# Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities.

What's the name of the category with parallel arrows collapsed into one?

Finally, consider a category of an appropriate "order type" that has undecomposable arrows, for example $(\mathbb{N},\leq)$. What's the name of the digraph (not a category then, of course) with undecomposable arrows only, e.g. $\mathbb{N}$ with the successor relation?

When I ask for names I mean the name of the construction, with respect to the original category (like in skeleton of).

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It is not true that the only isomorphisms in the skeleton of a category are the identities. – Qiaochu Yuan Feb 7 '12 at 8:30
@Qiaochu: Why not? – Hans Stricker Feb 7 '12 at 8:36
That would defeat the point of the construction of the skeleton (it's supposed to be a category equivalent to the original category, and getting rid of all the isomorphisms doesn't accomplish this), and it's easy to find counterexamples (take e.g. the skeleton of the category of finite sets). – Qiaochu Yuan Feb 7 '12 at 8:39
Sorry, but Wikipedia on skeletons says that no two distinct objects are isomorphic. What do I miss? – Hans Stricker Feb 7 '12 at 8:44
You miss that an object may have nontrivial automorphisms. – Qiaochu Yuan Feb 7 '12 at 8:47