nLab loop digraph object

Contents

Contents

Idea

A loop digraph object internal to a category is an object that behaves in that category like loop digraphs do in Set.

Β Definition

In a category with finite products

A loop digraph object in a category π’ž\mathcal{C} with finite products is

  • a object VV in π’ž\mathcal{C}

  • an object EE in π’ž\mathcal{C}

  • a morphism R:Eβ†’VΓ—VR:E \to V \times V

such that RR is monic: for every object AA in π’ž\mathcal{C} and morphisms f:Aβ†’Ef: A \to E and g:Aβ†’Eg: A \to E, R∘f=R∘gR \circ f = R \circ g implies that f=gf = g.

In general categories

A loop digraph object in a category π’ž\mathcal{C} is

  • a object VV in π’ž\mathcal{C}

  • an object EE in π’ž\mathcal{C}

  • a morphism s:Eβ†’Vs:E \to V

  • a morphism t:Eβ†’Vt:E \to V

such that ss and tt are jointly monic: for every object AA in π’ž\mathcal{C} and morphisms f:Aβ†’Ef: A \to E and g:Aβ†’Eg: A \to E, s∘f=s∘gs \circ f = s \circ g and t∘f=t∘gt \circ f = t \circ g imply that f=gf = g.

Properties

A loop digraph object is equivalently an object with an internal binary endorelation.

Last revised on May 13, 2022 at 22:27:04. See the history of this page for a list of all contributions to it.