nLab thin category

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 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.

Examples

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

Related concepts

• symmetric proset

• specialization order

• separation axioms in terms of lifting properties