nLab thin category

Thin categories

Thin categories


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

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

Another synonym is (0,1)-category.


Relation to order theory

Up to isomorphism, a small thin category is a preordered set (“proset”). Up to equivalence, a small 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.


Any preordered set is a thin category. In particular so are posets, (semi)lattices, Heyting algebras and frames.

Last revised on February 23, 2024 at 05:05:54. See the history of this page for a list of all contributions to it.