nLab symmetry shifting

Idea

Given a monoidal category $V$, we can form the category of monoids $Mon(V)$ in $V$.

• If $V$ is a braided monoidal category, then $Mon(V)$ admits a tensor product, making it a monoidal category. (The category of monoids in $Mon(V)$ is the category of commutative monoids $CMon(V)$ in $V$.)

• If $V$ is a symmetric monoidal category, then $Mon(V)$ is furthermore a braided monoidal category (moreover a symmetric monoidal category).

One dimension higher, given a monoidal bicategory $V$, we can form the bicategory of pseudomonoids $PsMon(V)$ in $V$.

• If $V$ is a braided monoidal bicategory, then $Mon(V)$ is a monoidal bicategory.

• If $V$ is a sylleptic monoidal bicategory, then $Mon(V)$ is furthermore a braided monoidal bicategory.

• If $V$ is a symmetric monoidal bicategory, then $Mon(V)$ is furthermore a sylleptic monoidal bicategory (moreover a symmetric monoidal bicategory).

This phenomenon extends to higher dimensions and is dubbed symmetry shifting in Stenzel 2026. The following is Corollary 2.10 ibid.

Theorem

Let $0 \leq k \leq \infty$ and let $V$ be an $E_k$-monoidal $(\infty, 1)$-category. For all $0 \leq m \leq k$, the $(\infty, 1)$-category of $E_m$-monoids in $V$ is an $E_{k - m}$-monoidal $(\infty, 1)$-category.

Related concepts

• k-tuply monoidal n-category

• microcosm principle

• Fox's theorem

References

• Raffael Stenzel: Symmetry shifting for monoidal bicategories [arXiv:2602.15358]