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I am interested in categories, whose objects are morphisms (in an other category).

I want to see examples of such categories.

I have examples of this in my research: Such as the categories of continuous functions between endo-funcoids (or endo-reloids). Endo-funcoids (and endo-reloids) are themselves morphisms in respective categories, but they are connected with "continuous function" morphisms.

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closed as unclear what you're asking by Bruno Joyal, Dennis Gulko, Jared, azimut, William Sep 30 '13 at 3:12

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Sorry but... what is the question being asked? –  Bruno Joyal Sep 27 '13 at 21:00
    
I added the question: "I want to see examples of such categories." –  porton Sep 27 '13 at 21:05

1 Answer 1

Every example comes from a 2-category. All natural examples probably come from natural examples of 2-categories, such as $\mathbf{Cat}$, the 2-category of all small categories (and functors and natural transformations). A more interesting example might be the 2-category of rings, bimodules, and bimodule homomorphisms. I.e. $\hom(R,S)$ is the category of all $(R,S)$-bimodules, and composition of bimodules is given by the tensor product), so the objects of the category $R-\mathbf{Mod}-S$ are the (1-)morphisms in the category I described.

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Can you prove that "every example comes from a 2-category"? –  porton Sep 27 '13 at 20:58
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@porton: In fact, every category is a hom-category in a 2-category. It's pretty much the same as the proof that every monoid is a hom-set in a category. Actually it's more like the more trivial proof that every set is a hom-set in a category: just define a 2-category with objects $X$ and $Y$, and $\hom(X,Y)$ to be whatever category you like. (and all other hom-categories empty or trivial as appropriate) –  Hurkyl Sep 27 '13 at 21:01

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