# Contents

## Idea

The 2-category of permutative categories. With respect to multi-linear functors (Deligne tensor product) this canonically becomes a multicategory (Elmendorf-Mandell 04, Section 3).

## Properties

###### Proposition

The construction of K-theory spectra of permutative categories constitutes a multifunctor

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

between multicategories 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.

## References

category: category