Naturally occurring examples of badly behaved categories

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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 $BS^1$.

• ...

Answer 5

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).

Answer 6

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.

Answer 7

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 $\mathbf{Top}$.

Answer 8

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.

Answer 9

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 $V$?