nLab (0,1)-category

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Idea

Following the general concept of ( n , r ) (n,r) -category, a (0,1)(0,1)-category is a category whose hom-objects are (-1)-groupoids, hence which for every pair of objects a,ba,b have either no morphism aba \to b or an essentially unique one.

More in detail, recall that:

Therefore:

Remark

(relation between preorders and (0,1)-categories)
An (0,1)(0,1)-category is equivalently a proset (hence a poset).

We may without restriction assume that every hom-\infty-groupoid is just a set. Then since this is (-1)-truncated it is either empty or the singleton. So there is at most one morphism from any object to any other.

Extra stuff, structure, property

Last revised on July 22, 2022 at 12:05:30. See the history of this page for a list of all contributions to it.