multifunctor

The canonical kind of morphisms between multicategories are *multifunctors*. This generalized the concept of functor between categories and that of monoidal functors between monoidal categories.

The construction of K-theory spectra of permutative categories constitutes a multifunctor

$\mathbb{K}
\;\colon\;
PermCat
\longrightarrow
Spectra$

between multicategories of permutative categories (under Deligne tensor product) and spectra (under smash product of spectra).

Hence for a multilinear functor of permutative categories

$F
\;\colon\;
\mathcal{A}_1 \times \cdots \times \mathcal{A}_n
\longrightarrow
\mathcal{B}$

there is a compatibly induced morphism of K-spectra out of the smash product

$\mathbb{K}(\mathcal{A}_1)
\wedge
\cdots
\wedge
\mathbb{K}(\mathcal{A}_n)
\longrightarrow
\mathbb{K}(\mathcal{B})
\,.$

This implies that the construction further extends to a 2-functor from the 2-category $PermCat Cat$ of enriched categories over the multicategory of permutative categories to that of enriched categories of spectra:

$\mathbb{K}_\bullet
\;\colon\;
PermCat Cat \longrightarrow Spectra Cat$

which applies $\mathbb{K}$ to each hom-object.

(May 80, theorem 1.6, Theorem 2.1, Elmendorf-Mandell 04, theorem 1.1, Guillou 10, Theorem 1.1

See for instance

- Ross Tate, p. 2 of
*Multicategories*, 2018 (pdf)

Created on September 16, 2018 at 04:11:55. See the history of this page for a list of all contributions to it.