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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.

MSC:
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
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