nLab morphism of multicategories

Idea

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 “multifunctors” (e.g. Elmendorf & Mandell 2004, p. 2, review in Tate 2018, p. 2) but see at multifunctor for other meanings of this term.

Examples

Proposition

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

References

Review:

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

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]