Thin categories

category theory

# Thin categories

## Definition

A thin category is a category in which, given any two objects $x$ and $y$ and any two morphisms $f$ and $g$ from $x$ to $y$, the morphisms $f$ and $g$ are equal:

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

## Properties

Up to isomorphism, a thin category is the same thing as a proset. Up to equivalence, a thin category is the same thing as a poset. So mostly we just talk about posets here, but some references want to distinguish these from thin categories. (It is really a question of whether you're working with strict categories, which are classified up to isomorphism, or categories as such, which are classified up to equivalence.)

## Examples

Since a poset is a thin category, in particular (semi)lattices, Heyting algebras, frames are, too.

Last revised on December 1, 2018 at 14:14:16. See the history of this page for a list of all contributions to it.