Rings, modules, and algebras in infinite loop space theory

Details Rings, modules, and algebras in infinite loop space theory Rings, modules, and algebras in infinite loop space theory

2004-03-23

arxiv.org/abs/math/0403403v1

Adv. in Math. 205 (2006), no. 1, 163-228

Mathematics

K-Theory and Homology

59 pages, 1 figure

Scientific paper

10.1016/j.aim.2005.07.007

We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in Smith's category of symmetric spectra, with an intermediate stop at a new category that may be of interest in its own right, whose objects we call symmetric functors.

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