(0,1)-category

(0,1)-topos

# Contents

## Idea

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. This view is an instance of negative thinking.

## Definition

###### Definition

An (n,1)-category is an (∞,1)-category such that every hom ∞-groupoid is (n-1)-truncated.

Notice that

###### Proposition

An $(0,1)$-category is equivalently a poset.

###### Proof

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 setoid (up to isomorphism).