Some Basics of Class Field Theory

Class field theory is one of the crown jewels of modern algebraic number theory. It can be viewed as a generalization of the “reciprocity laws” discovered by Carl Friedrich Gauss and other number theorists of the 19th century. However, as we have not yet discussed reciprocity laws in this blog, we will leave that point of view to the references for now. Instead, in order to describe class field theory, we rely on the following quote from the mathematician Claude Chevalley:

“The object of class field theory is to show how the abelian extensions of an algebraic number field K can be determined by objects drawn from our knowledge of K itself; or, if one prefers to present things in dialectic terms, how a field contains within itself the elements of its own transcending.”

Our approach in this post, as with much of the modern literature, will start from the “local” point of view, and then we will put together all these local pieces in order to have a “global” theory. Recall that there is an analogy between geometry and number theory, and that primes play the role of points in this analogy. Therefore, as in geometry “local” means “zooming in” on a point, in number theory “local” means “zooming in” on a prime. Putting these pieces together is akin to what we have done in Adeles and Ideles. This will become more clear when we define local fields and global fields.

Let K be a (nonarchimedean) local field. This means that K is complete with respect to a discrete valuation (see Valuations and Completions and Adeles and Ideles) and that its residue field is finite. These are either the fields \mathbb{Q}_{p} (the p-adic numbers), \mathbb{F}_{p}((t)) (the field of formal power series over a finite field \mathbb{F}_{p}), or their finite extensions. Let L be a finite extension of K.

We define the norm homomorphism as

\displaystyle N_{L|K}(x)=\prod_{\sigma}\sigma x

for x\in L and \sigma\in \text{Gal}(L|K) (note that there are many notations for the action of \sigma on x; in the book Algebraic Number Theory by Jurgen Neukirch, the notation x^{\sigma} is used instead). We let N_{L|K}L^{\times} stand for the image of the norm homomorphism in K. Then local class field theory tells us that we have the following isomorphism:

\displaystyle K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

We see that everything in the left-hand side belongs to the field K. This is related to Chevalley’s quote earlier. However, there is even more to this, as we shall see later.

Understanding more about this isomorphism requires the theory of Galois cohomology (see Etale Cohomology of Fields and Galois Cohomology). Namely, the Galois cohomology group H^{2}(\text{Gal}(L|K),L^{\times}) is isomorphic as a group to the group homomorphisms from \text{Gal}(L|K)^{\text{ab}} to K^{\times}/N_{L|K}L^{\times}. It is cyclic of degree equal to the degree of L over K.

There is an injective map from H^{2}(\text{Gal}(L|K),L^{\times}) to the quotient \mathbb{Q}/\mathbb{Z}, and the element of H^{2}(\text{Gal}(L|K),L^{\times}) that gets mapped to 1/n, where n is the degree of L over K, is precisely the element that corresponds to the inverse of the isomorphism K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

Now let K be a global field, which means that it is a finite extension either of \mathbb{Q} (the rational numbers) or of \mathbb{F}_{p}(t) (the function field over a finite field \mathbb{F}_{p}). Let L be a finite extension of K. Let C_{K} and C_{L} denote the idele class groups (see Adeles and Ideles) of K and L respectively. As in the local case, we will need a norm homomorphism, but this time it will be for idele class groups.

We will define this norm homomorphism “componentwise”. Writing an idele as (z_{w}), we take the norm N_{L_{w}|K_{v}}(z_{w}), and take the product for all primes w above v. We do this for every prime v of K, and thus we obtain an element of the group of ideles of K, and then we take the quotient to obtain an element of the idele class group of K. We denote by N_{L|K}C_{L} the image of this norm homomorphism in C_{K}.

Then global class field theory tells us that we have the following isomorphism:

\displaystyle C_{K}/N_{L|K}C_{L}\sim\text{Gal}(L|K)^{\text{ab}}

Again, as in the local case, everything in the left-hand side belongs to C_{K}.

As we have said earlier, we can obtain this isomorphism by putting together the local pieces from local class field theory, i.e. homomorphisms from K_{v}^{\times} to \text{Gal}(L_{w}|K_{v})^{\text{ab}} which come from the isomorphisms from K_{v}^{\times}/N_{L|K}L_{w}^{\times}, as ideles have components which are local fields, and then taking the quotient to obtain the isomorphism for idele class groups, similar to what we have done for the norm homomorphism.

However, in order to obtain the desired isomorphism, the map (called the Artin map)

\psi:I_{K}^{\times}\rightarrow\text{Gal}(L|K)^{\text{ab}}

from the group of ideles I_{K} of K to the group \text{Gal}(L|K)^{\text{ab}}, which is obtained from putting together the local pieces (before taking the quotients) must satisfy three properties:

(i) It has to be continuous with respect to the topologies on I_{K} and \text{Gal}(L|K) (the topology on the group of ideles is discussed in Adeles and Ideles, while the topology on \text{Gal}(L|K) is the so-called Krull topology – the latter is part of the theory of profinite groups).

(ii) The image of K^{\times} (as embedded in its group of ideles I_{K}) is equal to the identity.

(iii) It is equal to the Frobenius morphism for elements in I_{K}^{S} (see again Adeles and Ideles for the explanation of this notation), where S consists of the archimedean primes and those primes which are ramified in L (see Splitting of Primes in Extensions).

It is a challenging task in itself to prove that the Artin map does indeed satisfy these properties, and for now we leave it to the references. Instead, we mention a few more properties of class field theory. In the local case, class field theory also classifies the subgroups of K^{\times} which are of the form N_{L|K}L^{\times}, which correspond to the open subgroups of finite index in K^{\times}. Since the finite abelian extension L of K also obviously corresponds to the subgroup N_{L|K}L^{\times}, we then obtain a classification of the finite abelian extensions of K. Similarly, in the global case, class field theory classifies the subgroups of C_{K} which are of the form N_{L|K}C_{L}, which correspond to the open subgroups of finite index in C_{K}. The field which corresponds to the such a subgroup is called its class field. In the case that L is the maximal unramified abelian extension of K, L is called the Hilbert class field of K, and there we have the result that the ideal class group (see Algebraic Numbers) of K is isomorphic to the Galois group \text{Gal}(L|K). With the ideas discussed in this last paragraph, the goal of class field theory as expressed in the quote of Chevalley is fulfilled; we are able to describe the abelian extensions of K from knowledge only of K itself.

References:

Class Field Theory on Wikipedia

Artin Reciprocity Law on Wikipedia

Profinite Group on Wikipedia

Class Field Theory by J.S. Milne

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

Primes of the Form x^{2}+ny^{2} by David A. Cox

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Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

250px-TrefoilKnot_01.svg

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

Splitting of Primes in Extensions

In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.

Let \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_{K} of an algebraic number field K (we will sometimes also refer to \mathfrak{p} as a prime ideal of K – this is common practice and hopefully will not cause any confusion). In an algebraic number field L which contains K (we also say that L is an extension of K, and write L|K), this prime ideal \mathfrak{p} decomposes into a product of prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r} in \mathcal{O}_L, with respective exponents e_{1},e_{2}...e_{r}, i.e.

\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}.

The exponents e_{1},e_{2}...e_{r} are called the ramification indices of the prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}. If e_{i}=1, and the residue field extension \kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p}) (see below) is separable, we say that \mathfrak{P}_{i} is unramified over K. If e_{1}=e_{2}=...e_{r}=1, we say that the prime \mathfrak{p} is unramified. If all primes of K are unramified in L, we say that the extension L|K is unramified.

In the rest of this post we will continue to assume the factorization of \mathfrak{p} as shown above. The residue fields \kappa(\mathfrak{P}_{i}) and \kappa(\mathfrak{p}) of \mathcal{O}_{L} and \mathcal{O}_{K} at the primes \mathfrak{P}_{i} and \mathfrak{p} are defined as the quotients \mathcal{O}_{L}/\mathfrak{P}_{i} and \mathcal{O}_{K}/\mathfrak{p}, and the inertia degrees f_{i} are defined as the degrees of the fields \kappa(\mathfrak{P}_{i}) with respect to the field \kappa(\mathfrak{p}) (i.e. the dimensions of the vector spaces \kappa(\mathfrak{P}_{i}) over the field of scalars \kappa(\mathfrak{p})), i.e.

\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})].

The ramification indices e_{i}, the inertia degrees f_{i}, and the degree n=[L:K] of the field extension L with respect to K are related by the following “fundamental identity“:

\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n

In order to understand these concepts better, we can look at the following “extreme” cases:

If e_{i}=1 and f_{i}=1 for all i, then r=n, and we say that the prime \mathfrak{p} splits completely in L.

If r=1 and f_{1}=1, then e_{1}=n, and we say that the prime \mathfrak{p} ramifies completely in L.

If r=1 and e_{1}=1, then f_{1}=n, and we say that the prime \mathfrak{p}  is inert in L.

Consider for example, the field \mathbb{Q}(i) as a field extension of the field \mathbb{Q}. The ring of integers of \mathbb{Q}(i) is the ring of Gaussian integers \mathbb{Z}[i] (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of \mathbb{Q} is the ring of ordinary integers \mathbb{Z}. The degree [\mathbb{Q}(i):\mathbb{Q}] is equal to 2. The prime ideal (5) of \mathbb{Z} splits completely as the product (2+i)(2-i) in \mathbb{Z}[i], the prime ideal (2) of \mathbb{Q} ramifies completely as (1+i)^{2} in \mathbb{Z}[i], while the prime ideal (3) of \mathbb{Z} is inert in \mathbb{Z}[i].

We now bring in Galois groups. We assume that L is a Galois extension of K. This means that the order of G(L|K), the Galois group of L over K, is equal to the degree of L over K. In this case, it turns out that we will have

\displaystyle e_{1}=e_{2}=...=e_{r}

and

\displaystyle f_{1}=f_{2}=...=f_{r}.

The fundamental identity then becomes

efr=n.

This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal \mathfrak{P} of \mathcal{O}_{K}, we define the decomposition group G_{\mathfrak{P}} as the subgroup of the Galois group G that fixes \mathfrak{P}, i.e.

\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}.

The elements of L that are fixed by the decomposition group G_{\mathfrak{P}} form what is called the decomposition field of K over \mathfrak{P}, denoted Z_{\mathfrak{P}}:

 \displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}

Every element \sigma of G_{\mathfrak{P}} automorphism \bar{\sigma} of \kappa(\mathfrak{P}) sending the element given by a\text{ mod }\mathfrak{P} to the element given by \sigma a\text{ mod }\mathfrak{P}. The residue field of the decomposition field Z_{\mathfrak{P}} with respect to \mathfrak{p} is the same as the residue field of the field K with respect to \mathfrak{p}, which is \kappa(\mathfrak{p}). Therefore we have a surjective homomorphism

\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))

which sends the element \sigma of G_{\mathfrak{P}} to the element \bar{\sigma} of G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})). The kernel of this homorphism is called the inertia group of \mathfrak{P} over K. Once again, the elements of L fixed by the inertia group I_{\mathfrak{P}} form what we call the inertia field of K over \mathfrak{P}, denoted T_{\mathfrak{P}}:

 \displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}

The groups G_{\mathfrak{P}}, I_{\mathfrak{P}}, G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})) are related by the following exact sequence:

\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0

Meanwhile, the relationship between the fields K, Z_{\mathfrak{P}}, T_{\mathfrak{P}}, and L can be summarized as follows:

\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L

The ramification index, inertia degree, and the number of primes in K into which a prime \mathfrak{p} in L splits are given in terms of the degrees of the aforementioned fields as follows:

\displaystyle e=[L:T_{\mathfrak{P}}]

\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]

\displaystyle r=[Z_{\mathfrak{P}}:K]

Let \mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}, and \mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}. We also refer to \mathfrak{P}_{Z} (resp. \mathfrak{P}_{T}) as the prime ideal of Z_{\mathfrak{P}} (resp. T_{\mathfrak{P}}) below \mathfrak{P}.

The ramification index of \mathfrak{P} over \mathfrak{P}_{T} is equal to e, and its inertia degree is equal to 1. Meanwhile, the ramification index of \mathfrak{P}_{T} over \mathfrak{P}_{Z} is equal to 1, and its inertia degree is equal to e. Finally, the ramification index and inertia degree of \mathfrak{P}_{Z} over \mathfrak{p} are both equal to 1.

We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let \mathfrak{P} be a prime that is unramified over K. Then G_{\mathfrak{P}} is isomorphic to G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})), it is cyclic, and it is generated by the unique automorphism

\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}    for all    \displaystyle a\in \mathcal{O}_{K}

where q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]. The automorphism \varphi_{\mathfrak{P}} is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.

References:

Splitting of Prime Ideals in Galois Extensions on Wikipedia

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Number Fields by Daniel Marcus

Algebraic Theory of Numbers by Pierre Samuel

Algebraic Number Theory by Jurgen Neukirch

SEAMS School Manila 2017: Topics on Elliptic Curves

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

Southeast Asian Mathematical Society (SEAMS) School Manila 2017: Topics on Elliptic Curves

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.

Some Useful Links: Quantum Gravity Seminar by John Baez

I have not been able to make posts tackling physics in a while, since I have lately been focusing my efforts on some purely mathematical stuff which I’m trying very hard to understand. Hence my last few posts have been quite focused mostly on algebraic geometry and category theory. Such might perhaps be the trend in the coming days, although of course I still want to make more posts on physics at some point.

Of course, the “purely mathematical” stuff I’ve been posting about is still very much related to physics. For instance, in this post I’m going to link to a webpage collecting notes from seminars by mathematical physicist John Baez on the subject of quantum gravity – and much of it involves concepts from subjects like category theory and algebraic topology (for more on the basics of these subjects from this blog, see Category TheoryHomotopy Theory, and Homology and Cohomology).

Here’s the link:

Seminar by John Baez

As Baez himself says on the page, however, quantum gravity is not the only subject tackled on his seminars. Other subjects include topological quantum field theory, quantization, and gauge theory, among many others.

John Baez also has lots of other useful stuff on his website. One of the earliest mathematics and mathematical physics blogs on the internet is This Week’s Finds in Mathematical Physics, which apparently goes back all the way to 1995, and is one of the inspirations for this blog:

This Week’s Finds in Mathematical Physics by John Baez

Many of the posts on This Week’s Finds in Mathematical Physics show the countless fruitful, productive, and beautiful interactions between mathematics and physics. This is also one of the main goals of this blog – reflected even by the posts which have been focused on mostly “purely mathematical” stuff.

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):

Monoidal_category_pentagon.svg

Monoidal_category_triangle.svg

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:

Monoid_multiplication.svg

Monoid_unit_svg.svg

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{Cat},\times,\mathbf{1})

(\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

Rings

K-algebras

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):

Braid_category_hexagon.svg

Braid_category_inverse_hexagon.svg

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.

References:

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.

References:

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

Adeles and Ideles

In Valuations and Completions we introduced the p-adic numbers \mathbb{Q}_{p}, which, like the real numbers, are the completion of the rational numbers under a certain kind of valuation. There is one such valuation for each prime number p, and another for the “infinite prime”, which is just the usual absolute value. Each valuation may be thought of as encoding number theoretic information related to the prime p, or to the “infinite prime”, for the case of the absolute value (more technically, the p-adic valuations are referred to as nonarchimedean valuations, while the absolute value is an example of an archimedean valuation).

We can consider valuations not only for the rational numbers, but for more general algebraic number fields as well. In its abstract form, given an algebraic number field K, a (multiplicative) valuation of K is simply any function |\ | from K to \mathbb{R} satisfying the following properties:

(i) |x|\geq 0, where x=0 if and only if x=0

(ii) |xy|=|x||y|

(iii) |x+y|\leq|x|+|y|

If this seems reminiscent of the discussion in Metric, Norm, and Inner Product, it is because a valuation does, in fact, define a metric on K, and by extension, a topology. Two valuations are equivalent if they define the same topology; another way to phrase this statement is that two valuations |\ |_{1} and |\ |_{2} are equivalent if |x|_{1}=|x|_{2}^{s} for some positive real number s, for all x\in K.  The valuation is nonarchimedean if |x+y|\leq\text{max}\{|x|,|y|\}; otherwise, it is archimedean.

Just as in the case of rational numbers, we also have an exponential valuation, defined as a function v from the field K to \mathbb{R}\cup \infty satisfying the following conditions:

(i) v(x)=\infty if and only if x=0

(ii) v(xy)=v(x)+v(y)

(iii) v(x+y)\geq\text{min}\{v(x),v(y)\}

Two exponential valuations v_{1} and v_{2} are equivalent if v_{1}(x)=sv_{2}(x) for some real number s, for all x\in K.

The idea of valuations allows us to make certain concepts in algebraic number theory (see Algebraic Numbers) more abstract. We define a place v of an algebraic number field K as an equivalence class of valuations of K. We write K_{v} to denote the completion of K under the place v; these are the generalizations of the p-adic numbers and real numbers to algebraic number fields other than \mathbb{Q}. The nonarchimedean places are also called the finite places, while the archimedean places are also called the infinite places. To express whether a place v is a finite place or an infinite place, we write v|\infty or v\nmid\infty respectively.

The infinite places are of two kinds; the ones for which K_{v} is isomorphic to \mathbb{R} are called the real places, while the ones for which K_{v} is isomorphic to \mathbb{C} are called the complex places. The number of real places and complex places of K, denoted by r_{1} and r_{2} respectively, satisfy the equation r_{1}+2r_{2}=n, where n is the degree of K over \mathbb{Q}, i.e. n=[K:\mathbb{Q}].

By the way, in some of the literature, such as in the book Algebraic Number Theory by Jurgen Neukirch, “places” are also referred to as “primes“. This is intentional – one may actually think of our definition of places as being like a more abstract replacement of the definition of primes. This is quite advantageous in driving home the concept of primes as equivalence classes of valuations; however, to avoid confusion, we will stick to using the term “places” here, along with its corresponding notation.

When v is a nonarchimedean valuation, we let \mathfrak{o}_{v} denote the set of all elements x of K_{v} for which |x|_{v}\leq 1. It is an example of a ring with special properties called a valuation ring. This means that, for any x in K, either x or x^{-1} must be in \mathfrak{o}_{v}. We let \mathfrak{o}_{v}^{*} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}=1, and we let \mathfrak{p}_{v} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}< 1. It is the unique maximal ideal of \mathfrak{o}_{v}.

Now we proceed to consider the modern point of view in algebraic number theory, which is to consider all these equivalence classes of valuations together. This will lead us to the language of adeles and ideles.

An adele \alpha of K is a family (\alpha_{v}) of elements \alpha_{v} of K_{v} where \alpha_{v}\in K_{v}, and \alpha_{v}\in\mathfrak{o}_{v} for all but finitely many v. We can define addition and multiplication componentwise on adeles, and the resulting ring of adeles is then denoted \mathbb{A}_{K}. The group of units of the ring of adeles is called the group of ideles, denoted I_{K}. For a finite set of primes S that includes the infinite primes, we let

\displaystyle \mathbb{A}_{K}^{S}=\prod_{v\in S}K_{v}\times\prod_{v\notin S}\mathfrak{o}_{v}

and

\displaystyle I_{K}^{S}=\prod_{v\in S}K_{v}^{*}\times\prod_{v\notin S}\mathfrak{o}_{v}^{*}.

We denote the set of infinite primes by S_{\infty}. Then \mathfrak{o}_{K}, the ring of integers of the number field K, is given by K\cap\mathbb{A}_{K}^{S_{\infty}}, while \mathfrak{o}_{K}^{*}, the group of units of \mathfrak{o}_{K}, is given by K^{*}\cap I_{K}^{S_{\infty}}.

Any element of K is also an element of \mathbb{A}_{K}, and any element of K^{*} (the group of units of K) is also an element of I_{K}. The elements of I_{K} which are also elements of K^{*} are called the principal ideles. This should not be confused with the concept of principal ideals; however the terminology is perhaps suggestive on purpose. In fact, ideles and fractional ideals are related. Any fractional ideal \mathfrak{a} can be expressed in the form

\displaystyle \mathfrak{a}=\prod_{\mathfrak{p}}\mathfrak{p}^{\nu_{\mathfrak{p}}}.

Therefore, we have a mapping

\displaystyle \alpha\mapsto (\alpha)=\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_v)}

from the group of ideles to the group of fractional ideals. This mapping is surjective, and its kernel is I_{K}^{S_{\infty}}.

The quotient group I_{K}/K^{*} is called the idele class group of K, and is denoted by C_{K}. Again, this is not to be confused with the ideal class group we discussed in Algebraic Numbers, although the two are related; in the language of ideles, the ideal class group is defined as I_{K}/I_{K}^{S_{\infty}}K^{*}, and is denoted by Cl_{K}. There is a surjective homomorphism C_{K}\mapsto Cl_{K} induced by the surjective homomorphism from the group of ideles to the group of fractional ideals that we have described in the preceding paragraph.

An important aspect of the concept of adeles and ideles is that they can be equipped with topologies (see Basics of Topology and Continuous Functions). For the adeles, this topology is generated by the neighborhoods of 0 in \mathbb{A}_{K}^{S_{\infty}} under the product topology. For the ideles, this topology is defined by the condition that the mapping \alpha\mapsto (\alpha,\alpha^{-1}) from I_{K} into \mathbb{A}_{K}\times\mathbb{A}_{K} be a homeomorphism onto its image. Both topologies are locally compact, which means that every element has a neighborhood which is compact, i.e. every open cover of that neighborhood has a finite subcover. For the group of ideles, its topology is compatible with its group structure, which makes it into a locally compact topological group.

In this post, we have therefore seen how the theory of valuations can allow us to consider a more abstract viewpoint for algebraic number theory, and how considering all the valuations together to form adeles and ideles allows us to rephrase the usual concepts related to algebraic number fields, such as the ring of integers, its group of units, and the ideal class group, in a new form. In addition, the topologies on the adeles and ideles can be used to obtain new results; for instance, because the group of ideles is a locally compact topological (abelian) group, we can use the methods of harmonic analysis (see Some Basics of Fourier Analysis) to study it. This is the content of the famous thesis of the mathematician John Tate. Another direction where the concept of adeles and ideles can take us is class field theory, which relates the idele class group to the other important group in algebraic number theory, the Galois group (see Galois Groups). The language of adeles and ideles can also be applied not only to algebraic number fields but also to function fields of curves over finite fields. Together these fields are also known as global fields.

References:

Adele Ring on Wikipedia

Tate’s Thesis on Wikipedia

Class Field Theory on Wikipedia

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

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

F:\textbf{Sets}\rightarrow\textbf{Vect}_{K}

\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.

References:

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.

References:

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