The tensor product as well as the product in any category is determined by the corresponding limit diagram. So I should be able to find the product and tensor product in the category of super vector spaces together with grading preserving linear maps if I look at the usual diagrams. One finds the product pretty easily, by noting that the usual projection morphisms can only preserve the grading if we set $P(V,W) = (V_0\times W_0)\oplus (V_1\times W_1)$. Now I am stuck at the tensor product diagram; in particular, I do not quite know how we can get to the result that $V_1\otimes W_1$ belongs to the even component of the tensor product. Can someone help me?
2026-02-23 01:19:15.1771809555
Tensor product of super vector spaces
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