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If $\mathcal{D}$ is a category, one may define a category with the same objects but having morphisms precisely all possible families of parallel morphisms of $\mathcal{D}$, in the obvious way.
It appears practically certain that this is a standard construction with a usual technical name; would you please tell what it is?
In short: it is possible to define compositions of one hom-set with another hom-set, and thus get categories having the hom-sets themselves as the morphisms; is there a usual technical term for this?
(In a sense, it is the straightforward generalization of what are called “Minkowski sums” or “sum-sets” in the special situation of commutative monoids, but this question concerns the general construction for any category. It can probably also made into an endofunctor of the category of all small categories.)
Perhaps needlessly, details:
Suppose $\mathcal{D}$ is a category.
Let $\mathcal{D}^+$ denote the category which has
for arbitrary objects $O,O'$ of $\mathcal{D}^+$ we define the hom-‘set’ to be
$\mathcal{D}^+(O,O') :=$ class of all class-indexed families $\mathcal{M}=\{ h_i\colon i\in I\}$,
with $I$ a class and each $h_i\in\mathcal{D}(O,O')$.
(We note that each morphism is a class of parallel morphisms.)
Composition is defined in the obvious way: if $O,O',O''$ are objects,
and if $\{ h_i\colon i\in I\}\in\mathcal{D}^+(O,O')$ and $\{ h_i'\colon i\in I'\}\in\mathcal{D}^+(O',O'')$, then
$\{ h_i'\colon i\in I'\}\circ\{ h_j\colon j\in I\} := \{ h_i'\circ h_j\colon (i,j)\in I'\times I\}$
(We note that $\mathrm{dom}(h_i')=\mathrm{cod}(h_j)$ for all indices, so all the compositions are defined.)
(We note that the identity morphism at an object of $\mathcal{D}^+$ is the singleton-indexed class containing only the singleton-morphism of $\mathcal{D}$ at that object.)
Isn’t this the change of base (in the sense of enriched categories) of the power set functor (which is a monoidal monad wrt the cartesian monoidal structure on Set)?
Oh yes, good point Ulrik.
Thanks for the hint. Will look into this.
To be fair, Ulrik’s suggestion when applied to a category $C$ produces a category $D$ where the elements of hom-sets $D(x, y)$ are subsets of $C(x, y)$. So if the indexing is important to you, you’d need a slightly different construction.
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