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.