On the Commutative Algebra of Categories

TL;DR

This paper develops a unified framework using commutative semiring categories to understand various categorical structures, bridging algebraic theories and stable homotopy theory through module categories.

Contribution

It introduces a novel perspective that generalizes algebraic theories and stable homotopy theory via modules over commutative semiring categories.

Findings

Categories of cartesian monoidal, semiadditive, and connective spectra are recoverable as modules over semiring categories.

Provides a language that unifies algebraic theories and stable homotopy theory.

Connects algebraic K-theory with these categorical frameworks.

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.