Equivalence versus Isomorphism of Categories [duplicate]

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Can skeleta simplify category theory?

When one first learns about categories, one confusing fact is that the nicest notion of two categories being the same is not isomorphism - it is rather equivalence. But it's occurred to me that when two categories are equivalent but not isomorphic, it seems that the only reason why this is so is because one category has multiple copies of the same object.

Could we make a notion of some type of category in which no two different objects are isomorphic? Let's call such a category succinct, since I don't know any other term for it (I hope succinct doesn't have another meaning). Then would it be true that any two succinct categories which are equivalent are also isomorphic? And if we have any category, we can form a new succinct category by identifying isomorphic objects. Then the question arises, why don't we just always work with succinct categories? Why, if we're interested in the category of sets, don't we say that any two sets with the same cardinality are not only isomorphic, but are actually the same object in the category of sets?

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marked as duplicate by S. Carnahan♦Jun 30 '10 at 12:23

This seems to be more or less a duplicate of mathoverflow.net/questions/11674/… – Reid Barton Jun 30 '10 at 11:00
The problem is that categories, as they appear in nature, are not succint/skeletal. – Mariano Suárez-Alvarez Jun 30 '10 at 11:26
I've closed this question, since the answers to the earlier question render it somewhat redundant. – S. Carnahan Jun 30 '10 at 12:26
Sorry, you are right that it is the same. I searched but didn't find it, probably in part because I didn't know the term "skeletal." – David Corwin Jun 30 '10 at 12:29
As a side remark, I don't see why having multiple copies of isomorphic objects should be a defect. They naturally arise in many situations. Think e.g. to tangent spaces of a manifold. Also, non trivial categorical constructions may be done on families of isomorphic objects (e.g. tangent bundles, to stay in theme). – Pietro Majer Jun 30 '10 at 16:59

These categories are known as skeletons, or skeletal categories. Unfortunately, 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 way, try to define the arrows). What one can do is to use the axiom of choice to choose a single object in each isomorphism class, and thus define a skeletal full subcategory of $\mathcal C$ (the skeleton of $\mathcal C$).

Identifying isomorphic objects is often an extremely bad idea. The question whether two sets are equal is very different from that of whether they have the same cardinality.

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And choosing which wasteful objects to discard is generally a waste of time. – Tom Goodwillie Jun 30 '10 at 11:18

A groupoid is a category where every morphism is an isomorphism. Taking its skeleton throws the baby with the water, as you are left with a category where there are no morphisms between different objects, i.e., a bunch of groups or a diagonal groupoid.

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Not true. Let this groupoid be a group. Then it's already skeletal. – David Carchedi Jun 30 '10 at 12:08
LOL, you are right, I will edit in a sec – Bugs Bunny Jun 30 '10 at 12:09
(My objection is no longer valid now that the answer has been edited). – David Carchedi Jun 30 '10 at 12:17