Homotopy Type Theory dagger 2-preorder > history (Rev #2)

Contents

Definition

A dagger 2-preorder is a dagger precategory CC such that

  • For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), a propositional binary relation R A,BSR \leq_{A, B} S
  • For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), R A,BRR \leq_{A, B} R.
  • For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), R A,BSR \leq_{A, B} S and S A,BRS \leq_{A, B} R implies R=SR = S.
  • For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), T:Hom(A,B)T:Hom(A, B), R A,BSR \leq_{A, B} S and S A,BTS \leq_{A, B} T implies R A,BTR \leq_{A, B} T.
  • For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), R A,BSR \leq_{A, B} S implies R B,AS R^\dagger \leq_{B, A} S^\dagger.

See also

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