Coherence for pseudo commutative 2-monads

Under mild conditions, the category of algebras (and strict maps) $T{\text{-alg}}_s$ of a commutative monad on a symmetric monoidal closed category $T:𝒱\to 𝒱$ is symmetric monoidal closed [4]. By merging the concept of commutative monad with that of 2-monad one is led to the definition of a strictly commutative 2-monad over a symmetric monoidal 2-category. With the motivation of studying some 2-monads $T:𝒦\to 𝒦$ which are not strictly commutative, such as the 2-monad in $\mathrm{𝐂𝐚𝐭}$ for symmetric strict monoidal categories, as well as their categories of algebras and pseudo maps $T\text{-}alg$, Hyland and Power defined the concept of a pseudo commutative 2-monad $T:𝒦\to 𝒦$ on a symmetric monoidal 2-category [3]. These are 2-monads which are strictly commutative up to coherent isomorphisms in a precise sense. For such 2-monads $T:𝒦\to 𝒦,$ they proved that $T\text{-}alg$ can be enhanced to a $\mathrm{𝐂𝐚𝐭}$-enriched non-symmetric multicategory, and that when $T$ satisfies the extra condition of being symmetric, $T\text{-}alg$ is a symmetric $\mathrm{𝐂𝐚𝐭}$-enriched multicategory [3].

Our first result is that when $T$ is a symmetric, pseudo commutative 2-monad, the free algebra functor $T:𝒦\to T\text{-}alg$ can be enhanced to a non-symmetric $\mathrm{𝐂𝐚𝐭}$-enriched multifunctor [7]. This $\mathrm{𝐂𝐚𝐭}$-enriched multifunctor fails to preserve the action of the symmetric group on multilinear maps by swapping inputs, but it does so up to coherent isomorphisms. Such $\mathrm{𝐂𝐚𝐭}$-enriched multifunctors are called pseudo symmetric and they were defined by Yau, who proved that inverse $K$-theory gives one example [8]. Our second result is that when $T$ is a symmetric pseudo commutative 2-monad, the $\mathrm{𝐂𝐚𝐭}$-enriched multifunctor $T:𝒦\to T\text{-}alg$ is pseudo symmetric [7]. By using results from [6], we can rigidify $T:𝒦\to T\text{-}alg$ to get a symmetric $\mathrm{𝐂𝐚𝐭}$-enriched multifunctor $T:𝒦\times E{\mathrm{\Sigma }}_{\ast }\to T\text{-}alg,$ where $E{\mathrm{\Sigma }}_{\ast }$ is the categorical Barrat-Eccles operad. This can be considered as a coherence theorem for symmetric pseudo commutative 2-monads. Our proof also implies a coherence theorem conjectured by Hyland and Power [3] for the case when $T$ is not symmetric.

Our results will find applications in Algebraic $K$-theory since symmetric pseudo commutative 2-operads [1, 2], 2-operads whose associated 2-monads are symmetric pseudo commutative, are used to parameterize the input of several equivariant algebraic $K$-theory constructions [2, 9]. They also apply to KZ 2-monads [5], which include 2-monads whose algebras are categories with a given class of colimits [5].