nLab morphism of multicategories




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 β€œmultifunctors” (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

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



  • Ross Tate, p. 2 of: Multicategories (2018) [pdf, pdf]

See also:

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