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Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).

The Dirichlet pigeonhole principle, even if it belonged to the theory of categories (it does not), it would not provide the answer to my question. Indeed, its applications are striking. But the principle itself is obvious and natural.

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    $\begingroup$ What I learned from the answers to that question is that theorems in category theory are often simple and not very surprising, but when you apply them to specific cleverly chosen categories you can get surprising and beautiful constructions almost for free. $\endgroup$ – Paul Siegel Mar 16 '14 at 13:33
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    $\begingroup$ So we don't have to show that the least natural theorem exists uniquely up to isomorphism? $\endgroup$ – Allen Knutson Mar 16 '14 at 16:39
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    $\begingroup$ "Unexpected/least natural" is not the same as "applied", unless such applications per se are unexpected. $\endgroup$ – Wlodek Kuperberg Mar 16 '14 at 17:04
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    $\begingroup$ @WlodzimierzHolsztynski - CW = Community Wiki, because there is no single correct answer to this question. $\endgroup$ – David Roberts Mar 17 '14 at 3:23
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    $\begingroup$ It also means that people can more freely edit the answers, as they are not competing to be the one right answer, and so people don't get reputation points for answers. The original asker doesn't either. $\endgroup$ – David Roberts Mar 17 '14 at 8:48
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In topos theory, it is not at all obvious that an elementary topos has finite colimits: there is nothing in the usual definition which would suggest this. And the standard proof, which passes through monadicity theorems and Beck-Chevalley conditions, winds up looking highly technical to most people, and (for most people) doesn't shed much intuitive light on why the result is true.

It's possible to prove this result by other means. There is some information on constructing coproducts in a topos in a hands-on way here, but as one can see it takes a while to set out. One can do this for coequalizers as well, but I don't know where this has been written down (yet).

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  • $\begingroup$ I can't comment on your "hands-on way" because the link is broken (is there a problem with ncatlab?). However, I used to give these constructions (i.e. coproducts and coequalisers in the internal logic of a topos) as a take-home exam exercise for undergraduate students. So I guess these constructions can't be really difficult :-) $\endgroup$ – Michal R. Przybylek Mar 20 '14 at 0:03
  • $\begingroup$ But in order to get an interpretation of logic in an elementary topos you need to know that it has finite colimits, don't you? $\endgroup$ – Peter Arndt Mar 20 '14 at 0:25
  • $\begingroup$ @PeterArndt, no --- you are given positive connectives and quantifications, so it suffices to define finite disjunctions in the internal logic. But this is easy when you can quantify over all propositions. $\endgroup$ – Michal R. Przybylek Mar 20 '14 at 0:47
  • $\begingroup$ @MichalR.Przybylek Sounds like undergraduate students in your country are better than undergraduates in mine. :-) In some sense you're right, although I imagine most people would need a few hints to carry out all the details. $\endgroup$ – Todd Trimble Mar 20 '14 at 0:51
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    $\begingroup$ The first proof that coproducts are derivable from the now standard axioms for an elementary topos was given by Christian Mikkelsen in 1976 in his PhD thesis under the supervision of Anders Kock in Aarhus. I forget the details but they are indeed simpler than the construction that one would obtain by unwinding Bob Pare's theorem that the contravariant powerset functor is monadic. $\endgroup$ – Paul Taylor Mar 20 '14 at 15:49
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In a similar vein to Todd Trimble's remark on elementary toposes, let's recall the following fact:

Theorem. An accessible category is complete if and only if it is cocomplete.

The quickest proof I know of uses the accessible adjoint functor theorem. The curious thing is this: from a theoretical perspective, the cocompleteness hypothesis is more useful; yet in practice it is the completeness hypothesis that is easiest to verify. For instance, it is easy to see that many categories of essentially algebraic structures (e.g. groups, rings, categories) are accessible and complete, but the construction of colimits in these categories is often non-obvious.

The only instance of the "if" direction I am aware of is a corollary of Giraud's theorem: any category satisfying Giraud's axioms is necessarily complete. This is obvious if we assume Giraud's theorem, because any such category is a topos of sheaves on a Grothendieck site, hence, a reflective subcategory of a complete category.

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If Yoneda's Lemma counts as "unexpected", then this would be my favorite (but you might argue that it is too elementary to be unexpected).

It has numerous applications, and I particularly like the framework it provides to define affine varieties as representable functors from the category of commutative $k$-algebras to the category of sets, and similarly to define linear algebraic groups as representable functors from the category of commutative $k$-algebras to the category of groups (more precisely those functors represented by a finitely generated $k$-algebra).

In particular, Yoneda's Lemma gives, without any additional effort, the duality between the category of linear algebraic $k$-groups and the category of finitely generated commutative Hopf $k$-algebras.

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  • $\begingroup$ I wouldn't call it unexpected, I'd say that it it exactly as expected. Any category, a priori, only has data about objects and arrows. So all way know about any given object is the arrows going into it and coming out of it. We should expect to be able to recover the object, knowing all of the arrows going into it, because that's all there is to know about it! $\endgroup$ – fhyve Mar 21 '14 at 2:10

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