Following the general concept of -category, a -category is a category whose hom-objects are (-1)-groupoids, hence which for every pair of objects have either no morphism 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 -category is equivalently a proset (hence a poset).
We may without restriction assume that every hom--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 -category with the structure of a site is a (0,1)-site: a posite.
A -category with the structure of a topos is a (0,1)-topos: a Heyting algebra.
A -category with the structure of a Grothendieck topos is a Grothendieck (0,1)-topos: a frame or locale.
A -category which is also a groupoid (that is, every morphism is an isomorphism) is a -category (which may think of as either a -category or as a -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.