Thin categories

category theory

# Thin categories

## Definition

A thin category or posetal category is a category in which, given a pair of objects $x$ and $y$ and any two morphisms $f, g \,\colon\, x \to y$, the morphisms $f$ and $g$ are equal:

$x \underoverset{\quad g \quad}{f}{\rightrightarrows} y \;\;\;\implies\;\;\; f = g$

Another synonym is (0,1)-category.

## Properties

### Relation to order theory

Up to isomorphism, a thin category is a preordered set (“proset”). Up to equivalence, a thin category is the same thing as a partially ordered set (“poset”).

For more on this see at relation between preorders and (0,1)-categories.

## Examples

A poset is a thin category. In particular so are (semi)lattices, Heyting algebras and frames.

Last revised on September 22, 2022 at 18:42:24. See the history of this page for a list of all contributions to it.