Category of super vector spaces

Definition

Let $k$ be a field of characteristic different from $2$ . The category of super vector spaces $\mathbf{SuperVect}_k$ has super vector spaces as objects and even linear maps as morphisms. Its tensor product is graded by

(V\otimes W)_{\bar r} =\bigoplus_{\bar i+\bar j=\bar r}V_{\bar i}\otimes W_{\bar j},

and its symmetry is the Koszul braiding

\tau_{V,W}(v\otimes w)=(-1)^{|v||w|}w\otimes v

for homogeneous $v,w$ . With this tensor product, unit $k$ in even degree, and braiding, $\mathbf{SuperVect}_k$ is a symmetric monoidal category. The signs forced by this braiding are recorded in the Koszul sign rule.

Odd maps

Only even maps are ordinary morphisms in $\mathbf{SuperVect}_k$ . The internal mapping object, which includes both even and odd maps, is the super internal Hom. The parity shift turns an odd map into an even map to or from a shifted object.

Why the morphism convention matters

Allowing all homogeneous maps as ungraded morphisms does not produce the ordinary symmetric monoidal category used to define superalgebras: composition and tensoring must retain degrees and Koszul signs. Keeping even maps as the categorical morphisms and all degrees in the internal Hom makes those roles explicit.

References

P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, pp. 41–97. Relevant: Sections 1–2.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. DOI record. Relevant: Chapter 1.