The canonical kind of morphisms between multicategories generalize the concept of *functors* between categories and that of *monoidal functors* between monoidal categories.

It is common to speak here of βmultifunctorsβ (e.g. Elmendorf & Mandell 2004, p. 2, review in Tate 2018, p. 2) but see at *multifunctor* for other meanings of this term.

The construction of K-theory spectra of permutative categories constitutes a morphism of multicategories

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

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

Review:

See also:

- Anthony Elmendorf, Michael Mandell,
*Rings, modules and algebras in infinite loop space theory*, Adv. in Math.**205**1 (2006) 163-228 [arXiv:math/0403403, doi:10.1016/j.aim.2005.07.007]

Last revised on July 22, 2022 at 12:57:24. See the history of this page for a list of all contributions to it.