arXiv subjects
Publications and source records attributed to M. A. Mandell.
We show that the $K$-theory construction of arXiv:math/0403403, which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of arXiv:math/0403403, whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories.
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.
We extend to the non-connective case a lemma of Bokstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH.