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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.

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With regard to the answer that you accepted in the earlier question (and with all due respect to the answerer), I don't think that answer quite hit the mark. (I won't say that it was 'wrong', just that much more perspicuous answers were given). Thus, I think that the "nice" reason you cite above isn't exactly to the point. –  Todd Trimble May 6 '11 at 20:49
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Indeed. I don't see in what way cartesian closedness helps in answering that question. –  Mariano Suárez-Alvarez May 6 '11 at 20:56
    
It is not clear to me what you actually want. You say 'it would be nice if there were some kind of categorical invariant capable of distinguishing them'. Are you asking for categorical invariants that distinguish amongst these categories or between these categories and others? As some of the structures you mention are sort of 'algebraic' there are obstruction theoretic methods to give invariants that the 'obstruction' to a given category having a given structure. Is that what you are looking for? –  Tim Porter May 7 '11 at 6:18

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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.

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