Super internal Hom - Knowlpedia

Definition

For super vector spaces $V$ and $W$ , their super internal Hom is the super vector space defined by

\underline{\operatorname{Hom}}(V,W)_{\bar\epsilon} = \left\{ f:V\to W: f(V_{\bar i})\subseteq W_{\bar i+\bar\epsilon} \right\}.

Its even part consists of parity-preserving maps and is the morphism space $\operatorname{Hom}_{\mathbf{SuperVect}}(V,W)$ . Its odd part consists of parity-reversing maps.

Evaluation is an even map

\underline{\operatorname{Hom}}(V,W)\otimes V\longrightarrow W, \qquad f\otimes v\longmapsto f(v).

Together with the Koszul sign rule, this object makes the category of super vector spaces closed symmetric monoidal. The parity shift identifies odd maps $V\to W$ with even maps $V\to\Pi W$ , up to the chosen sign convention for the natural identifications.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. Publisher record. Relevant: Chapter 1.
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. Relevant: Sections 1–2.