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What are examples of properties of particular categories that can be formulated in categorical language and "feel" like they ought to be provable formally but they actually are not? I think that this question is inherently imprecise but one can not seriously say that every question on this site is precise.

To clarify, the existence of fiber products in the category of schemes would not count IMHO because I personally do not expect a random category to have fiber products.

A non-example is the fact that for a morphism of schemes $X\rightarrow Y$, the diagonal morphism $X\rightarrow X\times_Y X$ is a monomorphism. At first I thought that this could only proven non-formally by noticing that the diagonal morphism is an immersion and immersions are monomorphisms but it turns out that in any category with fiber products the diagonal morphism is a monomorphism.

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    $\begingroup$ Just to be precise, you'd like a statement that can be expressed in the language of categories (with enough structure), and which holds in any category (with enough structure), but the proof itself cannot be carried out in the language of categories? It's almost as if you're asking about a non-completeness result about of category theory (seen as a formal system). $\endgroup$ – Andrej Bauer Mar 28 '19 at 7:39
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    $\begingroup$ I believe this question is really too imprecise, in the sense that it really makes it impossible to answer it : To me "Abstract nonsense" refers to anythings that is going to be true in any category, but you are explicitly rejecting all properties that are not true in all category... So there just cannot be any answer to your question. $\endgroup$ – Simon Henry Mar 28 '19 at 9:32
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    $\begingroup$ @SimonHenry I think the question is more like can you immediately recognize a statement provable by abstract nonsense as such and the question is not about properties true in any category. So has categorical intuition ever lied to you? $\endgroup$ – user74900 Mar 28 '19 at 12:16
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    $\begingroup$ Does "filtered colimits preserve exactness in $R-\mathrm{Mod}$" fall into your scope ? It sounds like it should be provable by abstract nonsense, but then the proof should work in any abelian category with filtered colimits, but it doesn't : it's actually specific to $R$-modules (and certain abelian categories, which receive an additional name if they satisfy the property) $\endgroup$ – Maxime Ramzi Mar 28 '19 at 19:38
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It's unclear to me both what counts as "abstract nonsense" and what counts as "looks like it should be provable by abstract nonsense", so I think the question is rather subjective. Nonetheless, I think it has merit, so here's a possible example under my own subjective interpretation. Taking inspiration from Aknazar Kazhymurat's comment, my own take is that one's understanding of "what is likely to be provable by abstract nonsense" should be constantly evolving based on new evidence, so maybe the question is really more like "what are some interesting examples of subtleties in the scope of abstract nonsense?".

Consider the category $Top_\Delta$ of Delta-generated spaces. This is the smallest full subcategory of the category $Top$ of topological spaces which is closed under colimits and contains the unit interval. Since it is generated under colimits by a small amount of data, it is natural to think that $Top_\Delta$ should be a locally presentable category (which is indeed the case). Indeed, until recently it was beleived that any cocomplete category with a small colimit-generator was locally presentable under the set-theoretical hypothesis known as Vopenka's Principle. So it was thought that, modulo set-theoretical subtleties, the local presentability of $Top_\Delta$ should be viewed as following directly from its definition.

Unfortunately, there turned out to be an error in the proof of this consequence of Vopenka's principle (and it turns out that $Set^{op}$ is a counterexample), so this line of reasoning is actually not valid. Even before this was realized, though, it was desirable to have a proof that worked without strong set-theoretical assumptions.

Perhaps surprisingly it turns out that the proof in the literature that $Top_\Delta$ is locally presentable relies on notions of infinitary logic. Now, in some sense, this just means that the proof is "doubly-abstract nonsense", but in another sense, it's a type of abstract nonsense that even sophisticated category theorists are not generally familiar with. An unwinding of this proof avoids explicitly using infinitary logic, but still requires one to think about topological spaces in terms of ultrafilters rather than open sets.

What is the moral of the story? I'm not quite sure. Maybe it's simply "accessibility questions sometimes require you to think hard and creatively about the specifics of your situation", even though they are often treated as an afterthought (I've certainly been guilty of this attitude before!).

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  • $\begingroup$ Where is the error noted? Or is that in arXiv:1812.10649 also? $\endgroup$ – David Roberts Apr 2 '19 at 21:53
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    $\begingroup$ @DavidRoberts Yes, that paper is the first place it's noted. Here's the link again. As far as I know, nobody was aware of this error 8 months ago. $\endgroup$ – Tim Campion Apr 2 '19 at 22:10
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A universal central extensions looks like an object defined by a universal property similar to the universal property of the reflector of a reflective subcategory. Thus, one might expect that its functoriality can be proved abstractly in the same way the functoriality of a reflector is proved, but this is not the case (see this question for references for a proof). I have to admit that the proof can be formulated in an abstract way, but it still requires some specific facts about central extensions.

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  • $\begingroup$ I think it can be done pretty abstractly. If $G$ is a perfect group, by the universal coefficient theorem we have $H^2(G, K) \cong \text{Hom}(H_2(G), K)$. Hence the central extension functor is representable, by $H_2(G)$, and has a universal element, namely the identity map $H_2(G) \to H_2(G)$, corresponding to the universal central extension by $H_2(G)$. Furthermore this argument is functorial in $G$. Am I missing something? $\endgroup$ – Qiaochu Yuan Mar 29 '19 at 20:06
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    $\begingroup$ @QiaochuYuan I would say that a proof is abstract if it applies to any category (with additional structure/properties). Your proof is not abstract in this sense since it uses many specific constructions on groups. We can formulate this problem in any category equipped with a notion of a "central extension" of an object $X$, which is just a map $Y \to X$ satisfying some property. Then a "universal central extension" is the initial such map. The problem is not provable in this generality. My last remark refers to the fact that it is true if "central extensions" are stable under pullbacks. $\endgroup$ – Valery Isaev Mar 29 '19 at 22:33

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