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})

and

\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})

and

\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.

References:

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

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