nLab dagger 2-poset

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

A dagger 2-poset is a dagger category CC such that

  1. for each object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphisms RHom(A,B)R \in Hom(A, B) and SHom(A,B)S \in Hom(A, B) there is a binary relation R A,BSR \leq_{A, B} S
  2. for each object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphism RHom(A,B)R \in Hom(A, B), R A,BRR \leq_{A, B} R.
  3. for each object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphisms RHom(A,B)R \in Hom(A, B) and SHom(A,B)S \in Hom(A, B), R A,BSR \leq_{A, B} S and S A,BRS \leq_{A, B} R implies that R=SR = S.
  4. for each object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphism RHom(A,B)R \in Hom(A, B), SHom(A,B)S \in Hom(A, B), and THom(A,B)T \in Hom(A, B), R A,BSR \leq_{A, B} S and S A,BTS \leq_{A, B} T implies that R A,BTR \leq_{A, B} T.
  5. for each object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphisms RHom(A,B)R \in Hom(A, B) and SHom(A,B)S \in Hom(A, B), R A,BSR \leq_{A, B} S implies R B,AS R^\dagger \leq_{B, A} S^\dagger.

A dagger 2-poset which only satisfies 1., 2., 4., and 5. is called a dagger 2-preorder or a dagger 2-proset.

A dagger 2-poset is a univalent dagger 2-poset if the underlying dagger-category is a univalent dagger category.

Morphisms

Examples

See also

Last revised on June 7, 2022 at 11:07:58. See the history of this page for a list of all contributions to it.