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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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