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What are some examples of naturally occurring badly behaved (possibly higher) categories?

When working with a specific category like ${\bf Set}$ or ${\bf Cat}$, we usually understand/explain them by lauding structural properties they posess -- ${\bf Set}$ is an autological topos, ${\bf Cat}$ is Cartesian closed, etc.

What structures can be arranged naturally/canonically into (possibly higher) categories, despite the resulting categories having few/none of the structural properties we would usually like to have in order to carry out category-theoretic-type proofs?

A natural example is ${\bf Field}$, the category of fields and field homomorphisms, since it has no terminal object, no initial object, no finite products, is not algebraic and is not presentable. (It is, however, accessible with a multi-initial object given by the set of prime fields).

I suspect that it gets worse than this, but I can't think of anything further off the top of my head. Any examples are appreciated.

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    $\begingroup$ This conversation has been moved to chat. I think they were getting long enough; this way any further comments will not be buried. $\endgroup$ – Tim Campion Apr 16 at 0:32
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The discussion in the comments kind of went off the rails, but the point I meant to make by linking to dichotomy between nice objects and nice categories is that you can get lots of examples by starting with a nice category and restricting to a full subcategory by imposing some condition on the objects that isn't preserved by categorical operations. Fields are one example; manifolds are another. So are CW-complexes and Kan complexes, or more generally the category of cofibrant and/or fibrant objects in any model category.

There's a certain genericity to this class of examples, of course, since any (small) category can be embedded in its presheaf category, which is almost maximally nice.

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    $\begingroup$ Yes, it seems like one could come up with many explicit examples of categories with "undesirable" properties by taking some "modern" category that was introduced to have nice categorical properties, and looking at its predecessor. Actually, the category doesn't even have to be that "modern." For example, CW complexes behave better than simplicial complexes. $\endgroup$ – Timothy Chow Apr 15 at 13:33
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The category $\mathbf{Sch}$ of schemes is an example of an ill-behaved category (and "nice" schemes, such as finite type over a field, are not better). There are coproducts, but no coequalizers. There are finite limits, but no infinite products. It is not clear how epimorphisms of schemes look like (the monomorphisms are better understood).

By contrast, the categories $\mathbf{RS}$ and $\mathbf{LRS}$ of (locally) ringed spaces are complete and cocomplete, and the epimorphisms are what you expect.

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  • $\begingroup$ Worse: how about the category of separated schemes (the old "schemas" before "preshemas" became the norm)? $\endgroup$ – David Roberts Apr 16 at 0:35
  • $\begingroup$ I don't see why the category of separated schemes is worse with respect to the universal constructions I mentioned. Coproducts and finite limits still exist. $\endgroup$ – Martin Brandenburg Apr 16 at 8:02
  • $\begingroup$ I was thinking that maybe even fewer colimits exist... One could throw in more adjectives $\endgroup$ – David Roberts Apr 16 at 10:03
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The category $Rel$ of sets and relations between them fails to have finite (co)limits.

So does the $(2,1)$-category $Span$ of sets and spans between them (the pushout of the forward map $2 \to 1$ against itself fails to exist).

These are examples illustrating the point that self-dual categories often fail to have nice categorical properties (Hilbert Spaces, as mentioned by Andre, would be another).

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  • $\begingroup$ I should add one precisification of this point -- the opposite of a locally presentable category is never locally presentable unless the category is a preorder. So a self-dual category which is accessible can never be complete / cocomplete. This statement encompasses all 3 examples mentioned in this answer -- $Rel, Span, Hilb$. $\endgroup$ – Tim Campion Apr 21 at 18:05
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I'm not sure if this really counts, but sometimes an $n$-category is sufficiently strict to have an underlying $k$-category for some $k<n$, and in this case that $k$-category will generally be poorly behaved. For instance, bicategories and pseudofunctors happen to form a 1-category, but it's not a very well-behaved 1-category: it's not complete or cocomplete, although it does have products. But this 1-category underlies the (weak) 3-category of bicategories, which is perfectly well-behaved as a 3-category. More generally, for any 2-monad $T$ there is a 1-category of (strict or pseudo) $T$-algebras and pseudo $T$-morphisms, which is not well-behaved as a 1-category, although as a (weak) 2-category it is complete and cocomplete (as long as $T$ is nice, e.g. accessible on a locally presentable 2-category).

Relatedly, the homotopy $k$-category of an $n$-category for $k<n$ is not generally very well-behaved. For instance, the 1-category whose objects are categories and whose morphisms are natural isomorphism classes of functors, or the 1-category whose objects are topological spaces and whose morphisms are homotopy classes of continuous maps. Of course, once again the higher category is more "natural/canonical".

Perhaps a better example is the 1-category of bicategories and lax functors, which is quite poorly behaved, but does not underlie any 3-category. It does underlie a 2-category whose 2-cells are icons, but even that 2-category is not particularly well-behaved; it has some 2-limits but not all. More generally, algebras and lax morphisms for any 2-monad are important (e.g. lax monoidal functors), but are not a very well-behaved 1-category or even 2-category from an abstract point of view.

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Here's a somewhat general form of obstruction: many nice categorical properties imply that a category's classifying space is contractible. For example any category with any of the following structures has a contractible classifying space:

  • an initial or terminal object

  • binary products or coproducts

  • even just any functorial way to embed two objects $X,Y$ into a common object $X \to F(X,Y) \leftarrow Y$

  • ...

Here's a note by Omar Antolin-Camarena exploring some of these properties.

So if you have a category whose classifying space is not contractible, then chances are it's not very "nice" from a categorical perspective. For example:

  • The category of fields has a disconnected classifying space. So does the category of algebraically closed fields.

  • The category of algebraically closed fields of characteristic $p$ has a classifying space $BGal(k)$ where $k$ is the algebraic closure of $\mathbb F_p$ if $p \neq 0$ and $k = \overline{\mathbb Q}$ if $p=0$.

  • Connes' cyclic category $\Lambda$ has classifying space $BS^1 = \mathbb C\mathbb P^\infty$.

  • It follows that $Ind(\Lambda)$, a sort of "category of cyclic sets" also has classifying space $BS^1$.

  • ...

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The category of Hilbert spaces and bounded linear maps is pretty badly behaved:
No limits, no colimits, not even (co)products.

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    $\begingroup$ Banach spaces and bounded linear maps is similarly bad (I think one has $\aleph_1$-filtered colimits at least.) Of course, both examples are improved by restricting to maps of norm $\leq 1$. $\endgroup$ – Tim Campion Apr 18 at 20:51
  • $\begingroup$ @TimCampion Restricting to maps of norm $\le 1$ is a good idea for Banach spaces, but doesn't do much good when working with Hilbert spaces (binary coproducts exist in the former case, but not in the latter case). $\endgroup$ – André Henriques Apr 19 at 20:47
  • $\begingroup$ We have finite limits in $\mathbf{Hilb}$, right? And $\mathbf{Hilb}$ has a nice symmetric monoidal structure (Hilbert space tensor product). $\endgroup$ – Martin Brandenburg Apr 20 at 21:04
  • $\begingroup$ @MartinBrandenburg Yes and yes. Finite limits and colimits exist in Hilb. (Note however that the functor which sends a Hilbert space to its underlying vector does not preserve coequalisers.) The Hilbert space tensor product is very nice, but I don't know how to characterise is in category-theory language. $\endgroup$ – André Henriques Apr 21 at 14:17
  • $\begingroup$ This is interesting. So Hilb has finite colimits and $\aleph_1$-filtered colimits, which leaves a sort of "gap in its colimit spectrum", where it specifically lacks filtered colimits which are not $\aleph_1$-filtered. I wonder if other "categories of objects with metric-like completeness conditions" follow this pattern. $\endgroup$ – Tim Campion Apr 21 at 18:04
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The category $\mathbf{Met}$ of metric spaces and metric maps is another example. There are finite limits, but no infinite products. Countable products exist at least when we use continuous maps as the morphisms instead. $\mathbf{Met}$ has no binary coproducts, and this can be seen as the starting point of the Gromov-Hausdorff distance, where we consider all possible metrics on a disjoint union. Coequalizers do not exist either. It is worth pointing out that the injective objects of $\mathbf{Met}$ have gathered some interest.

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  • $\begingroup$ As with Andre's example, it's worth noting that if you restrict to the subcategory $Met_{\leq 1} \subset Met$ of distance non-increasing (i.e. 1-Lipschitz, or contractive maps), then you obtain a complete and cocomplete category which is generally well-behaved. $\endgroup$ – Tim Campion Apr 20 at 22:51
  • $\begingroup$ @TimCampion My answer precisely refers to that category. non-increasing = metric (see the link). So I do not think that your statements are correct. $\endgroup$ – Martin Brandenburg Apr 20 at 23:24
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    $\begingroup$ Oh I see. You disallow infinite distances. Then my point is that if you "compactify" the category, by allowing distances to be infinite, you do get a complete and cocomplete category. $\endgroup$ – Tim Campion Apr 21 at 0:30
  • $\begingroup$ @TimCampion Good remark! So in a disjoint union for points $x,y$ in different spaces we let $d(x,y) = \infty$. (Which is just the general definition $\mathrm{Hom}(x,y)=0$ in a coproduct of enriched categories.) $\endgroup$ – Martin Brandenburg Apr 21 at 9:17
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Someone mentioned that the category $\mathbf{Man}$ of (topological) manifolds is really badly behaved, but $\mathbf{Top}$ (the category of topological spaces) isn't much better eithter. It's both complete and cocomplete, but it is not cartesian closed (and thus not a topos).

There are two solutions to this problem: either you generalize what is meant by a space, or you restrict your attention to a collection of "nice" spaces. nLab has a list of "nice" categories of "spaces" which behave better than $\mathbf{Top}$.

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    $\begingroup$ I would say that $\mathbf{Top}$ behaves much better than $\mathbf{Man}$. Notice that $\mathbf{Man}$ has finite products, but fiber products exist only in very good cases (cf. regular value theorem). Countable colimits exist, but larger colimits do not exist when you insist that manifolds should be secound-countable (so better choose paracompactness), and coequalizers usually don't exist either. For topological spaces, we just need to switch to one of the well-behaved substitutes (like compactly generated Hausdorff spaces) to get a bicomplete ccc. This is not possible for manifolds. $\endgroup$ – Martin Brandenburg Apr 16 at 12:18
  • $\begingroup$ I wanted to say "countable coproducts" and "larger coproducts" - not general colimits of course. $\endgroup$ – Martin Brandenburg Apr 20 at 21:17
  • $\begingroup$ @MartinBrandenburg I guess it really depends on what counts as "well-behaved substitutes". For manifolds you can also pass to smooth spaces, although this is arguably a bigger stretch than passing to compactly-genenrated Hausdorff spaces. $\endgroup$ – xuq01 Apr 21 at 4:23
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Wouldn't pretty much any everyday mathematical object provide an example of a poorly-behaved category? For example, the category of all bases in some finite-dimensional vector space $V$?

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    $\begingroup$ What are the morphisms between bases in a fixed $V$? This seems to me to be something like the groupoid with one object and automorphism group $GL(V)$, but you may have had something else in mind. $\endgroup$ – David Roberts Apr 15 at 9:45
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    $\begingroup$ Well, any set can be regarded as a discrete category, which will then generally not have any nice categorical properties! $\endgroup$ – Mike Shulman Apr 15 at 13:41
  • $\begingroup$ @MikeShulman Besides accessibility of course (if the set is small) (-; $\endgroup$ – Alexander Campbell Apr 15 at 23:28

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