nLab
thin category

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Category theory

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Thin categories

Definition

A thin category is 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:

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 April 30, 2017 at 08:36:42. See the history of this page for a list of all contributions to it.