# The main theorems of category theory and their applications

This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more focused version of the question Most striking applications of category theory?.

It seems that category theory began as an organizational tool in topology and algebraic geometry, but by now it has grown into an area of research in its own right with applications all over the place in mathematics. I realized as I was thinking about my answer to the question above that I did not know the statements of the main results in category theory, or even if there are "main results". In a comment, Martin Brandenburg suggested several examples:

• The general adjoint functor theorem
• Freyd's representability criterion
• Recognition theorems for locally presentable categories
• Brown's representability theorem

He also indicated that these results have numerous unsung applications to other areas of mathematics. This question is essentially an invitation for Martin and anyone else add to this list and explain some of the applications of the items on it. I would not have asked this question on mathoverflow if I believed that such a list already existed; if I am wrong then the question should probably be closed.

Here is what I have in mind for an answer to this question. It should include the statement of a theorem in pure category theory (ideally using language which is friendly to outsiders) and at least one application to another area of mathematics. The community wiki rule "one theorem per answer" makes sense here, particularly so that others can conveniently add applications to your list.

When I say "theorem in pure category theory", I don't insist that the result be incredibly nontrivial, just that it is a result which is stated and proved in the language of category theory. For example, the statement that $\pi_1$ is a functor belongs to topology, not category theory; on the other hand the Yoneda lemma counts even though it is a "lemma" instead of a "theorem".

When I say "application to another area of mathematics" I am ideally looking for statements which can be formulated without using the language of categories and functors. I want to be clear that I am interested in applications of specific results in category theory, not just results for which categorical thinking is useful (such results are everywhere).

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By the way, Martin's examples are fair game for answers to this question because I don't know the precise statements or any applications. –  Paul Siegel Dec 14 '11 at 16:52
This question is impossible to answer –  Fernando Muro Dec 14 '11 at 21:17
@Martin: That question is mentioned in the question. –  quid Dec 14 '11 at 21:48
Yes. And I have to admit that it is a little bit more focused. But basically the questions are the same: What are the applications of "pure" category theory (in form of ideas or theorems) "outside of" category theory? –  Martin Brandenburg Dec 14 '11 at 21:59
It seems that most of those who answered the other question interpreted it to mean "Where does category theory show up in mathematics?" I am already convinced that categorical considerations often help one organize certain mathematical constructions, but I am beginning to understand that some people use category theory as a tool for proving theorems rather than simply as a language for articulating them. Analogously, the topological proof of the infinitude of primes is a "striking application of topology" to number theory, but it only uses the language and not the substance of topology. –  Paul Siegel Dec 14 '11 at 23:18

The small object argument. Essentially this states that if you have a collection of maps $f_\alpha$ in a presentable category (actually, you only need the domains of the $f_\alpha$ to be compact, along with cocompleteness of the category), then any map in the category can be functorially factored as the composite of two maps:

1. A map which is a transfinite pushout of coproducts of the $f_\alpha$.
2. A map which has the right lifting property with respect to the $f_\alpha$.

This was first used by Grothendieck to show (in his Tohoku paper) that a Grothendieck abelian category always has enough injectives (which, as far as I know, is not directly obvious for abelian sheaves on a site, for instance). Later it became the main tool in constructing model structures on categories, because it lets you show that the factorizations needed in the definition exist.

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Using the Barr cover it is easy to show sheaves of abelian groups on a site have enough injectives. See Johnstone's topos theory book. –  Benjamin Steinberg Dec 14 '11 at 21:26
Thanks for the reference. –  Akhil Mathew Dec 14 '11 at 21:37

Monadicity theorems like Beck's give sufficient conditions on a functor $U \colon B \to A$ with a left adjoint F to be equivalent (in the slice category Cat/A) to the forgetful functor out of the category $A^{U F}$ of algebras for the monad $U F$.

Aside from their obvious usefulness in universal algebra, one important application is in descent theory: suppose you have a category C and a C-indexed category E (i.e. a pseudofunctor $E \colon C^{\mathrm{op}} \to \mathrm{Cat}$) such that

1. Each $f^* = Ef$ has a right adjoint $f_!$, and

2. The Beck--Chevalley condition holds: E takes any pullback square in C to an isomorphism in Cat whose mate is again an isomorphism.

Then (this is due to Bénabou and Roubaud) for a morphism f in C, the category of descent data for f is equivalent to the category of algebras for the monad $f_!f^*$. In particular, f is of effective descent if $f_!$ is monadic. See the nLab page on monadic descent for details.

I believe the original application had C the category of commutative rings and $E \colon R \mapsto R\mathrm{-Mod}$.

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This application comes also into life in the great paper "Descent Theory for Schemes" by Bachuki Mesablishvili. There the author generalizes Grothendieck's descent theory from fpqc morphisms to pure morphisms, which are morphisms which are schematically dense and stay so after arbitrary base change. Very recommended! The author also uses general theorems of factorization systems in categories - although in this case can be done also in a ad hoc manner without any additional effort. See also my answer here: mathoverflow.net/questions/48047/… –  Martin Brandenburg Dec 14 '11 at 22:16

There was some discussion here about special cases of Yoneda's lemma, including the usual examples of Cayley's theorem for groups and Dedekind's embedding theorem for posets.

It also seems that a good chunk of Tannaka duality can be seen as an application of the (enriched) Yoneda lemma -- see the nLab page for discussion.

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I've heard people say before that Cayley's theorem for groups is a special case of the Yoneda Lemma, but I think it's more correct to say that the permutation representation of G as acting on itself by translations is a special case of the Yoneda embedding. The Yoneda lemma, then says that for any G-set X, G-equivariant maps G &rarr; X are in bijection with the elements of X. –  Omar Antolín-Camarena Mar 4 '13 at 4:37

I think of Gabriel-Ulmer duality as one of the early key results in categorical model theory.

Many structures in mathematics are what are called "essentially algebraic". This includes all algebraic structures (which model theories given by a functional signature and universally quantified equations): groups, rings, Lie algebras, etc. It also includes algebra-like structures which don't quite fit in this mold, for example categories, where some of the operations may only be partially defined (but the domains of the operations are given by equations involving other operations). There are various nice ways of expressing the syntax of an essentially algebraic theory: one is via limit sketches, another is just to use finitely complete categories, in much the same way that (following Lawvere) classical algebraic theories can be expressed as certain categories with products.

If we think of a small finitely complete category as a "theory" $T$, then a model of $T$ in this way of thinking is a finitely continuous functor $T \to Set$. The category of models would then be the category $Cont(T, Set)$ of such functors and transformations between them.

What is amazing and very nice is that the theory $T$ can be recovered from the category of models $M$, by considering functors

$$M \to Set$$

which preserve all limits (have left adjoints) and all filtered colimits. This is essentially Gabriel-Ulmer duality, giving a perfect dual correspondence between theories or syntax of finitary essentially algebraic type (small finitely complete categories) and semantic categories of models of such (which can be described in pure category terms as locally finitely presentable categories).

This perfect duality between syntax and semantics is part of a long story in categorical model theory, culminating the theory of accessible and locally presentable categories. Spiritually, it is reminiscent of many other dualities such as the Tannaka-Krein duality mentioned by Finn Lawler earlier.

Edit: This is in response to Martin's comment. I'll give a sample application and some nontrivial consequences (which, to be sure, can be arrived at via other avenues). The moral for me is that Gabriel-Ulmer duality is a relatively simple statement that can become, after a while, a useful part of one's daily thinking.

My example is the category $\text{Cocomm}$ of cocommutative coalgebras over (let's say) a field $k$. What can we say about it? Well, it's easy to see it is cocomplete: one constructs colimits (coproducts, coequalizers, etc.) as one would on the underlying vector spaces, lifting the vector-space colimit to a coalgebra structure in a canonical way. Much more significantly, there is a fundamental theorem about (cocommutative) coalgebras: each is the union (in particular a filtered colimit) of its finite-dimensional subcoalgebras. Meanwhile, a finite-dimensional coalgebra $C$ has the property that $\hom(C, -): \text{Cocomm} \to Set$ preserves filtered colimits.

People who have taken Gabriel-Ulmer duality into their hearts will immediately recognize the import of these results: $Cocomm$ is locally finitely presentable. This implies something at first unexpected: not only are coalgebras coalgebraic over vector spaces: they are models of an essentially algebraic theory! Indeed, Gabriel-Ulmer duality implies that $Cocomm$ is equivalent to the category of finitely continuous (left exact) functors

$$\text{Cocomm}_{\text{fin.dim.}}^{op} \to Set$$

or if you prefer, to the category of finitely continuous functors $\text{CommAlg}_{fd} \to Set$ (so in this case, the relevant "theory" is the category of finite-dimensional commutative algebras).

This unexpected recognition has a host of useful consequences. For example, the category $Cocomm$ is complete. Those who think this is obvious are invited to construct infinite products and equalizers explicitly with their bare hands -- it is not trivial. Also, the category is cartesian closed. Indeed $C$ and $D$ are viewed as cocommutative coalgebras, then the exponential $D^C$ is identified with the left exact functor that takes a finite-dimensional algebra $A$ to $\hom(A^\ast \otimes_k C, D)$, a very pretty and explicit formula.

Similar considerations apply to (not necessarily cocommutative) coalgebras, and to differential graded coalgebras (cocommutative or not).

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Dear Todd, while I think this is a significant and profound answer, I don't see any application "outside of" category in it. Perhaps you know one and add it (and I delete this silly comment). It would be great if some people which have not heard so far about presentable categories (or categories at all?) read your answer and are convinced that they have to learn more about them right away ;). –  Martin Brandenburg Dec 14 '11 at 22:08
Actually, you can only recover base category up to Cauchy completion. A trivial example is two different, but Morita equivalent rings. Or do you mean something more intricate? –  Anton Fetisov Dec 15 '11 at 0:18
Fetisov Anton: not sure what you're referring to. The "theories" here are finitely complete, and that implies they are Cauchy complete. But in any event I'm not referring to the process of trying to recover a small category from its category of presheaves (which would indeed involve Cauchy completion); I'm referring to something similar but different enough to be significant. –  Todd Trimble Dec 15 '11 at 0:54
Thanks for the edit. But why should one care about products in this category when you are not interested in the category theoretic properties in the first place? What is a specific application which can be formulated without the notions of category theory? [This was also one of the requirements which Paul Siegel has mentioned in his question] –  Martin Brandenburg Dec 15 '11 at 19:51
Yeah, okay. I think Paul said, "ideally...", whereas I was giving myself a little leeway here. I mean, let's be fair: what about monadicity theorems (as mentioned by Finn)? Has the intended application been purged of all categorical language? :-) Same question for the small object argument mentioned by Akhil. Is my answer being subjected to a different standard? It seems to me that an algebraist interested in studying coalgebras but who didn't know a lot of category theory (e.g., Gabriel-Ulmer duality) might, just might, find this sort of thing useful to know. :-) –  Todd Trimble Dec 15 '11 at 22:43

The method of forcing in mathematical logic.

If you want to prove the consistency of axiom systems, you can just explicitly present a model of it. To prove results about set theory itself, like the independence of the continuum hypothesis or the axiom of choice, you need to find something like a "modified set theory". According to general philosophy, it is enough to present a category, similar to the category of sets. Category theory gives an abundance of such categories: elementary toposes. They are a special class of cartesian closed categories. They come equipped with a natural internal intuitionistic logic, describing the properties of morphisms. Now the problem of finding the required model transforms into the problem of finding a topos, satisfying our set of axioms, interpreted using the internal logic. It can be done categorically. An example of the construction, proving the independence of the continuum hypothesis, can be found in P.T. Johnstone's "Topos theory".

Technically one can construct a model in some category of sheaves, but topos-theoretic approach is much more simple and flexible (besides, sheaf is also a categorical notion and any topos is roughly a category of sheaves).

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Any Grothendieck topos is a category of sheaves. –  David Roberts Dec 15 '11 at 0:57
More explicitly, any topos that has all small (not just finitary) coproducts and satisfies a size condition is a category of sheaves. –  Toby Bartels Dec 15 '11 at 1:56
This sounds like a nice application of the existence of certain categories, but is there a particular theorem or set of theorems in category theory that is relevant here? It sounds like the hard work here might really be in logic or set theory. –  Paul Siegel Dec 15 '11 at 3:59
@David: Yes, of course. I just didn't feel like discussing technical conditions. The existence of such internal language is, in my opinion, a prominent categorical fact by itself. It is completely non-obvious. The study of semantics for this language also takes many pages and is categorical in nature. The problems of set theory are of little importance here. –  Anton Fetisov Dec 15 '11 at 5:54
@Paul: in category theory constructions have their own value, they need not always be accompanied by mind-blowing theorems. The constructions can already be mind-blowing. –  Andrej Bauer Dec 16 '11 at 8:18

Freyd-Mitchell and Gabriel-Popescu theorems, and also the characterization of co-Grothendieck cats.

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Do you know any good applications of these results? –  Paul Siegel Dec 16 '11 at 17:13
Freyd-Mitchell is quite convenient if you are lazy and don't want to prove things like the Snake lemma in an arbitrary abelian category by categorical arguments but by the usual diagram chase with elements. –  Lennart Meier Feb 26 '13 at 16:20

This is that a small category may be regarded as a simplicial set with unique inner horn fillers.

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