hom-category in nLab

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Contents

Idea

Recall that in a category CC, each pair of objects xx and yy determines a set C(x,y)C(x,y), the hom-set of xx and yy. In a 22-category BB, each pair of objects determines a category B(x,y)B(x,y). This category is the hom-category of xx and yy.

The objects in the hom-category C(x,y)C(x,y) are the 1-morphisms in CC from xx to yy, while the morphisms in the hom-category C(x,y)C(x,y) are the 2-morphisms of CC that are horizontally between xx and yy.

As a 22-category is enriched over Cat, a hom-category is a special case of a hom-object. But the hom-category makes sense also for the weakly enriched concept of bicategory.

• hom-object

• hom-set

• hom-category

• hom-space, derived hom-space