Volume 18, issue 5 (2018)

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On the commutative algebra of categories

John D Berman

Algebraic & Geometric Topology 18 (2018) 2963–3012

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.

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higher algebra, Lawvere theory, operad
Mathematical Subject Classification 2010
Primary: 18C10, 55U40
Secondary: 13C60, 19D23
Received: 17 August 2017
Revised: 5 February 2018
Accepted: 4 May 2018
Published: 22 August 2018
John D Berman
Department of Mathematics
University of Virginia
Charlottesville, VA
United States