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

𝕂:PermCat⟢Spectra \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:π’œ 1Γ—β‹―Γ—π’œ nβŸΆβ„¬ 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

𝕂(π’œ 1)βˆ§β‹―βˆ§π•‚(π’œ n)βŸΆπ•‚(ℬ). \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 PermCatCatPermCat Cat of enriched categories over the multicategory of permutative categories to that of enriched categories of spectra:

𝕂 β€’:PermCatCat⟢SpectraCat \mathbb{K}_\bullet \;\colon\; PermCat Cat \longrightarrow Spectra Cat

which applies 𝕂\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.