nLab
PermCat

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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

𝕂:PermCatSpectra \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:𝒜 1××𝒜 n 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(𝒜 1)𝕂(𝒜 n)𝕂(). \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 :PermCatCatSpectraCat \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

References

category: category

Created on September 16, 2018 at 04:33:02. See the history of this page for a list of all contributions to it.