# Equivalent Categories

I have never really done any category theory before, and am looking to use it to somehow classify some of my work. In particular I am trying to determine wheter two given categories, say $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are equivalent or not. Are there any efficient ways of doing this?

Obviously if I define a functor $F:\mathcal{C}_{1}\rightarrow\mathcal{C}_{2}$, then I can check whether this is an equivalence of categories (either by using the definition, or by checking whether $F$ is full, faithful and every element of $\mathcal{C}_{2}$ is the isomorphic image of an element of $\mathcal{C}_{1}$), but this relies on defining the functor $F$. Are there any methods of determining whether two categories are equivalent/inequivalent without defining any functors?

Edit/Motivation: The case that I am dealing with involves trying to see if given two simplicial complexes $\Delta_{1}\subseteq\Delta_{2}$ with an associated $G$-action for some group $G$, the category of presheaves defined on $\Delta_{1}$ and the category of presheaves defined on $\Delta_{2}$ are equivalent (see Ronan and Smith's paper for details about presheaves). The motivation stems from the fact that there is a notion of simplicial complexes being $G$-homotopy equivalent, and I am trying to see whether I can generalise this to presheaves.

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Well, if your categories are finite and have an Euler characterisic $\chi$ (as defined by Tom Leinster in The Euler characteristic of a category), then equivalence yields $\chi(\mathcal{C}_1) = \chi(\mathcal{C}_2)$. Note that there are Euler characteristics defined for larger classes of categories, which might yield similar results. –  roman Nov 9 '12 at 9:04
Can you tell us what are your categories? If they are indeed equivalent, an equivalence functor between them should not be hard to find. –  Berci Nov 9 '12 at 11:16
@Berci I've added this to the original question. I had originally left it out to try not to complicate things, but I appreciate that this was an error in hindsight. –  David Ward Nov 9 '12 at 11:52

There are some indirect methods of checking if two categories are equivalent without producing an explicit equivalence. For instance, given two categories $C,D$, the presheaf categories $Hom(C^{op},Set)$ and $Hom(D^{op},Set)$ are equivalent if, and only if, the Karoubi envelopes of $C$ and $D$ are equivalent. So, if the categories you are interested in are presheaf categories you can try to find the indexing categories, compute their Karoubi envelopes, and (as may happen) hope that these will clearly be equivalent. All of these notions are well-known so a quick search for the term Karoubi envelope will get you many relevant hits (e.g., http://ncatlab.org/nlab/show/Karoubi+envelope).

This situation extends to base categories other than $Set$ but this is somewhat less known and less worked out.

Related to that, if your categories arise either as algebras for monads or as algebras for operads you can try to show that the monads (resp. operads) must yield equivalent categories of algebras. There is not a lot of machinery to do that but the techniques of the Karoubi envelope business can be extended to some extent to monads and operads.

Still in the same realm but more generally still, if your categories arise as models for Lawvere theories then you can try to apply the general techniques of the theory of Lawvere theories to your categories.

If your categories admit a nice representation in terms of generators and relations you can of course try to show that the generators of one category yield the generators of the other, and vice versa, and that the relations essentially agree.

Two categories can be shown to be equivalent by showing the categories arise as localizations of two Quillen equivalent categories.

If you can show that each of your categories is a universal solution to the same problem in $Cat$ then they must be isomorphic. Similarly, if you can show that each of your categories is a weak universal solution to the same problem in $Cat$ as a 2-category then they must be equivalent.

That's all I can think of right now. It would greatly help to know what are the categories you are interested in.

Addendum after edit to original question: Certainly try to see if the opposites of your indexing categories are Karoubi equivalent then. Presheaf categories are equivalent iff the indexing categories are Karoubi equivalent, so you completely transfer the problem to the indexing categories. Computing the Karoubi envelope is typically not that difficult.

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Thanks for a very detailed response. As I've now mentioned in the question, the categories in question are presheaf categories, but not presheaves defined on topological spaces. –  David Ward Nov 9 '12 at 11:57
The setting for the Karoubi envelope is presheaf with base category Set and an arbitrary category as exponent. So you don't need to worry about about presheaves defined on topological spaces at all. –  Ittay Weiss Nov 9 '12 at 11:59

To determine that two categories are not equivalent, a sensible strategy is to see if you can find a categorical property one satisfies that the other doesn't. This is the same strategy you'd use to determine whether two groups are isomorphic or not (for example an abelian and a nonabelian group are not isomorphic), whether two topological spaces are homotopy equivalent or not (for example a connected and a disconnected space are not homotopy equivalent), etc.

What kind of categorical properties depends strongly on what kind of categories you're looking at, but a general thing to look for when your categories look like "categories of mathematical objects" is the existence of various kinds of limits and colimits.

If you strongly suspect that your categories are equivalent, it seems to me like the cleanest approach would be to construct an equivalence between them, although again that depends on what kind of categories you're looking at.

Note that this problem contains, say, the isomorphism problem for monoids as a subproblem. So you really should specify what kind of categories you're looking at.

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Isomorphisms between objects lead to an equivalence relation on objects. The skeleton of a category $\mathcal{C}$ is morally the quotient of $\mathcal{C}$ by this relation. Two categories being equivalent iff they have isomorphic skeletons (chapter IV.4. of Saunders Mac Lane's book), you may try to check if $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have isomorphic skeletons.