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    • CommentRowNumber1.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 7th 2017
    • (edited Jun 10th 2017)

    If 𝒟\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 𝒟\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 𝒟\mathcal{D} is a category.

    Let 𝒟 +\mathcal{D}^+ denote the category which has

    • precisely the same objects as 𝒟\mathcal{D}
    • for arbitrary objects O,OO,O' of 𝒟 +\mathcal{D}^+ we define the hom-‘set’ to be

      𝒟 +(O,O):=\mathcal{D}^+(O,O') := class of all class-indexed families ={h i:iI}\mathcal{M}=\{ h_i\colon i\in I\},

    with II a class and each h i𝒟(O,O)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,OO,O',O'' are objects,

    and if {h i:iI}𝒟 +(O,O)\{ h_i\colon i\in I\}\in\mathcal{D}^+(O,O') and {h i:iI}𝒟 +(O,O)\{ h_i'\colon i\in I'\}\in\mathcal{D}^+(O',O'') , then

    {h i:iI}{h j:jI}:={h ih j:(i,j)I×I}\{ 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 dom(h i)=cod(h j)\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 𝒟 +\mathcal{D}^+ is the singleton-indexed class containing only the singleton-morphism of 𝒟\mathcal{D} at that object.)

    • CommentRowNumber2.
    • CommentAuthorUlrik
    • CommentTimeJun 7th 2017
    • (edited Jun 7th 2017)

    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)?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 7th 2017

    Oh yes, good point Ulrik.

    • CommentRowNumber4.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 10th 2017

    Thanks for the hint. Will look into this.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 10th 2017

    To be fair, Ulrik’s suggestion when applied to a category CC produces a category DD where the elements of hom-sets D(x,y)D(x, y) are subsets of C(x,y)C(x, y). So if the indexing is important to you, you’d need a slightly different construction.

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