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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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    OK, now I got it, thank you.2012-02-07

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I guess I would call the first construction the "preorderization" since it is the universal map from a category to a preorder. I guess I would call the second construction the Hasse diagram.