nLab symmetric monoidal functor

Idea

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

Definition

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

$$\array{ F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A) }$$

commutes, where $\sigma$ denotes the symmetry isomorphism both of $C$ and $D$.

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

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

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

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

Examples

Example

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

The category of chain complexes of super vector spaces $Ch(Supervect)$ equipped with the tensor product of chain complexes carries two symmetric braidings, $\tau_{Deligne}$ and $\tau_{Bernst}$ (this Prop.). The identity functor on $Ch(SuperVect)$ 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

$$dgcsAlg_{Deligne} \;\simeq\; dgcsAlg_{Bernstein}$$

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

	$\phantom{A}$Deligne’s convention$\phantom{A}$	$\phantom{A}$Bernstein’s convention$\phantom{A}$
$\phantom{A}$$\alpha_i \cdot \alpha_j =$$\phantom{A}$	$\phantom{A}$$(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$	$\phantom{A}$$(-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$
$\phantom{A}$common in$\phantom{A}$ $\phantom{A}$discussion of$\phantom{A}$	$\phantom{A}$supergravity$\phantom{A}$	$\phantom{A}$AKSZ sigma-models$\phantom{A}$
$\phantom{A}$representative$\phantom{A}$ $\phantom{A}$references$\phantom{A}$	$\phantom{A}$Bonora et. al 87,$\phantom{A}$ $\phantom{A}$Castellani-D’Auria-Fré 91,$\phantom{A}$ $\phantom{A}$Deligne-Freed 99$\phantom{A}$	$\phantom{A}$AKSZ 95,$\phantom{A}$ $\phantom{A}$Carchedi-Roytenberg 12$\phantom{A}$

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

• braided monoidal functor

• symmetric monoidal functor

References

An exposition is in

• John Baez, Some definitions everyone should know.