Cup product in sheaf cohomology

Assume the Axiom of Choice (The Axiom of Choice) and the standing smallness or supplied cofinal-denominator hypothesis of Derived category of an abelian category. Let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), let $H^q(X,-)$ be sheaf cohomology computed from the supplied functorial injective resolution datum (Sheaf cohomology as right derived global sections), let ${\mathbb{Z}}_X$ be the constant sheaf with value $\mathbb{Z}$ (The constant sheaf is the sheaf of locally constant functions), let ${\otimes }_{\mathbb{Z}}^{\mathbf{L}}$ be the derived tensor product of abelian sheaves on the bounded-above derived category, with comparison $c_{\mathcal{F},\mathcal{G}}:\mathcal{F}{\otimes }_{\mathbb{Z}}^{\mathbf{L}}\mathcal{G}\to \mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$, of clauses 2 and 4 of Derived tensor product of abelian sheaves, where ${\otimes }_{\mathbb{Z}}$ is the tensor product of abelian sheaves of Tensor product of abelian sheaves and its total complex, and let $\alpha ⟼\tilde{\alpha }:H^p(X,\mathcal{F})\stackrel{\sim }{\to }{\mathrm{Hom}}_{D(\mathrm{A}\mathrm{b}(X))}({\mathbb{Z}}_X[-p],\mathcal{F})$ be the canonical isomorphism of Sheaf cohomology classes as derived morphisms, so that $\tilde{\alpha }$ is the morphism of $D(\mathrm{A}\mathrm{b}(X))$ corresponding to a class $\alpha \in H^p(X,\mathcal{F})$; the representing morphism $\tilde{\alpha }$ is uniquely determined by $\alpha$, and it is the morphism that the bijection $\phi \mapsto {\phi }_X(1_X)$ of Morphisms from the constant sheaf are global sections describes in degree zero. Write $\kappa (p,q):{\mathbb{Z}}_X[-p-q]\to {\mathbb{Z}}_X[-p]{\otimes }_{\mathbb{Z}}^{\mathbf{L}}{\mathbb{Z}}_X[-q]$ for the canonical isomorphism of clause 2 of Koszul coherence of derived sheaf tensor.

Let $\mathcal{F},\mathcal{G},\mathcal{H}$ be abelian sheaves on $X$ and let $\mu :\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\to \mathcal{H}$ be a morphism of abelian sheaves, called a tensor pairing. For $p,q\ge 0$ the cup product with respect to $\mu$ is the pairing ${\cup }_{\mu }:H^p(X,\mathcal{F})\times H^q(X,\mathcal{G})⟶H^{p+q}(X,\mathcal{H})$ that assigns to $\alpha \in H^p(X,\mathcal{F})$ and $\beta \in H^q(X,\mathcal{G})$ the class $\alpha {\cup }_{\mu }\beta$ corresponding, under the isomorphism above taken in degree $p+q$ and for the sheaf $\mathcal{H}$, to the composite ${\mathbb{Z}}_X[-p-q]\stackrel{\kappa (p,q)}{\to }{\mathbb{Z}}_X[-p]{\otimes }_{\mathbb{Z}}^{\mathbf{L}}{\mathbb{Z}}_X[-q]\stackrel{\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta }}{\to }\mathcal{F}{\otimes }_{\mathbb{Z}}^{\mathbf{L}}\mathcal{G}\stackrel{c_{\mathcal{F},\mathcal{G}}}{\to }\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\stackrel{\mu }{\to }\mathcal{H}$ in $D(\mathrm{A}\mathrm{b}(X))$; here the second arrow is the derived tensor product of the representing morphisms, which is defined because the derived tensor product of clause 2 of Derived tensor product of abelian sheaves is a bifunctor on $D^{-}(\mathrm{A}\mathrm{b}(X))$, and the last two arrows are the comparison of clause 4 of that lemma and the pairing $\mu$.

Conventions and special cases.

1. (Sign convention.) The identification ${\mathbb{Z}}_X[-p-q]\cong {\mathbb{Z}}_X[-p]{\otimes }_{\mathbb{Z}}^{\mathbf{L}}{\mathbb{Z}}_X[-q]$ used in the definition is the one of the Stacks Project (footnote 3 to Section 31 of Cohomology of Sheaves), realised here by the canonical isomorphism $\kappa (p,q)$ of clause 2 of Koszul coherence of derived sheaf tensor; the Koszul sign in the symmetry of the tensor-product total complex is the one of clause 3 of Associator, symmetry and unitors of the abelian sheaf tensor product, under which the hidden sign of that footnote is $+1$. Thus no further sign is inserted by this definition.
2. (Identity pairing.) For $\mu ={\mathrm{id}}_{\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}}$ and $\mathcal{H}=\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$ the cup product is a pairing $H^p(X,\mathcal{F})\times H^q(X,\mathcal{G})\to H^{p+q}(X,\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G})$, written $\alpha \cup \beta$. In degree zero, $p=q=0$, the definition reduces to the section pairing $a{\cup }_{\mu }b={\mu }_X(a\otimes b)$, where the two global sections first give a section $a\otimes b$ of the tensor sheaf via the canonical map $\mathrm{\Gamma }(X,\mathcal{F}){\otimes }_{\mathbb{Z}}\mathrm{\Gamma }(X,\mathcal{G})\to \mathrm{\Gamma }(X,\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G})$, under $H^0(X,-)\cong \mathrm{\Gamma }(X,-)$.
3. (Sheaves of rings.) Let $\mathcal{R}$ be an abelian sheaf on $X$ together with a morphism $\mu :\mathcal{R}{\otimes }_{\mathbb{Z}}\mathcal{R}\to \mathcal{R}$ that is associative, that is, $\mu \circ (\mu \otimes \mathrm{id})=\mu \circ (\mathrm{id}\otimes \mu )$ after the canonical associativity identifications of the tensor product of abelian sheaves, and with a unit section $1_{\mathcal{R}}\in \mathrm{\Gamma }(X,\mathcal{R})$ satisfying ${\mu }_U(s\otimes 1_{\mathcal{R}}{\mathrm{\mid }}_U)={\mu }_U(1_{\mathcal{R}}{\mathrm{\mid }}_U\otimes s)=s$ for every open $U\subseteq X$ and $s\in \mathcal{R}(U)$; when in addition $\mu \circ {\sigma }_{\mathcal{R},\mathcal{R}}=\mu$ for the symmetry ${\sigma }_{\mathcal{R},\mathcal{R}}$ of clause 2 of Associator, symmetry and unitors of the abelian sheaf tensor product we call $\mathcal{R}$ commutative. Then ${\cup }_{\mu }$ is the multiplication of the graded product on $H^{\ast }(X,\mathcal{R})={⨁}_{q\ge 0}{H}^q(X,\mathcal{R})$ whose associativity, unitality and graded commutativity, together with the naturality of the cup product, are established by the cup-product theorem below on this page; the tensor-product form $H^p(X,\mathcal{F}){\otimes }_{\mathbb{Z}}{H}^q(X,\mathcal{G})\to H^{p+q}(X,\mathcal{H})$ of ${\cup }_{\mu }$ is likewise recorded there (its clause 1), since additivity in each variable is not asserted by this definition.