Categorical Invariants

Question

I apologize in advance if this question seems too vague.

In many topology courses, concepts like the fundamental group and homology groups are introduced as a means of distinguishing non-homeomorphic spaces - for instance, $\mathbb{T}^2$ and $S^2$. Similarly, things like the rank of an abelian group, and the Krull dimension of a ring are (relatively) interesting ways of taking an object and capturing useful information in a number. Of course, the most interesting invariants are those that are functorial in some way or another. In my earlier question, I asked for a reason that the category of topological spaces cannot be embedded in the category of groups. Now it turns out that one nice reason is that the category of groups is not cartesian closed, while the category of (compactly generated weakly Hausdorff) spaces is. I found this to be rather nice, as cartesian closedness is a rather global property. On the other hand, lots of categories are cartesian closed, and it would be nice if there were some kind of categorical invariant capable of distinguishing them. So my question is: Are there any nice categorical invariants? Preferably a categorical invariant would take the form of a functor $\mathfrak{C}\mathfrak{A}\mathfrak{T}\to\mathscr{C}$, where $\mathfrak{C}\mathfrak{A}\mathfrak{T}$ is the (meta) category of all categories and $\mathscr{C}$ is some nice category (abelian groups, sets, etc). But I'd be interested in any interesting way of capturing global information about a category.

Answer 1

Many invariants are given by topological invariants composed with the nerve functor from (small) categories to topological spaces. For example, you can talk of $\pi_0(C)$ of a category $C$ and it is exactly what you might guess, the set of connected components. But also higher homotopy groups are defined. This article by Tom Leinster studies the Euler characteristic of a finite category.

The center of a category provides a functor from $Cat$ (but only equivalences are allowed here) to the category of commutative monoids. This recovers, for example, $R$ from the category $R$-Mod. The automorphism class group of a category $C$ is the group of all equivalences $C \cong C$ up to isomorphism. This provides a functor $Cat \to Groups$.

An instance of decategorification is the functor $Cat \to Set$ which takes every (essentially small) category to the set of isomorphism classes of its objects.

There are certainly more invariants when you restrict to special categories. For example, Gabriel defined in his thesis the dimension of an abelian category, which has the property dim $R$-Mod = dim $R$ for noetherian $R$. But you can also define the global projective / injective dimension of an abelian category. Monoidal categories have an underlying monoid, which consists of the endomorphisms of the unit object. Every finite groupoid has a cardinality.

Answer 2

I think the existence of limits for different types of diagrams serve as categorical invariants. Higher category theory I think is the search of more fine grained invariants but I don't know if the practitioners and developers of that program think of it that way. When comparing two categories, if all the same diagrams have limits then for all intents and purposes those categories are equivalent as far as category theory is concerned because there will be no way to distinguish between them categorically.

The Yoneda Lemma (https://ncatlab.org/nlab/show/Yoneda+lemma) says something to this effect already. If two objects in a category are isomorphic as presheaves then their objects were already isomorphic to begin with (isomorphism of representables). In other words, if two objects of a category have the same structure for their incoming arrows then they are isomorphic (and presumably there is the dual version with outgoing arrows and their structure).