In category theory, monoidal functors are functors between monoidal categories that preserve the monoidal structure, that is, multiplication and the identity element.
Let be {\ displaystyle ({\ mathcal {C}}, \ otimes, I _ {\ mathcal {C}})} and {\ displaystyle ({\ mathcal {D}}, \ bullet, I _ {\ mathcal {D}})} - monoidal categories. Monoidal functor from {\ displaystyle {\ mathcal {C}}} at {\ displaystyle {\ mathcal {D}}} consists of a functor {\ displaystyle F: {\ mathcal {C}} \ to {\ mathcal {D}}} natural transformation
- {\ displaystyle \ phi _ {A, B}: FA \ bullet FB \ to F (A \ otimes B)}
and morphism
- {\ displaystyle \ phi: I _ {\ mathcal {D}} \ to FI _ {\ mathcal {C}}} ,
called structural morphisms , such that for any {\ displaystyle A} , {\ displaystyle B} , {\ displaystyle C} at {\ displaystyle {\ mathcal {C}}} charts
-
- and
commutative in the category {\ displaystyle {\ mathcal {D}}} . Standard notation is used here. {\ displaystyle \ alpha, \ rho, \ lambda} for monoidal category structure {\ displaystyle {\ mathcal {C}}} and {\ displaystyle {\ mathcal {D}}} .
A strongly monoidal functor is a monoidal functor such that structural morphisms {\ displaystyle \ phi _ {A, B}, \ phi} reversible.
A strictly monoidal functor is a monoidal functor whose structural morphisms are identical.
Forgetting functor {\ displaystyle U: (\ mathbf {Ab}, \ otimes _ {\ mathbf {Z}}, \ mathbf {Z}) \ rightarrow (\ mathbf {Set}, \ times, \ {* \})} from the category of abelian groups to the category of sets. Here is structural morphism {\ displaystyle \ phi _ {A, B} \ colon U (A) \ times U (B) \ to U (A \ otimes B)} Is a surjection induced by the standard mapping {\ displaystyle A \ times B \ to A \ otimes B \ to} ; display {\ displaystyle \ phi \ colon \ {* \} \ to \ mathbb {Z}} translates singleton * to 1.