Following the general concept of $(n,r)$-category, a $(0,1)$-category is (up to equivalence) a poset or (up to isomorphism) a proset. We may also call this a $1$-poset.
An (n,1)-category is an (∞,1)-category such that every hom ∞-groupoid is (n-1)-truncated.
Notice that
a 0-truncated ∞-groupoid is equivalently a set;
a (-1)-truncated ∞-groupoid is either contractible or empty.
An $(0,1)$-category is equivalently a poset.
We may without restriction assume that every hom-$\infty$-groupoid is in fact a set on the nose. 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.
0-category, (0,1)-category