nLab
symmetric monoidal functor

Contents

Idea

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.

Definition

A monoidal functor F:(C,)(D,)F : (C,\otimes) \to (D, \otimes) 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.

Properties

Proposition

(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)

Examples

Example

(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}representativeA\phantom{A}
A\phantom{A}referencesA\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}

References

An exposition is in

Last revised on July 27, 2018 at 06:58:29. See the history of this page for a list of all contributions to it.