nLab
multifunctor

Idea

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.

Examples

Proposition

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

References

See for instance

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