Supersymmetric monoidal categories

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 ‫-2/ޚ‬graded, equipped with isomorphisms X ⊗ Y → Y ⊗ 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 √ 2 in the theory of Q-symmetric functions and Schur-Sergeev duality.1. Introduction 1479 2. Supercategories 1486 3. Monoidal structures 1489 4. Some 2-category theory 1503 5. Spin-symmetric groups 1511 6. Linear spin-species 1517 7. Isomeric vector spaces 1522 8. Clifford eversion 1526 9.

The authors' abstract, as published at the source. Algebra & Number Theory, 2026 · DOI ↗