Following the general concept of $(n,r)$-category, a $(0,1)$-category is a category whose hom-objects are (-1)-groupoids, hence which for every pair of objects $a,b$ have either no morphism $a \to b$ or an essentially unique one.
More in detail, recall that:
an (n,1)-category is an (∞,1)-category such that every hom ∞-groupoid is (n-1)-truncated.
a 0-truncated ∞-groupoid is equivalently a set;
a (-1)-truncated ∞-groupoid is either contractible or empty.
Therefore:
(relation between preorders and (0,1)-categories)
An $(0,1)$-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.
A $(0,1)$-category with the structure of a site is a (0,1)-site: a posite.
A $(0,1)$-category with the structure of a topos is a (0,1)-topos: a Heyting algebra.
A $(0,1)$-category with the structure of a Grothendieck topos is a Grothendieck (0,1)-topos: a frame or locale.
A $(0,1)$-category which is also a groupoid (that is, every morphism is an isomorphism) is a $(0,0)$-category (which may think of as either a $0$-category or as a $0$-groupoid), which is the same as a set (up to equivalence) or a symmetric proset (up to isomorphism).
(0,1)-category
Last revised on September 22, 2022 at 18:54:47. See the history of this page for a list of all contributions to it.