Supersymmetric monoidal categories

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 $\mathbf{Z}\mathrm{/}2$-graded, equipped with isomorphisms $X\otimes Y\to Y\otimes X$ of parity $\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }$ on homogeneous objects. There are two fundamental examples: the groupoid of spin-sets, and the category of queer 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–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 queer superalgebra, such as certain factors of $\sqrt{2}$ in the theory of Q-symmetric functions and Schur–Sergeev duality.