nLab category of monic maps

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

Given a dagger 2-poset AA, the category of monic maps MonoMap(A)MonoMap(A) is the sub-2-poset whose objects are the objects of AA and whose morphisms are the injective maps of AA.

In every dagger 2-poset, given two injective maps f:hom A(a,b)f:hom_A(a,b) and g:hom A(a,b)g:hom_A(a,b), if fgf \leq g, then f=gf = g. This means that the sub-2-poset Map(A)Map(A) is a category and trivially a 2-poset.

Examples

  • For the dagger 2-poset Rel of sets and relations, the category of monic maps MonoMap(Rel)MonoMap(Rel) is equivalent to the category of sets and injections.

See also

Last revised on July 6, 2023 at 18:08:45. See the history of this page for a list of all contributions to it.