nLab (0,1)-category

Idea

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:

Remark

(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.

Extra stuff, structure, property

• 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).

Related concepts

• relation between preorders and (0,1)-categories

• (-2)-groupoid

• (-1)-groupoid

• 0-groupoid

• 1-groupoid

• 2-groupoid

• 3-groupoid

• 4-groupoid

• n-groupoid

• ∞-groupoid

• (0,0)-category

• (1,0)-category

• (n,0)-category

• (∞,0)-category

• 0-category

• 1-category

• 2-category

• 3-category

• 4-category

• n-category

• ∞-category

• (0,1)-category

• (1,1)-category

• (2,1)-category

• (n,1)-category

• (∞,1)-category

• (1,2)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category