nLab cocartesian monoidal dagger category

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Definition

A cocartesian monoidal dagger category is a monoidal dagger category (C,+,0)(C, +, 0) with

  • a morphism i AHom C(A,A+B)i_A \in Hom_C(A,A + B) for AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C).
  • a morphism i BHom C(B,A+B)i_B \in Hom_C(B,A + B) for AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C).
  • a morphism d A+BHom C(A+B,D)d_{A + B} \in Hom_C(A + B,D) for an object DOb(C)D \in Ob(C) and morphisms d AHom C(A,D)d_A \in Hom_C(A,D) and d BHom C(B,D)d_B \in Hom_C(B,D)
  • a morphism 0 AHom C(0,A)0_A \in Hom_C(0,A) for every object ACA \in C

such that

  • for every object DOb(C)D \in Ob(C) and morphisms d AHom C(A,D)d_A \in Hom_C(A,D) and d BHom C(B,D)d_B \in Hom_C(B,D), d A+Bi A=d Ad_{A + B} \circ i_A = d_A
  • for every object DOb(C)D \in Ob(C) and morphisms d AHom C(A,D)d_A \in Hom_C(A,D) and d BHom C(B,D)d_B \in Hom_C(B,D), d A+Bi B=d Bd_{A + B} \circ i_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), f0 A=0 Bf \circ 0_A = 0_B.

In a cocartesian monoidal dagger category, the tensor product is called a coproduct and the tensor unit is called an initial object.

Examples

See also

Created on May 3, 2022 at 21:56:42. See the history of this page for a list of all contributions to it.