nLab
(0,1)-category

Contents

Idea

Following the general concept of (n,r)(n,r)-category, a (0,1)(0,1)-category is (up to equivalence) a poset or (up to isomorphism) a proset. We may also call this a 11-poset. This view is an instance of negative thinking.

Definition

Definition

An (n,1)-category is an (∞,1)-category such that every hom ∞-groupoid is (n-1)-truncated.

Notice that

Proposition

An (0,1)(0,1)-category is equivalently a poset.

Proof

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