On the Commutative Algebra of Categories

Abstract

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or $\infty$-category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K-theory bridging between the two.

51 pages; some proofs and notation clarified; this is the final version to appear in Algebraic and Geometric Topology

References in corpus (8)

• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• Universality of multiplicative infinite loop space machines
• Models of G-spectra as presheaves of spectra
• An Inverse K-Theory Functor
• Stable bundles over rig categories
• The fundamental pro-groupoid of an affine 2-scheme
• Stratified categories, geometric fixed points and a generalized Arone-Ching theorem
• An Equivariant Tensor Product on Mackey Functors

Cited by in corpus (1)

• Higher Lawvere theories