mathematically proven in certain special cases) expresses the idea of the previous paragraph as follows (quoted verbatim from the paper Homological Algebra of Mirror Symmetry by Maxim Kontsevich):

Let (V,\omega) be a 2n-dimensional symplectic manifold with c_{1}(V)=0 and W be a dual n-dimensional complex algebraic manifold.

The derived category constructed from the Fukaya category F(V) (or a suitably enlarged one) is equivalent to the derived category of coherent sheaves on a complex algebraic variety W.

The statement makes use of the language of category theory and homological algebra (see Category TheoryMore Category Theory: The Grothendieck ToposEven More Category Theory: The Elementary ToposExact SequencesMore on Chain Complexes, and The Hom and Tensor Functors), but the idea that it basically expresses is that there exists a relation between the symplectic aspects of the Calabi-Yau manifold V, as encoded in its Fukaya category, and the complex aspects of the Calabi-Yau manifold W, as encoded in its category of coherent sheaves (see Sheaves and More on Sheaves). As we have said earlier, the subjects of algebraic geometry and complex geometry are closely related, and hence the language of sheaves show up in (and is an important part of) both subjects. The concept of derived categories, which generalize derived functors like the Ext and Tor functors, allow us to relate the two categories, which otherwise would be expressing different concepts. Inspired by string theory, therefore, we have now a deep and beautiful idea in geometry, relating its different aspects.

Is string theory the correct way towards a complete theory of quantum gravity, or the so-called “theory of everything”? As of the moment, we don’t know. Quantum gravity is a very difficult problem, and the scales involved are still far out of our reach – in order to probe smaller and smaller scales we need particle accelerators with higher and higher energies, and right now the technologies that we have are still very, very far from the scales which are relevant to quantum gravity. Still, it is hoped for that whatever we find in experiments in the near future, not only in the particle accelerators but also in the radio telescopes that look out into space, will at least guide us towards the correct path.

There are some who believe that, in the absence of definitive experimental evidence, mathematical beauty is our next best guide. And, without a doubt, string theory is related to, and has inspired, some very beautiful and very interesting mathematics, including that which we have discussed in this post. Still, physics, like all natural science, is empirical (based on evidence and observation), and hence it is ultimately physical evidence that will be the judge of correctness. It may yet turn out that string theory is wrong, and that it is a different theory which describes the fundamental physical laws of nature, or that it needs drastic modifications to its ideas. This will not invalidate the mathematics that we have described here, anymore than the discoveries of Copernicus invalidated the mathematics behind the astronomical model of Ptolemy – in fact this mathematics not only outlived the astronomy of Ptolemy, but served the theories of Copernicus, and his successors, just as well. Hence we cannot really say that the efforts of Ptolemy were wasted, since even though his scientific ideas were shown to be wrong, still his mathematical methods were found very useful by those who succeeded him. Thus, while our current technological limitations prohibit us from confirming or ruling out proposals for a theory of quantum gravity such as string theory, there is still much to be gained from such continued efforts on the part of theory, while experiment is still in the process of catching up.

Our search for truth continues. Meanwhile, we have beauty to cultivate.


String Theory on Wikipedia

Mirror Symmetry on Wikipedia

Homological Mirror Symmetry on Wikipedia

Calabi-Yau Manifold on Wikipedia

Kahler Manifold on Wikipedia

Riemannian Geometry on Wikipedia

Symplectic Geometry on Wikipedia

Complex Geometry on Wikipedia

Fukaya Category on Wikipedia

Coherent Sheaf on Wikipedia

Derived Category on Wikipedia

Image by User Andrew J. Hanson of Wikipedia

Homological Algebra of Mirror Symmetry by Maxim Kontsevich

The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory by Brian Greene

String Theory by Joseph Polchinski

String Theory and M-Theory: A Modern Introduction by Katrin Becker, Melanie Becker, and John Schwarz

Even More Category Theory: The Elementary Topos

In More Category Theory: The Grothendieck Topos, we defined the Grothendieck topos as something like a generalization of the concept of sheaves on a topological space. In this post we generalize it even further into a concept so far-reaching it can even be used as a foundation for mathematics.

I. Definition of the Elementary Topos

We start by discussing the generalization of the universal constructions we defined in More Category Theory: The Grothendieck Topos, called limits and colimits.

Given categories \mathbf{J} and \mathbf{C}, we refer to a functor F: \mathbf{J}\rightarrow \mathbf{C} as a diagram in \mathbf{C} of type \mathbf{J}, and we refer to \mathbf{J} as an indexing category. We write the functor category of all diagrams in \mathbf{C} of type \mathbf{J} as \mathbf{C^{J}}.

Given a diagram F: \mathbf{J}\rightarrow \mathbf{C}, a cone to F is an object N of \mathbf{C} together with morphisms \psi_{X}: N\rightarrow F(X) indexed by the objects X of \mathbf{J} such that for every morphism f: X\rightarrow Y in  \mathbf{J}, we have F(f)\circ \psi_{X}=\psi_{Y}.

A limit of a diagram F: \mathbf{J}\rightarrow \mathbf{C} is a cone (L, \varphi) to F such that for any other cone (N, \psi)  to F there exists a unique morphism u: N\rightarrow L such that \varphi_{X}\circ \psi_{X} for all X in J.

For example, when \mathbf{J} is a category with only two objects A and B and whose only morphisms are the identity morphisms on each of these objects, the limit of the diagram F: \mathbf{J}\rightarrow \mathbf{C} is just the product. Similarly, the pullback is the limit of the diagram F: \mathbf{J}\rightarrow \mathbf{C} when \mathbf{J} is the category with three objects A, B, and C, and the only morphisms aside from the identity morphisms are one morphism A\xrightarrow{f}C and another morphism B\xrightarrow{g}C. The terminal object is the limit of the diagram F: \mathbf{J}\rightarrow \mathbf{C} when \mathbf{J} is the empty category, and the equalizer is the limit of the diagram F: \mathbf{J}\rightarrow \mathbf{C} when \mathbf{J} is the category with two objects A and B and whose only morphisms aside from the identity morphisms are two morphisms A\xrightarrow{f}B and A\xrightarrow{g}B.

A colimit is the dual concept to a limit, obtained by reversing the directions of all the morphisms in the definition. In the same way that the limit generalizes the concepts of product, pullback, terminal object, and equalizer, the colimit generalizes the concepts of coproduct, pushout, initial object, and coequalizer.

Next we discuss the concept of adjoint functors. Consider two categories \mathbf{C} and \mathbf{D}, and two functors F: \mathbf{C}\rightarrow \mathbf{D} and G: \mathbf{D}\rightarrow \mathbf{C}. We say that F is right adjoint to G, and that G is left adjoint to F, if for all objects C in \mathbf{C} and D in \mathbf{D} there exist bijections

\theta: \text{Hom}_{\mathbf{C}}(C, G(D))\xrightarrow{\sim}\text{Hom}_{\mathbf{D}}(F(C), D)

which are natural in the sense that given morphisms \alpha: C\rightarrow C' in \mathbf{C} and \xi: D'\rightarrow D in \mathbf{D}, we have

\theta(G(\alpha)\circ f\circ \xi)=\alpha\circ \theta(f)\circ F(\xi).

Suppose that products exist in \mathbf{C}. For a fixed object A of \mathbf{C}, consider the functor

A\times - : \mathbf{C}\rightarrow \mathbf{C}

which sends an object C of \mathbf{C} to the product A\times C in \mathbf{C}. If this functor has a right adjoint, we denote it by

(-)^{A}: \mathbf{C}\rightarrow \mathbf{C}.

We refer to the object A as an exponentiable object. We refer to the object B^{A} for some B in \mathbf{C} as an exponential object in \mathbf{C}. A category is called Cartesian closed if it has a terminal object and binary products, and if every object is an exponentiable object.

In the category \mathbf{Sets}, the exponential object B^{A} corresponds to the set of all functions from A to B. This also explains our notation for functor categories such as \mathbf{Sets^{C^{op}}} and \mathbf{C^{J}}.

Finally, we discuss the concept of subobject classifiers. We start by defining two important kinds of morphisms, monomorphisms and epimorphisms. A monomorphism (also called a mono, or monic) is a morphism f: X\rightarrow Y such that for all morphisms g_{1}: Y\rightarrow Z and g_{2}: Y\rightarrow Z, whenever the compositions f\circ g_{1} and f\circ g_{2} are equal, then it is guaranteed that g_{1} and g_{2} are also equal. An epimorphism (also called an epi, or epic)  is the dual of this concept, obtained by reversing the directions of all the morphisms in the definition of a monomorphism.

Two monomorphisms f: A\rightarrow D and g: B\rightarrow D are called equivalent if there is an isomorphism h: A\rightarrow B such that g\circ h=f. A subobject of D is then defined as an equivalence class of monomorphisms with domain D.

A subobject classifier is an object \Omega and a monomorphism \text{true}: 1\rightarrow \Omega such that to every monic j: U\rightarrow X there is a unique arrow \chi_{j}: X\rightarrow \Omega such that if u: U\rightarrow 1 is the unique morphism from U to the terminal object 1, then we have

\chi_{j}\circ j=\text{true}\circ u.

The significance of the subject classifier can perhaps best be understood by considering the category \mathbf{Sets}. The characteristic function \chi_{j} of the subset U of X is defined as the function on X that gives the value 1 if x\in U and gives the value 0 if x\notin U. Then we can set the terminal object 1 to be the set \{0\} and the object \Omega as the set \{0,1\}. The morphism \text{true} then sends 0\in \{0\} to 0\in \{0,1\}. The idea is that subobjects, i.e. subsets of sets in \mathbf{Sets}, can be obtained as pullbacks of \text{true} along the characteristic function \chi_{j}.

For the category \text{Sh }(X) of sheaves on a topological space X, the subobject classifier is the sheaf on X where for each open subset U of X the set \mathcal{F} (U) is given by the set of open subsets of U. The morphism \text{true} then “selects” the “maximal” open subset U of U.

Now we define our generalization of the Grothendieck topos. An elementary topos is a category \mathcal{E} satisfying the following conditions.

(i) \mathcal{E} has all finite limits and colimits.

(ii) \mathcal{E} is Cartesian closed.

(iii) \mathcal{E} has a subobject classifier.

A Grothendieck topos satisfies all these conditions and is an example of an elementary topos. However, the elementary topos is a much more general idea, and whereas the Grothendieck topos can be considered as a “generalized space”, the elementary topos can be considered as a “generalized universe of sets”. The term “universe”, as used in mathematics, refers to the entirety of where our discourse takes place, such that any concept or construction that we will ever need to consider or discuss can be found in this universe.

Perhaps the most basic example of an elementary topos is the category \mathbf{Sets}. It is actually also a Grothendieck topos, with its underlying category the category with one object and one morphism, which is the identity morphism on its one object. An example of an elementary topos that is not a Grothendieck topos is the category \mathbf{FinSets} of finite sets. It is worth noting, however, that despite the elementary topos being more general, the Grothendieck topos still continues to occupy somewhat of a special place in topos theory, including its applications to logic and other branches of mathematics beyond its origins in algebraic geometry.

II. Logic and the Elementary Topos

Mathematics is formalized, as a language, using what is known as first-order logic (also known as predicate logic or predicate calculus). This involves constants and variables of different “sorts” or “types”, such as x or y, strung together by relations, usually written Q(x, y), expressing a statement such as x=y. We also have functions, usually written g(x, y) expressing something such as x+y. The variables and functions are terms, and when these terms and strung together by relations, they form formulas. These formulas in turn are strung together by binary connectives such as “and”, “or”, “not”, “implies” and quantifiers such as “for all” and “there exists” to form more complicated formulas.

We can associate with an elementary topos a “language”. The “types” of this language are given by the objects of the topos. “Functions” are given by morphisms of objects. “Relations” are given by the subobjects of the object. In addition to these we need a notion of quantifiers, “for all” (written \forall) and “there exists” (written \exists). These quantifiers are given, for the functors \text{Sub }(Y)\rightarrow \text{Sub }(X), by left and right adjoints \exists_{f}, \forall_{f}: \text{Sub }(X)\rightarrow \text{Sub }(Y). For the binary connectives such as “and”, or”, “not”, and “implies”, we rely on the Heyting algebra structure on the subobjects of an elementary topos.

The existence of a Heyting algebra structure means that there exist operations, called join (written \vee) and meet (written \wedge), generalizing unions and intersections of sets, supremum and infimum of elements, or binary connectives “and” and “or”, a least element (written 0), a greatest element (written 1), and an implication operation such that

z\leq(x\Rightarrow y) if and only if z\wedge x\leq y.

We also have the negation of an element x

\neg x=(x\Rightarrow 0).

This Heyting algebra structure for subobjects \text{Sub }(A) of an object A of an elementary topos is provided by taking pullbacks (for the meet) and coproducts (for the join), with 0\rightarrow A as the least element, A\rightarrow A as the greatest element, and the implication given by the exponential.

We have shown one way in which topos theory is related to logic. Now we show how topos theory is related to the most commonly accepted foundations of mathematics, set theory. More technically, these foundations come from a handful of axioms called the ZFC axioms. The letters Z and F come from the names of the mathematicians who developed it, Ernst Zermelo and Abraham Fraenkel, while the letter C comes from another axiom called the axiom of choice.

The elementary topos, with some additional conditions, can be used to construct a version of the ZFC axioms. The first condition is that whenever there are two morphisms f: A\rightarrow B and g: A\rightarrow B, and a morphism x: 1\rightarrow X from the terminal object 1 to A, we only have f\circ x=g\circ x if f=g. In this case we say that the topos is well-pointed. The second condition is that we have a natural numbers object, which is an object \mathbf{N} and morphisms 0:1\rightarrow \mathbf{N} ands:\mathbf{N}\rightarrow \mathbf{N}, such that for any other object X and morphisms x:1\rightarrow X and f:X\rightarrow X, we have a unique morphism h: \mathbf{N}\rightarrow X such that h\circ 0=x and h\circ s=f . The third condition is the axiom of choice; this is equivalent to the statement that for every epimorphism p:X\rightarrow I there exists s:I\rightarrow X such that s\circ p=1.

One of the issues that hounded set theory in the early days after the ZFC axioms were formulated where whether the axiom of choice could be derived from the other axioms (these axioms were simply called the ZF axioms) or whether it needed to be put in separately. Another issue concerned what was known as the continuum hypothesis, a statement concerning the cardinality of the natural numbers and the real numbers, and whether this statement could be proved or disproved from the ZFC axioms alone. The mathematician Paul Cohen showed that both the axiom of choice and the continuum hypothesis are independent of ZF and ZFC respectively. A topos-theoretic version of Cohen’s proof of the independence of the continuum hypothesis was then later developed by the mathematicians William Lawvere and Myles Tierney (both of whom also developed much of the original theory of elementary toposes).

We now discuss certain aspects of topos theory related to Cohen’s proof. First we introduce a construction in an elementary topos that generalizes the Grothendieck topology discussed in More Category Theory: The Grothendieck Topos. A Lawvere-Tierney topology on \mathcal{E} is a map: j: \Omega\rightarrow \Omega such that

(a) j\circ \text{true}=\text{true}

(b) j\circ j=j

(c) j\circ \wedge=\wedge \circ (j\times j)

The Lawvere-Tierney topology allows us to construct sheaves on the topos, and together with the Heyting algebra structure on the subobject classifier \Omega, allows us to construct double negation sheaves, which themselves form toposes that have the special property that they are Boolean, i.e. the Heyting algebra structure of its subobject classifier satisfies the additional property \neg \neg x=x. This is important because a well-pointed topos, which is necessary to formulate a topos-theoretic version of ZFC, is necessarily Boolean. Another condition for the topos to be well-pointed is for it to be two-valued, which means that there are only two morphisms from the terminal object 1 to \Omega. We can obtain such a two-valued topos from any other topos using the concept of a filter, which essentially allows us to take “quotients” of the Heyting algebra structure on \Omega.

There is yet another condition for an elementary topos to be well-pointed, namely that its “supports split” in the topos. This condition is automatically satisfied whenever the topos satisfies the axiom of choice.

It turns out that the topos of double negation sheaves over a partially ordered set is Boolean (as discussed earlier) and satisfies the axiom of choice. For proving the independence of the continuum hypothesis, a partially ordered set was constructed by Cohen, representing  “finite states of knowledge”, and we can use this to form a topos of double negation sheaves known as the Cohen topos. Using the concept of a filter we then obtain a two-valued topos and therefore satisfy all the requirements for a topos-theoretic version of ZFC. However, the continuum hypothesis does not hold in the Cohen topos, thus proving its independence of ZFC.

A similar strategy involving double negation sheaves was used by the mathematician Peter Freyd to develop a topos-theoretic version of Cohen’s proof of the independence of the axiom of choice from the other axioms ZF, using a different underlying category (since a partially ordered set would automatically satisfy the axiom of choice). In both cases the theory of elementary toposes would provide a more “natural” language for Cohen’s original proofs.

III. Geometric Morphisms

We now discuss morphisms between toposes. The elementary topos was inspired by the Grothendieck topos, which was in turn inspired by sheaves on a topological space, so we turn to the classical theory once more and look at morphisms between sheaves. Given a continuous function f: X\rightarrow Y, and a sheaf \mathcal{F} on X, we can define a sheaf, called the direct image sheaf, f_{*}\mathcal{F} on Y by setting f_{*}\mathcal{F}(V)=\mathcal{F}(f^{-1}(V)) for every open subset V\subseteq Y. Similarly, given a sheaf \mathcal{G} on Ywe also have the inverse image sheaf, however it cannot similarly be defined as f^{*}\mathcal{G}(U)=\mathcal{G}(f(U)) for an open subset U\subseteq X, since the image of U in Y may not be an open subset of Y.

This can be remedied by the process of “sheafification”; we think instead in terms of the “stalks” of the sheaf \mathcal{G}, i.e. sets that are in some way “parametrized” by the points y of Y. Then we can obtain sets “parametrized” by the points f(x); these sets then form the inverse image sheaf f^{*}\mathcal{G} on X. The points of a space are of course not open sets in the usual topologies that we use, so the definition of a stalk involves the “direct limit” of open sets containing the point. It is worth noting that the inverse image “preserves” finite limits.

The process of taking the direct image sheaf can be expressed as a functor between the category \text{Sh }(X) of sheaves on X to the category \text{Sh }(Y) of sheaves on Y. The inverse image sheaf is then the right adjoint to the direct image functor, and it has the property that it preserves finite limits.

A geometric morphism is a pair of adjoint functors between toposes such that the left adjoint preserves finite limits. This allows us to form the category \mathfrak{Top} whose objects are elementary toposes and whose morphisms are geometric morphisms. The natural transformations between geometric morphisms, called geometric transformations, give the category \mathfrak{Top} the extra structure of a 2-category. There are also logical morphisms between toposes, which preserve all structure, and with them and their natural transformations we can form the 2-category \mathfrak{Log}.

We can also define the topos \mathfrak{Top}/\mathcal{S} as the category whose objects are geometric morphisms p: \mathcal{E}\rightarrow \mathcal{S} and whose morphisms (p: \mathcal{F}\rightarrow \mathcal{S})\rightarrow (q: \mathcal{E}\rightarrow \mathcal{S}) are pairs (f, \alpha) where f: \mathcal{F}\rightarrow \mathcal{E} is a geometric morphism and \alpha: q\cong p\circ f is a geometric transformation. Together with “2-cells” (f, \alpha)\rightarrow (g, \beta) given by geometric transformations f\rightarrow g that are “compatible” in some sense with \alpha and \beta\mathfrak{Top}/\mathcal{S} also forms a 2-category.

Geometric morphisms can now be used to define the points of a topos. In the category of sets, we can use the morphisms of the set consisting of only one element to all the other sets to indicate the elements of these other sets. The same goes for topological spaces and their points. We have mentioned earlier the category \mathbf{Sets} as the topos of sheaves on a point. Therefore, we define the points of a topos \mathcal{E} as the geometric morphisms from \mathbf{Sets} to \mathcal{E}.

There exist, however, toposes (including Grothendieck toposes) without points. Sheaves, however, are defined only using open sets, therefore to deal with toposes satisfactorily we can make use of the concept of locales, which abstract the properties of open sets and the study of topological spaces, while “forgetting” the underlying sets of points. A topos which is equivalent to the category of sheaves on some locale is called a localic topos.

An important result in the theory of localic toposes is Barr’s theorem, which states that for every Grothendieck topos \mathcal{E} there exists a sheaf \text{Sh }(\mathbf{B}) on a locale \mathbf{B} with a “complete” Boolean algebra structure and an epimorphism \text{Sh }(\mathbf{B})\rightarrow \mathcal{E}. Another important results is Deligne’s theorem, which states that a coherent topos, i.e. a topos \mathcal{E} for which there is a  site (\mathbf{C}, J) where \mathbf{C} has finite limits and the Grothendieck topology has a “basis” which consists of finite covering families, has “enough points“, i.e. for any two arrows \alpha: E\rightarrow D and \alpha: E\rightarrow D in \mathcal{E} there exists a point p: \mathbf{Sets}\rightarrow \mathcal{E} such that the stalk p^{*}(\alpha) is not equal to the stalk p^{*}(\beta) .

We can also use geometric morphisms to define the idea of a classifying topos. A classifying topos is an elementary topos such that objects in any other topos can be “classified” by the geometric morphisms of the topos to the classifying topos. For example, ring objects in any topos \mathcal{E} are classified by the topos given by the opposite category of the category of “finitely presented” rings \mathbf{fp}\textbf{-}\mathbf{rings^{op}}. The object in \mathbf{fp}\textbf{-}\mathbf{rings^{op}} given by the polynomial ring \mathbf{Z}[X] is then a universal object, such that any ring object in \mathcal{E} can be obtained by constructing the pullback of \mathbf{Z}[X]\rightarrow \mathbf{fp}\textbf{-}\mathbf{rings^{op}} along \mathcal{E}\rightarrow \mathbf{fp}\textbf{-}\mathbf{rings^{op}}.

We now combine the idea of classifying toposes (which was inspired by the idea of classifying spaces in algebraic topology) with the applications of topos theory to first-order logic discussed earlier. A theory \mathbb{T} is a set of formulas, called the axioms of the theory, and a model of \mathbb{T} in a topos \mathcal{E} is an interpretation, i.e. an assignment of an object of \mathcal{E} to every type of the first-order language, a subobject of \mathcal{E} to every relation, and a morphism of \mathcal{E} to every function, with quantifiers and binary connectives provided by the corresponding adjoint functors and Heyting algebra structures respectively.

A theory is called a coherent theory if it is of the form \forall x (\phi(x)\Rightarrow \psi(x)), where \phi(x) and \psi(x) are coherent formulas, i.e. formulas which are built up using only the operations of finitary conjunction \wedge, finitary disjunction \vee, and existential quantification \exists. If we also allow as well the operation of infinitary disjunction \bigvee, then we will obtain a geometric formula, and a theory of the form \forall x (\phi(x)\Rightarrow \psi(x)), where \phi(x) and \psi(x) are geometric formulas is called a geometric theory.

Most theories in mathematics are coherent theories. For those which are not, however, there is a certain process called Morleyization which associates to those theories a coherent theory.

For any model of a coherent theory \mathbb{T} in an elementary topos \mathcal{E}, there exists a classifying topos \mathcal{E}_\mathbb{T} and a universal object (in this context also called a universal model) such that said model can be obtained as a pullback of U\rightarrow \mathcal{E}_\mathbb{T} along the geometric morphism \mathcal{E}\rightarrow \mathcal{E}_\mathbb{T}.

We mention yet another aspect of topos theory where logic and geometry combine. We have earlier mentioned the theorems of Deligne and Barr in the context of studying toposes as sheaves on locales. Combined with the logical aspects of the toposes, and the theory of classifying toposes, Deligne’s theorem implies that a statement of the form \forall x (\phi(x)\Rightarrow \psi(x)) where \phi(x) and \psi(x) are coherent formulas holds in all models of the coherent theory \mathbb{T} in any topos if and only if it holds in all models of \mathbb{T} in \mathbf{Sets}.

Meanwhile, Barr’s theorem implies that a statement of the form \forall x (\phi(x)\Rightarrow \psi(x)) where \phi(x) and \psi(x) are geometric formulas holds in all models of the geometric theory \mathbb{T} in any topos if  and only if it holds in all models of \mathbb{T} in Boolean toposes.

In this context, Deligne’s theorem and Barr’s theorem respectively correspond to finitary and infinitary versions of a famous theorem in classical logic called Godel’s completeness theorem.


Topos on Wikipedia

Topos on the nLab

What is … a Topos? by Zhen Lin Low

An Informal Introduction to Topos Theory by Tom Leinster

Topos Theory by Peter T. Johnstone

Sketches of an Elephant: A Compendium of Topos Theory by Peter T. Johnstone

Handbook of Categorical Algebra 3: Categories of Sheaves by Francis Borceux

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk

More Category Theory: The Grothendieck Topos

In Category Theory, we generalized the notion of a presheaf (see Presheaves) to denote a contravariant functor from a category \mathbf{C} to sets. In this post, we do the same to sheaves (see Sheaves).

We note that the notion of an open covering was necessary in order to define the concept of a sheaf, since this was what allowed us to “patch together” the sections of the presheaf over the open subsets of a topological space. So before we can generalize sheaves we must first generalize open coverings and other concepts associated to it, such as intersections.

product, which is a diagram of objects P, X, Y, and morphisms p_{1}: P\rightarrow X and p_{2}: P\rightarrow Y, and if there is another object Q and morphisms q_{1}: Q\rightarrow X and q_{2}: Q\rightarrow Y, then there is a unique morphism u from Q to P such that p_{1}\circ u=q_{1} and p_{2}\circ u=q_{2}. The object P is often also referred to as the product and written X\times Y.

A related notion is that of a  pullback (also called a fiber product) is a diagram of objects P, X, Y, and Z, and morphisms p_{1}: P\rightarrow X, p_{2}: P\rightarrow Y, f: X\rightarrow Z, and g: Y\rightarrow Z, such that f\circ p_{1}=g\circ p_{2}, and if there is another object Q and morphisms q_{1}: Q\rightarrow X and q_{2}: Q\rightarrow Y with f\circ q_{1}=g\circ q_{2}, then there is a unique morphism u from Q to P such that p_{1}\circ u=q_{1} and p_{2}\circ u=q_{2}. The object P is often also referred to as the fibered product and written X\times_{Z}Y.

Another related concept is that of a terminal object. A terminal object T in a category \mathbf{C} is just an object such that for every other object C in \mathbf{C} there is a unique morphism C\rightarrow T.

Finally, we give the definition of an equalizer. We will need this notion when we construct sheaves on our generalization of the open covering of a topological space. An equalizer is a diagram of objects E, X, Y and morphisms eq: E\rightarrow X, f: X\rightarrow Y, and g: X\rightarrow Y, such that f\circ eq=g\circ eq and if there is another object O and morphism m: O\rightarrow X such that f\circ m=g\circ m, there is a unique morphism u: O\rightarrow E such that eq\circ u=m.

By simply reversing the directions of the morphisms on these definitions, we obtain the “dual” notions of coproductpushout (also called fiber coproduct), initial object, and coequalizer.

The objects that we have defined above are called universal constructions and are subsumed under the more general concepts of limits and colimits. These universal constructions are unique up to unique isomorphism (An isomorphism in a category \mathbf{C} is a morphism f: C\rightarrow D in \mathbf{C} for which there exists a necessarily unique morphism g: D\rightarrow C in \mathbf{C}, called the inverse of f, such that f\circ g=1_{\mathbf{D}} and g\circ f=1_{\mathbf{C}}).

These universal constructions are generalizations of familiar concepts. For example, the product in the category of sets corresponds to the cartesian product, while its dual, the coproduct, corresponds to the disjoint union. The terminal object in the category of sets is any set composed of a single element, since every other set has only one function to it, while its dual, the initial object, is the empty set, which has only one function to every other set.

We now proceed with our generalization of an open covering. Let \mathbf{C} be a category and C an object of \mathbf{C}. A sieve S on C is given by a family of morphisms on \mathbf{C}, all with codomain C, such that whenever a morphism f is in S, it is guaranteed that the composition g\circ f is also in S for all morphisms g for which the composition g\circ f makes sense.

If S is a sieve on C and h: D\rightarrow C is any morphism with codomain C, then we denote by h^{*}(S) the family of morphisms g with codomain D such that the composition h\circ g is in S. h^{*}(S) is a sieve on D.

 We now quote from the book Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk the axioms for a Grothendieck topology.

A (Grothendieck) topology on a category \mathbf{C} is a function J which assigns to each object C of \mathbf{C} a collection J(C) of sieves on C, in such a way that

(i) the maximal sieve t_{C}=\{f|\text{cod}(f) =C\} is in J(C);

(ii) (stability axiom) if S\in J(C) , then h^{*}(S)\in J(D) for any arrow h: D\rightarrow C;

(iii) (transitivity axiom) if S\in J(C) and R is any sieve on C such that h^{*} (R)\in J(D) for all h: D\rightarrow C in S, then R\in  J(C).

The intuitions behind these axioms might perhaps best be seen by considering a category whose objects are open sets and whose morphisms are inclusions of these open sets. Axiom (i) essentially says that the open set C is covered by the collection of all its open subsets. Axiom (ii) says that the open subset D of C is covered by the intersections D\cap C_{i} of D with the open subsets C_{i} covering C. Axiom (iii) says that if a collection D_{i,j} of open subsets covers every open subset C_{j} covering C, then the collection D_{i,j} covers C.

We then quote from the same book the definition of a site:

A site will mean a pair (\mathbf{C}, J) consisting of a small category \mathbf{C} and a Grothendieck topology J on \mathbf{C}. If S\in J(C),  one says that S is a covering sieve, or that S covers C (or if necessary, that S J-covers C).

(The book uses the terminology of a small category to specify that the objects and morphisms of the category form a set, instead of a proper class. The terminology of sets and classes was developed to prevent what is known as “Russell’s paradox” and its variants. In many of the posts on this blog we will not need to explicitly specify whether a category is a small category or not.)

We already know how to construct a presheaf on \mathbf{C}; a presheaf is just a contravariant functor from \mathbf{C} to \mathbf{Sets}. Now we just need to generalize the conditions for a presheaf to become a sheaf.

We go back to the conditions that make a (classical) presheaf a sheaf. They can be summarized in the language of category theory by saying that

\displaystyle e:\mathcal{F}(U)\longrightarrow \prod_{i}\mathcal{F}(U_{i})

is the equalizer of

\displaystyle p: \prod_{i}\mathcal{F}(U_{i})\longrightarrow\prod_{i,j}\mathcal{F}(U_{i}\cap U_{j})


\displaystyle q: \prod_{i}\mathcal{F}(U_{i})\longrightarrow\prod_{i,j}\mathcal{F}(U_{i}\cap U_{j})

where for s\in \mathcal{F}(U), e(s)=\{s|_{U_{i}}|i\in I\} and for a family s_{i}\in \mathcal{F}(U_{i}),

p\{s_{i}\}=\{t_{i}|_{U_{i}\cap U_{j}}\}, \quad q\{s_{i}\}=\{s_{i}|_{U_{i}\cap U_{j}}\}.

The analogous condition for a (generalized) presheaf P on a category \mathbf{C} equipped with a Grothendieck topology J is for

\displaystyle e: P(C)\longrightarrow\prod_{f\in S}P(\text{dom f})

to be an equalizer for

\displaystyle p: \prod_{f\in S}P(\text{dom f})\longrightarrow\prod_{f, g f\in S\ \text{dom }f=\text{cod }g}P(\text{dom g})


\displaystyle q: \prod_{f\in S}P(\text{dom f})\longrightarrow\prod_{f, g f\in S\ \text{dom }f=\text{cod }g}P(\text{dom g})

We now introduce the notion of equivalent categories. We first establish some more notation. The set of morphisms from an object C to C' in a category \mathbf{C} will be denoted by \text{Hom}_{\mathbf{C}}(C, C'). A functor F: \mathbf{C}\rightarrow \mathbf{D} is called full (respectively faithful) if for any two objects C and C' of \mathbf{C}, the operation

\text{Hom}_{\mathbf{C}}(C, C')\rightarrow \text{Hom}_{\mathbf{C}}(F(C), F(C'))\quad f\rightarrow F(f)

is surjective (respectively injective). A functor F: C\rightarrow D is called an equivalence of categories if it is full and faithful and if any object D in \mathbf{D} is isomorphic to an object F(C) in the image of F in \mathbf{D}.

We again refer to the book of Mac Lane and Moerdijk for the definition of a Grothendieck topos:

A Grothendieck topos is a category which is equivalent to the category \text{Sh}(\mathbf{C}, J) of sheaves on some site (\mathbf{C}, J).

A Grothendieck topos is often referred to in the literature as some sort of a “generalized space”. In everyday life we think of “space” as something that objects occupy. Or perhaps we may think of a “place” as something that we live in (the word “topos” itself is the Greek word for “place”). The concept of sheaves expresses the idea that when we look at the objects on portions of a space, they can be “patched together” (it seems rather surreal, even unthinkable, for objects in everyday life not to patch together properly).

We have expressed the notion of a topology as being some sort of “arrangement” on a set. A Grothendieck topology is also an arrangement, but instead of making use of the “parts” (subsets) of a set, it instead makes use of the “relations” or “interactions” between objects in a category.

So we can think of the idea of a topos, perhaps, as making a “place” for our objects of interest (such as sets, groups, rings, modules, etc.) to “live in”. This place has an “arrangement” that our objects of interest “respect”, analogous to how open coverings are used to express how objects are “patched together” to form a sheaf on a topological space. This point of view has already become fruitful in algebraic geometry, where the geometry is described in terms of the algebra; so for instance, the “points” of a “shape” correspond to the prime ideals of a ring (see Rings, Fields, and Ideals and More on Ideals), so they may not correspond with the idea of a space we are usually used to, where the points are described by coordinates which are real numbers.

The idea of making a “place” for mathematical objects to “live in” is abstract enough, however, to not be confined to any one branch of mathematics. Thus, the idea of a topos, sufficiently generalized, has found many applications in everything from logic to differential geometry.


Topos in Wikipedia

Sheaf on Wikipedia

Grothendieck Topology on Wikipedia

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk

Category Theory

In many of the previous posts on this blog, we have seen that whenever there is some special property on our sets, then there are special kinds of functions between these sets that in some way “respect” this special property. For example, for topological spaces (see Basics of Topology and Continuous Functions), there are special kinds of functions called continuous functions that in some way respect the topology. Similarly, for vector spaces and modules (see Vector Spaces, Modules, and Linear Algebra), there are special kinds of functions between them called linear transformations that respect the closure under addition and scalar multiplication.

For groups, we also have the concept of a homomorphism (not to be confused with the concept of a homeomorphism in topology), which is a function f between groups such that whenever ab=c in the domain it is guaranteed that f(a)f(b)=f(c) in the range. A homomorphism is therefore a function between groups that respects the law of composition of the groups.

We now introduce in this post a kind of “language” that expresses these ideas in a neat way, with a mechanism that allows us to relate two different kinds of these special sets; this is useful, for example in algebraic topology, where we study topological spaces using groups. Information about topological spaces are reflected as information about groups.

In Homotopy Theory, we learned about the fundamental group of a space, formed by looking at loops starting and ending at a chosen basepoint, and identifying loops that can be deformed into one another under an equivalence relation. For a circle, the fundamental group is the abelian group of integers under addition, which intuitively corresponds to the number of times a loop winds around the circle taking into account the direction (clockwise or counterclockwise).

We now consider a torus. We can think of a torus as being constructed by “gluing” circles to every point of some other “base” circle. We can refer to these circles that are being glued to the “base” circle as the “fiber” circles. Thus the points of the torus can be specified by giving a pair of numbers, one referring to a coordinate of a point of the “base” circle, and the other a coordinate of a point of the “fiber” circle at that point on the base. More formally, we refer to the torus T^{2} as the Cartesian product S^{1}\times S^{1} of two circles.

The fundamental group of the torus is given by the set of ordered pairs of integers, one corresponding to the number of times a loop winds around the “body” of the torus, and the other corresponding to the number of times this same loop winds around the “hole” of the torus, once again taking into account the direction. The pair of integers can then be given a group structure by addition “componentwise”. This group of pairs of integers is an example of a direct product of groups and is written \mathbb{Z}\times \mathbb{Z} (Since the groups involved are all abelian, this is also sometimes referred to as an example of a direct sum and written \mathbb{Z}\oplus \mathbb{Z}).

That the torus is a Cartesian product of two circles and its fundamental group is a direct product of two copies of the fundamental groups of the circle is an example of how information about topological spaces is reflected as information about groups.

We now introduce the language of category theory. We quote from the book Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk:

A category \mathbf{C} consists of a collection of objects (often denoted by capital letters, A, B, C, ..., X, ...) a collection of morphisms (or maps or arrows) (f, g, ...), and four operations; two of the operations associate with each morphism f of \mathbf{C} its domain \text{dom}(f) or \text{d}_{0}(f) and its codomain  \text{cod}(f) or \text{d}_{1}(f), respectively, both of which are objects of \mathbf{C}. One writes f: C\rightarrow D or f: C\xrightarrow{f} D to indicate that f is a morphism on \mathbf{C} with domain C and codomain D, and one says that f is a morphism from C to D. The other two operations are an operation which associates with an object C of \mathbf{C} a morphism 1_{C} (or \text{id}_{C}) called the identity morphism of C and an operation of composition which associates to any pair (f,g) of \mathbf{C} such that \text{d}_{0}(f)=\text{d}_{1}(f) another morphism f\circ g, their composite. These operations are required to satisfy the following axioms.

(i) \text{d}_{0}(1_{C})=C=\text{d}_{1}(1_{C})

(ii) \text{d}_{0}(f\circ g)=\text{d}_{0}(g), \text{d}_{1}(f\circ g)=\text{d}_{1}(f),

(iii) 1_{D}\circ f=f, f\circ 1_{D}=f,

(iv) (f\circ g)\circ h=f\circ (g\circ h)

In (ii)-(iv), we assume that the compositions make sense; thus (ii) is required to hold for any pair of arrows f and g with \text{d}_{0}(f)=\text{d}_{1}(g), and (iii) is required to hold for any two objects C and D of \mathbf{C} and any morphism f from C to D, etc.

Many of the concepts we have already discussed form categories. We have, for example the categories

\mathbf{Sets}, where the objects are sets and the morphisms are functions,

\mathbf{Top}, where the objects are topological spaces and the morphisms are continuous functions,

\mathbf{Top_{*}}, where the objects are topological spaces with selected points, one for each space, called basepoints and the morphisms are continuous functions that take basepoints to basepoints,

\mathbf{Vct_{K}}, where the objects are vector spaces with field of scalars K and the morphisms are linear transformations,

\mathbf{R-Mod}, where the objects are modules with ring of scalars R and the morphisms are linear transformations, and

\mathbf{Grp}, where the objects are groups and the morphisms are homomorphisms.

Note that sometimes the notation for these categories varies depending on the author.

We now introduce the concept of a functor. From the same book as above,

Given two categories \mathbf{C} and \mathbf{D}, a functor from \mathbf{C} to \mathbf{D} is an operation F which assigns to each object C of \mathbf{C} an object F(C) of \mathbf{D}, and to each morphism f of {C} a morphism F(f) of \mathbf{D}, in such a way that F respects the domain and codomain as well as the identities and the composition: F(\text{d}_{0}(f))=\text{d}_{0}(F(f))F(\text{d}_{1}(f))=\text{d}_{1}(F(f))F(1_{C})=1_{F(C)}, and F((f\circ g))=F(f)\circ F(g), whenever this makes sense.

We have already discussed an example of a functor, namely the functor from the category \mathbf{Top_{*}} to the category \mathbf{Grp} which assigns to a topological space with a basepoint its fundamental group, and to basepoint-preserving continuous functions (continuous functions that take basepoint to basepoint) between these topological spaces homomorphisms between their fundamental groups.

In physics the concept of categories and functors is also used in topological quantum field theory. Roughly, the idea is that in classical mechanics (including relativity) the physics is expressed in terms of manifolds (shapes that have no sharp edges) while in quantum mechanics the physics is expressed in terms of vector spaces. So topological quantum field theory studies functors from the category whose objects are manifolds and whose morphisms are cobordisms (a cobordism between two manifolds is a manifold existing one dimension higher whose boundary is made up of those two original manifolds – this is used to express the idea of one manifold transforming into the other, with the one extra dimension of the cobordism representing time) to the category of vector spaces.

Going back to category theory, we discuss some more important concepts. The opposite category \mathbf{C^{op}} of a category \mathbf{C} is the category whose objects are the same as that of \mathbf{C} but whose morphisms are in the opposite direction of the morphisms of \mathbf{C}. A contravariant functor from  \mathbf{C} to \mathbf{D} is just a functor from  \mathbf{C^{op}} to \mathbf{D}.

We recall from our discussion in Presheaves that the classical notion of a presheaf on a topological space X is given by a set \mathcal{F}(U) (sometimes with the extra structure of groups, rings, or modules) for every open subset U of X and functions called restriction maps \rho_{UV}: \mathcal{F}(U)\rightarrow \mathcal{F}(V) for every inclusion of open subsets of X i: V\rightarrow U (we have written it this way instead of the more common V\subseteq U to make it look more symmetric) satisfying certain properties related to identity and composition. We now generalize this classical notion and simply define a presheaf on \mathbf{C} as just a contravariant functor from \mathbf{C} to \mathbf{Sets}.

We introduce one more concept in category theory, that of a natural transformation between functors. Quoting once more from the book of Mac Lane and Moerdijk,

Let F and G be two functors from a category \mathbf{C} to a category \mathbf{D}. A natural transformation \alpha from F to G, written \alpha: F\rightarrow G, is an operation associating with each object C of \mathbf{C} a morphism \alpha_{C}:FC\rightarrow GC of \mathbf{D}, in such a way that, for any morphism f: C'\rightarrow C in \mathbf{C}, G(f)\circ \alpha_{C'}=\alpha_{C}\circ F(f). The morphism \alpha_{C} is called the component of \alpha at C.

We have seen that a presheaf on \mathbf{C} is a contravariant functor from a category \mathbf{C} to the category of sets \mathbf{Sets} . We can now define the category of all presheaves on \mathbf{C}, written \mathbf{\hat{C}}, as the category whose objects are functors P: \mathbf{C^{op}}\rightarrow \mathbf{Sets} on \mathbf{C}, and whose morphisms are natural transformations \theta: P\rightarrow P' between such functors. \mathbf{\hat{C}} is an example of what is called a functor category, and in this context it is also sometimes written as \mathbf{Sets^{C^{op}}}.

We have now seen some basic constructions in category theory. There are many more. It can perhaps be said that in category theory the emphasis is on looking at the “relationships” or “interactions” between objects, as opposed to looking “inside” of these objects, as is the case with set theory. This allows us to make analogies between seemingly different objects, as what we have seen with topological spaces and groups. We end this post with a couple of quotes:

“Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies.”

-Stefan Banach (as quoted by Stanislaw Ulam)

“If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither ‘number’ nor ‘size’, but always form.”

-Alexander Grothendieck


Category Theory on Wikipedia

Functor on Wikipedia

Natural Transformation on Wikipedia

Topological Quantum Field Theory on Wikipedia

Categories for the Working Mathematician by Saunders Mac Lane

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk