Example 12.17.4. Let $F$ be a field. Let $\mathcal{C}$ be the category of graded $F$-vector spaces. Given graded vector spaces $V$ and $W$ we let $V \otimes W$ denote the graded $F$-vector space whose degree $n$ part is

$(V \otimes W)^ n = \bigoplus \nolimits _{n = p + q} V^ p \otimes _ F W^ q$

Given a third graded vector space $U$ as associativity constraint $\phi : U \otimes (V \otimes W) \to (U \otimes V) \otimes W$ we use the “usual” isomorphisms

$U^ p \otimes _ F (V^ q \otimes _ F W^ r) \to (U^ p \otimes _ F V^ q) \otimes _ F W^ r$

of vectors spaces. As unit we use the graded $F$-vector space $\mathbf{1}$ which has $F$ in degree $0$ and is zero in other degrees. There are two commutativity constraints on $\mathcal{C}$ which turn $\mathcal{C}$ into a symmetric monoidal category: one involves the intervention of signs and the other does not. We will usually use the one that does. To be explicit, if $V$ and $W$ are graded $F$-vector spaces we will use the isomorphism $\psi : V \otimes W \to W \otimes V$ which in degree $n$ uses

$V^ p \otimes _ F W^ q \to W^ q \otimes _ F V^ p,\quad v \otimes w \mapsto (-1)^{pq} w \otimes v$

We omit the verification that this works.

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