nLab cartesian monoidal dagger category

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Definition

A cartesian monoidal dagger category is category which is both a dagger category and a cartesian monoidal category in a compatible way, namely a monoidal dagger category (C,×,1)(C, \times, 1) with

  • a morphism p AHom C(A×B,A)p_A \in Hom_C(A \times B,A) for AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C).
  • a morphism p BHom C(A×B,B)p_B \in Hom_C(A \times B,B) for AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C).
  • a morphism p A×BHom C(D,A×B)p_{A \times B} \in Hom_C(D,A \times B) for an object DOb(C)D \in Ob(C) and morphisms d AHom C(D,A)d_A \in Hom_C(D,A) and d BHom C(D,B)d_B \in Hom_C(D,B)
  • a morphism ! AHom C(A,1)!_A \in Hom_C(A,1) for every object ACA \in C

such that

  • for every object DOb(C)D \in Ob(C) and morphisms d AHom C(D,B)d_A \in Hom_C(D,B) and d BHom C(D,B)d_B \in Hom_C(D,B), p Ad A×B=d Ap_A \circ d_{A \times B} = d_A
  • for every object DOb(C)D \in Ob(C) and morphisms d AHom C(D,B)d_A \in Hom_C(D,B) and d BHom C(D,B)d_B \in Hom_C(D,B), p Bd AB=d Bp_B \circ d_{A \otimes B} = d_B
  • for every object AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) and morphism fHom C(A,B)f \in Hom_C(A,B), ! Bf=! A!_B \circ f = !_A.

In a cartesian monoidal dagger category, the tensor product is called a cartesian product and the tensor unit is called a terminal object.

Examples

See also

Last revised on May 4, 2022 at 02:11:17. See the history of this page for a list of all contributions to it.