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.


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

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.

  • $\begingroup$ Can you prove that "every example comes from a 2-category"? $\endgroup$ – porton Sep 27 '13 at 20:58
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    $\begingroup$ @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) $\endgroup$ – Excluded and Offended Sep 27 '13 at 21:01

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