(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.

## 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 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.

## Extra stuff, structure, property

Revised on March 15, 2012 15:36:09 by Urs Schreiber (82.172.178.200)