Supersymmetric monoidal categories

Sam, Steven V; Snowden, Andrew

doi:10.2140/ant.2026.20.1479

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Supersymmetric monoidal categories

Steven V Sam and Andrew Snowden

Vol. 20 (2026), No. 8, 1479–1542

DOI: 10.2140/ant.2026.20.1479

Abstract

We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are $ℤ ∕ 2$ -graded, equipped with isomorphisms $X \otimes Y \to Y \otimes X$ of parity $| X | | Y |$ on homogeneous objects. There are two fundamental examples: the groupoid of spin-sets, and the category of isomeric vector spaces equipped with the half tensor product; other important examples can be derived from these (such as the category of linear spin species). There are also two general constructions. The first is the exterior algebra of a supercategory (due to Ganter and Kapranov). The second is a construction we introduce, called Clifford eversion. This defines an equivalence between a certain 2-category of supersymmetric monoidal supercategories and a corresponding 2-category of symmetric monoidal supercategories. We use our theory to better understand some aspects of the isomeric superalgebra, such as certain factors of $\sqrt{2}$ in the theory of Q-symmetric functions and Schur–Sergeev duality.

Keywords

superalgebra, supercategories

Mathematical Subject Classification

Primary: 18M05

Secondary: 05E10

Milestones

Received: 15 April 2021

Revised: 5 April 2024

Accepted: 5 August 2025

Published: 25 September 2026

Authors

Steven V Sam
Department of Mathematics University of California, San Diego La Jolla, CA United States
http://math.ucsd.edu/~ssam/
Andrew Snowden
Department of Mathematics University of Michigan Ann Arbor, MI United States
http://www-personal.umich.edu/~asnowden/

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Vol. 20, No. 8, 2026