nLab symmetric monoidal functor



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



A symmetric monoidal functor is a functor F:CDF : C \to D between symmetric monoidal categories that is a monoidal functor which respects the symmetry on both sides.


A (lax) monoidal functor F:(C,)(D,)F : (C,\otimes) \to (D, \otimes), with monoidal structure \nabla, between symmetric monoidal categories is symmetric if for all A,BCA,B \in C the diagram

FAFB σ FBFA A,B B,A F(AB) F(σ) F(BA) \array{ F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A) }

commutes, where σ\sigma denotes the symmetry isomorphism both of CC and DD.

As long as it goes between symmetric monoidal categories a symmetric monoidal functor is the same as a braided monoidal functor.



(symmetric monoidal functor induces functor on commutative monoids)

A symmetric monoidal functor

(𝒞 1, 1,τ 1)(𝒞 2, 2,τ 2) \left(\mathcal{C}_1, \otimes_1, \tau_1\right) \longrightarrow \left(\mathcal{C}_2, \otimes_2, \tau_2\right)

between two symmetric monoidal categories canonically preserves commutative monoids and extends to a functor between categories of commutative monoids (see here for more)

CMon(𝒞 1, 1,τ 1)CMon(𝒞 2, 2,τ 2) CMon\left(\mathcal{C}_1, \otimes_1, \tau_1\right) \longrightarrow CMon\left(\mathcal{C}_2, \otimes_2, \tau_2\right)



(identity functor on category of chain complexes of super vector spaces)

The category of chain complexes of super vector spaces Ch(Supervect)Ch(Supervect) equipped with the tensor product of chain complexes carries two symmetric braidings, τ Deligne\tau_{Deligne} and τ Bernst\tau_{Bernst} (this Prop.). The identity functor on Ch(SuperVect)Ch(SuperVect) carries the structure of a strong symmetric monoidal functor with respect to these two, making them equivalent. By Prop. this in turn induces an equivalence on the catories of commutative monoids, which in this case are differential graded-commutative superalgebras, with one of two equivalent grading conventions

dgcsAlg DelignedgcsAlg Bernstein dgcsAlg_{Deligne} \;\simeq\; dgcsAlg_{Bernstein}

sign rule for differential graded-commutative superalgebras
(different but equivalent)

A\phantom{A}Deligne’s conventionA\phantom{A}A\phantom{A}Bernstein’s conventionA\phantom{A}
A\phantom{A}α iα j= \alpha_i \cdot \alpha_j = A\phantom{A}A\phantom{A}(1) (n in j+σ iσ j)α jα i(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}A\phantom{A}(1) (n i+σ i)(n j+σ j)α jα i (-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}
A\phantom{A}common inA\phantom{A}
A\phantom{A}discussion ofA\phantom{A}
A\phantom{A}supergravityA\phantom{A}A\phantom{A}AKSZ sigma-modelsA\phantom{A}
A\phantom{A}Bonora et. al 87,A\phantom{A}
A\phantom{A}Castellani-D’Auria-Fré 91,A\phantom{A}
A\phantom{A}Deligne-Freed 99A\phantom{A}
A\phantom{A}AKSZ 95,A\phantom{A}
A\phantom{A}Carchedi-Roytenberg 12A\phantom{A}


An exposition is in

Last revised on January 31, 2023 at 18:14:15. See the history of this page for a list of all contributions to it.