Algebraic & Geometric Topology

On the commutative algebra of categories

John D Berman

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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 –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.

Article information

Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2963-3012.

Received: 17 August 2017
Revised: 5 February 2018
Accepted: 4 May 2018
First available in Project Euclid: 30 August 2018

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Zentralblatt MATH identifier

Primary: 18C10: Theories (e.g. algebraic theories), structure, and semantics [See also 03G30] 55U40: Topological categories, foundations of homotopy theory
Secondary: 13C60: Module categories 19D23: Symmetric monoidal categories [See also 18D10]

higher algebra Lawvere theory operad


Berman, John D. On the commutative algebra of categories. Algebr. Geom. Topol. 18 (2018), no. 5, 2963--3012. doi:10.2140/agt.2018.18.2963.

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