(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
meet, logical conjunction, and
join, logical disjunction, or
lattice of subobjects
complete lattice, algebraic lattice
distributive lattice, completely distributive lattice, canonical extension
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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.
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.
An (0,1)(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)(0,1)-category with the structure of a site is a (0,1)-site: a posite.
A (0,1)(0,1)-category with the structure of a topos is a (0,1)-topos: a Heyting algebra.
A (0,1)(0,1)-category with the structure of a Grothendieck topos is a Grothendieck (0,1)-topos: a frame or locale.