thin category

Thin categories

Thin categories


A thin or posetal category a category in which, given any two objects xx and yy and any two morphisms ff and gg from xx to yy, the morphisms ff and gg are equal:

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


Up to isomorphism, a thin category is the same thing as a proset (a preordered set). 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.)


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

Last revised on June 9, 2021 at 16:51:26. See the history of this page for a list of all contributions to it.