# A category's quotient category of isomorphism types

In an answer to another question on skeletal categories, it is said:

"[...] given a category $\mathcal C$, one can not define in general the structure of a category on the class of isomorphic classes of objects of $\mathcal C$ (to see why, try to define the arrows)."

On the other hand, in a paper entitled "A category's quotient category of isomorphism types versus its skeleton", the author seemingly constructs the quotient category of isomorphism types of an arbitrary category $\mathcal C$.

Question 1: How does this fit together? Is one of the authors wrong, or do they talk about different things?

A specific example where the quotient category of isomorphism types of objects of a category seems to be definable and is even isomorphic to its skeleton is FinSet with an arrow from $[A]$ to $[B]$ in the quotient category when there is an monomorphism from $A$ to $B$, i.e. when $A$ is smaller than $B$.

Question 2: Is this one of the exceptions the first author might have in mind when he says "not in general"? Can these exceptions be characterized?

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The first author has some implicit assumptions about what properties the category on the isomorphism classes should have; one could define a category whose objects are the isomorphism classes of objects of $\mathcal{C}$, by giving each object its identity morphism and then stopping. The result would not have a particularly interesting or useful relationship to $\mathcal{C}$ though. So one possibility is that whatever conditions the first author thinks the resulting category should meet are not met by the category defined by the second author. – Matthew Pressland Sep 2 '13 at 15:09
It is far from obvious whether or not the category $[\mathcal{C}]$ constructed in the cited paper is skeletal, and it is explicitly stated at the beginning of the paper that $[\mathcal{C}]$ is not even equivalent to $\mathcal{C}$. – Zhen Lin Sep 2 '13 at 15:38
@Matt: You are most certainly right. The implicit assumption the first author must have had in mind is: "a category on the class of isomorphism classes of objects of $\mathcal C$ that is isomorphic to the skeleton of $\mathcal C$". This, I have to admit, makes my question 1 trivial. My question 2 remains "valid", I hope. – Hans Stricker Sep 3 '13 at 15:02

(1): They’re talking about very different things.

The question you link originally asks whether we can use a quotient construction to replace any category by a skeletal category. Since it’s being proposed to justify always working with skeletal categories, presumably the quotient category is intended to be equivalent to the original one; and this (as per Angelo’s answer there) is not possible without the axiom of choice. (That is: one can show that if every category has an equivalent skeletal quotient, then AC holds.)

On the other hand, there are several ways that you could define a category structure on the class of iso classes of objects of a category — they just won’t usually be skeletal or equivalent to the original category. Trivially, you could use the discrete or codiscrete category structure.

Less trivially, the Fritsch 1985 paper you link considers the pushout, over the discrete category on $\mathrm{ob}(\mathbf{C})$, of $\mathbf{C}$ and the discrete category of iso classes of objects of $\mathbf{C}$. This is perhaps more clearly described by its universal property: functors out of it correspond to functors out of $\mathbf{C}$ sending isomorphic objects to equal objects. Taking $\mathcal{I}$ to be the “walking isomorphism”, the codiscrete category on two objects, one finds that its Fritsch quotient is the group $\mathbb{Z}$ viewed as a one-object category. (To see this, consider the universal properties.) This is neither skeletal, nor equivalent to $\mathcal{I}$ itself.

The Fritsch quotient is, while intriguing, a bit unnatural for many purposes: it doesn’t for instance respect equivalence of categories. The example above again shows this: $\mathcal{I}$ is equivalent to the terminal category $1$, whose Fritsch quotient is just itself, which is certainly not equivalent to $\mathbb{Z}$!

(2): Can the exceptions be classified?

(“The exceptions” here meaning, presumably, “categories where you can put a structure on the class of iso classes, such that the quotient map on objects $q : \mathrm{ob} \mathbf{C} \to \pi_0(\mathbf{C})$ becomes the object map of an equivalence of categories”.)

I don’t know any way to classify these in general, and the equivalence with AC vaguely suggests to me that one shouldn’t expect to (but that’s a very vague gut feeling; I’m not sure how meaningful it is). However, one very important class of examples is categories of rigid objects — categories where if two objects are isomorphic, they are uniquely so; or equivalently, where objects have no non-trivial automorphisms. I won’t give the details of how to do it for rigid categories; it’s a good exercise. But ideas related to this example arise naturally in many places — for instance, the construction of moduli spaces. If some kind of object — widgets, say — is always rigid, then one can usually construct a nice “space of widgets”. However, if they have non-trivial automorphisms/symmetries, the “space of widgets” will need to remember this, and so have to be some more complicated object — a stack of some sort. There are some very nice references on this which are eluding me right now; I’m trying to remember/track them down, and will give them when/if I do.

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