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Permutative categories, multicategories and algebraic \(K\)-theory. (English) Zbl 1205.19003
A. D. Elmendorf and M. A. Mandell [Adv. Math. 205, No. 1, 163–228 (2006; Zbl 1117.19001)] introduced a \(K\)-theory functor from permutative categories (symmetric strict monoidal categories) to symmetric spectra. The multiplicative structure was described by a multicategory structure. The present paper embeds the category \(\mathbf{P}\) of permutative categories fully and faithfully as a subcategory of a multicategory, namely the category \(\mathbf{Mult}_*\) of based multicategories, and extends the \(K\)-theory map to this new category. The multiplicative structure on \(\mathbf{P}\) is preserved in the embedding. The new source category for \(K\)-theory, \(\mathbf{Mult}_*\) is symmetric monoidal, closed, complete and cocomplete, and the \(K\)-theory map is a lax symmetric monoidal functor.

19D23 Symmetric monoidal categories
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
55P42 Stable homotopy theory, spectra
18D50 Operads (MSC2010)
55U99 Applied homological algebra and category theory in algebraic topology
Full Text: DOI arXiv
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