Cup-product laws

Fix a tensor pairing $\mu :\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}\to \mathcal{H}$. The assignment $\alpha \mapsto \tilde{\alpha }$ is a bijection and, being a group isomorphism [F1], additive: the representing morphism of $\alpha +{\alpha }^{\mathrm{\prime }}$ is $\tilde{\alpha }+{\tilde{\alpha }}^{\mathrm{\prime }}$. By the definition of the cup product the morphism representing $(\alpha +{\alpha }^{\mathrm{\prime }}){\cup }_{\mu }\beta$ is $\mu \circ c_{\mathcal{F},\mathcal{G}}\circ ((\tilde{\alpha }+{\tilde{\alpha }}^{\mathrm{\prime }}){\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)$; the derived tensor product is additive in each variable [F3] and composition is bilinear [F18], so this composite equals $\mu \circ c_{\mathcal{F},\mathcal{G}}\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)+\mu \circ c_{\mathcal{F},\mathcal{G}}\circ ({\tilde{\alpha }}^{\mathrm{\prime }}{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)$, which by additivity of [F1] represents $\alpha {\cup }_{\mu }\beta +{\alpha }^{\mathrm{\prime }}{\cup }_{\mu }\beta$; hence $(\alpha +{\alpha }^{\mathrm{\prime }}){\cup }_{\mu }\beta =\alpha {\cup }_{\mu }\beta +{\alpha }^{\mathrm{\prime }}{\cup }_{\mu }\beta$. The same computation with additivity of the derived tensor product in the second variable gives $\alpha {\cup }_{\mu }(\beta +{\beta }^{\mathrm{\prime }})=\alpha {\cup }_{\mu }\beta +\alpha {\cup }_{\mu }{\beta }^{\mathrm{\prime }}$. The assignment $(\alpha ,\beta )\mapsto \alpha {\cup }_{\mu }\beta$ is therefore additive in each variable and hence balanced over $\mathbb{Z}$ [F21]; by the universal property of the tensor product of abelian groups [F21], which is the tensor product over $R=\mathbb{Z}$ of the abelian groups $H^p(X,\mathcal{F})$ and $H^q(X,\mathcal{G})$ [F19], there is a unique homomorphism $H^p(X,\mathcal{F}){\otimes }_{\mathbb{Z}}{H}^q(X,\mathcal{G})\to H^{p+q}(X,\mathcal{H})$ with $\alpha \otimes \beta \mapsto \alpha {\cup }_{\mu }\beta$, which is the homomorphism of clause 1.

Let $f,g,h,\mu ,{\mu }^{\mathrm{\prime }}$ satisfy $h\circ \mu ={\mu }^{\mathrm{\prime }}\circ (f{\otimes }_{\mathbb{Z}}g)$. By naturality of [F1] the representing morphisms of $H^p(X,f)(\alpha )$ and $H^q(X,g)(\beta )$ are $f\circ \tilde{\alpha }$ and $g\circ \tilde{\beta }$, so by bifunctoriality of the derived tensor product [F3] the tensor $((f\circ \tilde{\alpha }){\otimes }^{\mathbf{L}}(g\circ \tilde{\beta }))$ equals $(f{\otimes }^{\mathbf{L}}g)\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })$. Naturality of the comparison [F4] gives $c_{{\mathcal{F}}^{\mathrm{\prime }},{\mathcal{G}}^{\mathrm{\prime }}}\circ (f{\otimes }^{\mathbf{L}}g)=(f{\otimes }_{\mathbb{Z}}g)\circ c_{\mathcal{F},\mathcal{G}}$; hence the morphism representing $H^p(X,f)(\alpha ){\cup }_{{\mu }^{\mathrm{\prime }}}{H}^q(X,g)(\beta )$ is ${\mu }^{\mathrm{\prime }}\circ (f{\otimes }_{\mathbb{Z}}g)\circ c_{\mathcal{F},\mathcal{G}}\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)$, which by the identity $h\circ \mu ={\mu }^{\mathrm{\prime }}\circ (f{\otimes }_{\mathbb{Z}}g)$ equals $Q(h)\circ \mu \circ c_{\mathcal{F},\mathcal{G}}\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)$. This is $Q(h)$ composed with the representing morphism of $\alpha {\cup }_{\mu }\beta$, so by naturality of [F1] applied to the sheaf morphism $h$ its class is $H^{p+q}(X,h)(\alpha {\cup }_{\mu }\beta )$, which is clause 2.

By [F2] the representing morphism of $1_X$ is ${\mathrm{id}}_{{\mathbb{Z}}_X}$. With the identity pairing of $({\mathbb{Z}}_X,\mathcal{F})$, the morphism representing $1_X\cup \alpha$ is $c_{{\mathbb{Z}}_X,\mathcal{F}}\circ ({\mathrm{id}}_{{\mathbb{Z}}_X}{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (0,p)$. The source of $\widetilde{1_X\cup \alpha }$ is $Z_p={\mathbb{Z}}_X[-p]$ and its target is the ordinary tensor sheaf ${\mathbb{Z}}_X\otimes \mathcal{F}$ in degree zero. Therefore the typed unit composite is $Q({\lambda }_{\mathcal{F}})\circ \widetilde{1_X\cup \alpha }={\lambda }^{\mathbf{L}}(\mathcal{F})\circ ({\mathrm{id}}_{{\mathbb{Z}}_X}{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (0,p),$ using ${\lambda }^{\mathbf{L}}(\mathcal{F})=Q({\lambda }_{\mathcal{F}})\circ c_{{\mathbb{Z}}_X,\mathcal{F}}$ [F14]. Naturality of the derived unitor [F13] moves $\tilde{\alpha }$ past it, and the shifted-unit law [F12] gives ${\lambda }^{\mathbf{L}}(Z_p)\circ \kappa (0,p)={\mathrm{id}}_{Z_p}$; hence this composite is $\tilde{\alpha }$. The same argument with ${\rho }^{\mathbf{L}}(\mathcal{F})=Q({\rho }_{\mathcal{F}})\circ c_{\mathcal{F},{\mathbb{Z}}_X}$ and ${\rho }^{\mathbf{L}}(Z_p)\circ \kappa (p,0)=\mathrm{id}$ proves $Q({\rho }_{\mathcal{F}})\circ \widetilde{\alpha \cup 1_X}=\tilde{\alpha }$.

For bounded-above complexes $\mathcal{F},\mathcal{G}$ define the natural comparison $C_{\mathcal{F},\mathcal{G}}:=Q\mathrm{Tot}({\alpha }_{\mathcal{F}}{\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{G}}):\mathcal{F}{\otimes }_{\mathbb{Z}}^{\mathbf{L}}\mathcal{G}\to Q\mathrm{Tot}(\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G})$. It is generally not an isomorphism: on a one-point space with both inputs $\mathbb{Z}\mathrm{/}2[0]$, the derived tensor has $H^{-1}={\mathrm{Tor}}_1^{\mathbb{Z}}(\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2)=\mathbb{Z}\mathrm{/}2$, whereas the ordinary tensor complex has no degree $-1$ term. If both inputs are bounded-above K-flat complexes, however, $C_{\mathcal{F},\mathcal{G}}$ is invertible. Factor its chain representative as $\mathrm{Tot}({\mathrm{id}}_{\mathcal{P}\mathcal{F}}\otimes {\alpha }_{\mathcal{G}})$ followed by $\mathrm{Tot}({\alpha }_{\mathcal{F}}\otimes {\mathrm{id}}_{\mathcal{G}})$: the first is a quasi-isomorphism because $\mathcal{P}\mathcal{F}$ is K-flat [F6, F7], and the second because $\mathcal{G}$ is K-flat [F8]. For cochain maps $f:\mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }}$ and $g:\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$, naturality of $\alpha$ [F5] gives the square $C_{{\mathcal{F}}^{\mathrm{\prime }},{\mathcal{G}}^{\mathrm{\prime }}}\circ (Qf{\otimes }^{\mathbf{L}}Qg)=Q\mathrm{Tot}(f\otimes g)\circ C_{\mathcal{F},\mathcal{G}}$. If all four complexes are K-flat, both comparisons are invertible, so this square may also be used in its conjugated form. No inverse of $C$ is used for arbitrary complexes.

Let $\mu :\mathcal{R}{\otimes }_{\mathbb{Z}}\mathcal{R}\to \mathcal{R}$ satisfy $\mu \circ {\sigma }_{\mathcal{R},\mathcal{R}}=\mu$ for the sheaf-level symmetry ${\sigma }_{\mathcal{R},\mathcal{R}}$ of clause 2 [F10], and let $\alpha \in H^p(X,\mathcal{R})$, $\beta \in H^q(X,\mathcal{R})$. The representing morphisms of $\alpha {\cup }_{\mu }\beta$ and $\beta {\cup }_{\mu }\alpha$ are $\mu \circ c_{\mathcal{R},\mathcal{R}}\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)$ and $\mu \circ c_{\mathcal{R},\mathcal{R}}\circ (\tilde{\beta }{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (q,p)$. By naturality of ${\sigma }^{\mathbf{L}}$ [F11] applied to $\tilde{\alpha }$ and $\tilde{\beta }$ one has $(\tilde{\beta }{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ {\sigma }^{\mathbf{L}}(Z_p,Z_q)={\sigma }^{\mathbf{L}}(\mathcal{R},\mathcal{R})\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })$; the symmetry at shifted units [F17] reads ${\sigma }^{\mathbf{L}}(Z_p,Z_q)=\kappa (q,p)\circ ((-1{)}^{pq}\mathrm{id})\circ \kappa (p,q{)}^{-1}$, so $\kappa (q,p)={\sigma }^{\mathbf{L}}(Z_p,Z_q)\circ \kappa (p,q)\circ ((-1{)}^{pq}{\mathrm{id}}_{Z_{p+q}})$ and the representing morphism of $\beta {\cup }_{\mu }\alpha$ equals $\mu \circ c_{\mathcal{R},\mathcal{R}}\circ {\sigma }^{\mathbf{L}}(\mathcal{R},\mathcal{R})\circ (\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta })\circ \kappa (p,q)\circ ((-1{)}^{pq}{\mathrm{id}}_{Z_{p+q}})$. By the first identity of clause 4(b) [F16] $c_{\mathcal{R},\mathcal{R}}\circ {\sigma }^{\mathbf{L}}(\mathcal{R},\mathcal{R})=Q({\sigma }_{\mathcal{R},\mathcal{R}})\circ c_{\mathcal{R},\mathcal{R}}$, and by $\mu \circ {\sigma }_{\mathcal{R},\mathcal{R}}=\mu$ the whole expression becomes the representing morphism of $\alpha {\cup }_{\mu }\beta$ postcomposed with multiplication by $(-1{)}^{pq}$ on $Z_{p+q}$. Composition is bilinear [F18] and the isomorphism of [F1] is additive, so that postcomposition multiplies the class by $(-1{)}^{pq}$; hence $\beta {\cup }_{\mu }\alpha =(-1{)}^{pq}(\alpha {\cup }_{\mu }\beta )$, the graded commutativity claimed in clause 5.

Let $\mathcal{F},\mathcal{G},\mathcal{H}$ be bounded-above K-flat complexes and put $T:=\mathrm{Tot}(\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G})$ and $U:=\mathrm{Tot}(\mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{H})$. The complexes $T,U$ are also K-flat by Stacks Project, Cohomology of Sheaves, Lemma 26, item 5 (tag 079R), so every comparison inverted below has K-flat source inputs [step 1.4]. By clause 3 [F15] ${\alpha }^{\mathbf{L}}(\mathcal{F},\mathcal{G},\mathcal{H})={\mathrm{\Xi }}^{-1}\circ Q(A(\mathcal{P}\mathcal{F},\mathcal{P}\mathcal{G},\mathcal{P}\mathcal{H}))\circ \mathrm{\Theta }$ with $\mathrm{\Theta }=Q\mathrm{Tot}({\alpha }_{\mathrm{Tot}(\mathcal{P}\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{P}\mathcal{G})}\otimes {\mathrm{id}}_{\mathcal{P}\mathcal{H}})$ and $\mathrm{\Xi }=Q\mathrm{Tot}({\mathrm{id}}_{\mathcal{P}\mathcal{F}}\otimes {\alpha }_{\mathrm{Tot}(\mathcal{P}\mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{P}\mathcal{H})})$. Naturality of $A$ in the three arguments [F9] applied to the augmentation cochain maps ${\alpha }_{\mathcal{F}},{\alpha }_{\mathcal{G}},{\alpha }_{\mathcal{H}}$ gives $A(\mathcal{F},\mathcal{G},\mathcal{H})\circ \mathrm{Tot}(\mathrm{Tot}({\alpha }_{\mathcal{F}}{\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{G}}){\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{H}})=\mathrm{Tot}({\alpha }_{\mathcal{F}}{\otimes }_{\mathbb{Z}}\mathrm{Tot}({\alpha }_{\mathcal{G}}{\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{H}}))\circ A(\mathcal{P}\mathcal{F},\mathcal{P}\mathcal{G},\mathcal{P}\mathcal{H})$; inverting the outer factors, which are quasi-isomorphisms [F5, F6, F8], expresses $Q(A(\mathcal{P}\mathcal{F},\mathcal{P}\mathcal{G},\mathcal{P}\mathcal{H}))$ through $Q(A(\mathcal{F},\mathcal{G},\mathcal{H}))$ and the two strict tensor factors. Naturality of $\alpha$ [F5] in the forms ${\alpha }_U\circ \mathcal{P}(\mathrm{Tot}({\alpha }_{\mathcal{G}}\otimes {\alpha }_{\mathcal{H}}))=\mathrm{Tot}({\alpha }_{\mathcal{G}}\otimes {\alpha }_{\mathcal{H}})\circ {\alpha }_{\mathrm{Tot}(\mathcal{P}\mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{P}\mathcal{H})}$ and ${\alpha }_T\circ \mathcal{P}(\mathrm{Tot}({\alpha }_{\mathcal{F}}\otimes {\alpha }_{\mathcal{G}}))=\mathrm{Tot}({\alpha }_{\mathcal{F}}\otimes {\alpha }_{\mathcal{G}})\circ {\alpha }_{\mathrm{Tot}(\mathcal{P}\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{P}\mathcal{G})}$, together with the functoriality of $\mathrm{Tot}$, gives $Q\mathrm{Tot}({\alpha }_{\mathcal{F}}{\otimes }_{\mathbb{Z}}\mathrm{Tot}({\alpha }_{\mathcal{G}}{\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{H}}))\circ \mathrm{\Xi }=C_{\mathcal{F},U}\circ ({\mathrm{id}}_{\mathcal{F}}{\otimes }^{\mathbf{L}}{C}_{\mathcal{G},\mathcal{H}})$ and $Q\mathrm{Tot}(\mathrm{Tot}({\alpha }_{\mathcal{F}}{\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{G}}){\otimes }_{\mathbb{Z}}{\alpha }_{\mathcal{H}})\circ \mathrm{\Theta }=C_{T,\mathcal{H}}\circ (C_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})$, where $({\mathrm{id}}_{\mathcal{F}}{\otimes }^{\mathbf{L}}{C}_{\mathcal{G},\mathcal{H}})=Q\mathrm{Tot}({\mathrm{id}}_{\mathcal{P}\mathcal{F}}\otimes \mathcal{P}\mathrm{Tot}({\alpha }_{\mathcal{G}}\otimes {\alpha }_{\mathcal{H}}))$ and $(C_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})=Q\mathrm{Tot}(\mathcal{P}\mathrm{Tot}({\alpha }_{\mathcal{F}}\otimes {\alpha }_{\mathcal{G}})\otimes {\mathrm{id}}_{\mathcal{P}\mathcal{H}})$ are the derived tensors of the cochain maps $C_{\mathcal{G},\mathcal{H}}$ and $C_{\mathcal{F},\mathcal{G}}$ with identities, by the formula of clause 2 [F3]. Substituting these two identities into the expression for ${\alpha }^{\mathbf{L}}$ and using that $C_{\mathcal{F},\mathcal{G}},C_{\mathcal{G},\mathcal{H}},C_{\mathcal{F},U},C_{T,\mathcal{H}}$ are isomorphisms [step 1.4] yields ${\alpha }^{\mathbf{L}}(\mathcal{F},\mathcal{G},\mathcal{H})=({\mathrm{id}}_{\mathcal{F}}{\otimes }^{\mathbf{L}}{C}_{\mathcal{G},\mathcal{H}}^{-1})\circ C_{\mathcal{F},U}^{-1}\circ Q(A(\mathcal{F},\mathcal{G},\mathcal{H}))\circ C_{T,\mathcal{H}}\circ (C_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})\mathrm{.}$

The shifted units $Z_n:={\mathbb{Z}}_X[-n]$ are bounded-above K-flat complexes (their sole term is the flat sheaf ${\mathbb{Z}}_X$), and their strict tensors $T,U$ below are K-flat by the same tensor-closure used in [step 2.1]. Write $Z_n:={\mathbb{Z}}_X[-n]$, write ${\kappa }_{p,q}:\mathrm{Tot}(Z_p{\otimes }_{\mathbb{Z}}{Z}_q)\to Z_{p+q}$ for the strict cochain map of clause 2, so that $\kappa (p,q)=C_{Z_p,Z_q}^{-1}\circ Q({\kappa }_{p,q}^{-1})$ [F12, step 1.4], and put $T:=\mathrm{Tot}(Z_p{\otimes }_{\mathbb{Z}}{Z}_q)$, $U:=\mathrm{Tot}(Z_q{\otimes }_{\mathbb{Z}}{Z}_r)$. Each ${\kappa }_{p,q}$ is an isomorphism of cochain complexes: by the shift convention $Z_n$ has ${\mathbb{Z}}_X$ as its only nonzero term, in degree $n$ [F22], so by the degree formula [F20] its source and target have ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X$ resp. ${\mathbb{Z}}_X$ as only nonzero term, in degree $p+q$, and its only nonzero component is the unit identification ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X\to {\mathbb{Z}}_X$ of clause 3 [F12], which is an isomorphism [F10]; hence ${\kappa }_{p,q}^{-1}$ is a cochain map. Both sides of the identity to be proved, ${\alpha }^{\mathbf{L}}(Z_p,Z_q,Z_r)\circ (\kappa (p,q){\otimes }^{\mathbf{L}}{\mathrm{id}}_{Z_r})\circ \kappa (p+q,r)=({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}\kappa (q,r))\circ \kappa (p,q+r),$ are morphisms $Z_{p+q+r}\to Z_p{\otimes }_{\mathbb{Z}}^{\mathbf{L}}(Z_q{\otimes }_{\mathbb{Z}}^{\mathbf{L}}{Z}_r)$. By [step 1.4] in the two variables and [step 2.1], the left-hand side equals $({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}{C}_{Z_q,Z_r}^{-1})\circ C_{Z_p,U}^{-1}\circ Q(A(Z_p,Z_q,Z_r)\circ \mathrm{Tot}({\kappa }_{p,q}^{-1}\otimes {\mathrm{id}}_{Z_r})\circ {\kappa }_{p+q,r}^{-1})$ and the right-hand side equals $({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}{C}_{Z_q,Z_r}^{-1})\circ C_{Z_p,U}^{-1}\circ Q(\mathrm{Tot}({\mathrm{id}}_{Z_p}\otimes {\kappa }_{q,r}^{-1})\circ {\kappa }_{p,q+r}^{-1})$. The two strict composites agree: both are cochain maps $Z_{p+q+r}\to \mathrm{Tot}(Z_p{\otimes }_{\mathbb{Z}}U)$ between complexes whose only nonzero term is ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}({\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X)$ in degree $p+q+r$ [F20, F22], so by [F23] they are determined by their component in that degree, a morphism ${\mathbb{Z}}_X\to {\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}({\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X)$, and by [F24] that morphism is determined by the image of $1_X$. There the first composite sends $1_X$ to $\alpha ((1\otimes 1)\otimes 1)$ and the second sends $1_X$ to $1\otimes (1\otimes 1)$, and these agree by the defining property of the sheaf-level associator [F9, F10]; hence $A(Z_p,Z_q,Z_r)\circ \mathrm{Tot}({\kappa }_{p,q}^{-1}\otimes {\mathrm{id}}_{Z_r})\circ {\kappa }_{p+q,r}^{-1}=\mathrm{Tot}({\mathrm{id}}_{Z_p}\otimes {\kappa }_{q,r}^{-1})\circ {\kappa }_{p,q+r}^{-1}$, and applying $Q$ [F5, F18] and the invertible outer factors [F3] gives the displayed identity.

Put $\mathcal{W}:=\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}$, $\mathcal{V}:=\mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{H}$, $T_1:=(\tilde{\alpha }{\otimes }^{\mathbf{L}}\tilde{\beta }){\otimes }^{\mathbf{L}}\tilde{\gamma }$ and $T_2:=\tilde{\alpha }{\otimes }^{\mathbf{L}}(\tilde{\beta }{\otimes }^{\mathbf{L}}\tilde{\gamma })$. By the definition of the cup product and bifunctoriality [F3] the morphisms representing $(\alpha \cup \beta )\cup \gamma$ and $\alpha \cup (\beta \cup \gamma )$ are ${\mathrm{\Psi }}_1:=c_{\mathcal{W},\mathcal{H}}\circ (c_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})\circ T_1\circ (\kappa (p,q){\otimes }^{\mathbf{L}}{\mathrm{id}}_{Z_r})\circ \kappa (p+q,r)$ and ${\mathrm{\Psi }}_2:=c_{\mathcal{F},\mathcal{V}}\circ ({\mathrm{id}}_{\mathcal{F}}{\otimes }^{\mathbf{L}}{c}_{\mathcal{G},\mathcal{H}})\circ T_2\circ ({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}\kappa (q,r))\circ \kappa (p,q+r)$. By [step 3.1], $(\kappa (p,q){\otimes }^{\mathbf{L}}{\mathrm{id}}_{Z_r})\circ \kappa (p+q,r)={\alpha }^{\mathbf{L}}(Z_p,Z_q,Z_r{)}^{-1}\circ ({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}\kappa (q,r))\circ \kappa (p,q+r)$, and by naturality of ${\alpha }^{\mathbf{L}}$ in the three arguments [F15] applied to $\tilde{\alpha },\tilde{\beta },\tilde{\gamma }$ one has ${\alpha }^{\mathbf{L}}(\mathcal{F},\mathcal{G},\mathcal{H})\circ T_1=T_2\circ {\alpha }^{\mathbf{L}}(Z_p,Z_q,Z_r)$; substituting both into ${\mathrm{\Psi }}_1$ gives ${\mathrm{\Psi }}_1=c_{\mathcal{W},\mathcal{H}}\circ (c_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})\circ {\alpha }^{\mathbf{L}}(\mathcal{F},\mathcal{G},\mathcal{H}{)}^{-1}\circ T_2\circ ({\mathrm{id}}_{Z_p}{\otimes }^{\mathbf{L}}\kappa (q,r))\circ \kappa (p,q+r)$. The second identity of clause 4(b) [F16] rearranges to $c_{\mathcal{W},\mathcal{H}}\circ (c_{\mathcal{F},\mathcal{G}}{\otimes }^{\mathbf{L}}{\mathrm{id}}_{\mathcal{H}})\circ {\alpha }^{\mathbf{L}}(\mathcal{F},\mathcal{G},\mathcal{H}{)}^{-1}=Q({\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}}{)}^{-1}\circ c_{\mathcal{F},\mathcal{V}}\circ ({\mathrm{id}}_{\mathcal{F}}{\otimes }^{\mathbf{L}}{c}_{\mathcal{G},\mathcal{H}})$, so ${\mathrm{\Psi }}_1=Q({\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}}{)}^{-1}\circ {\mathrm{\Psi }}_2$, that is $Q({\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}})\circ {\mathrm{\Psi }}_1={\mathrm{\Psi }}_2$. By naturality of [F1] applied to the sheaf morphism ${\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}}:\mathcal{W}{\otimes }_{\mathbb{Z}}\mathcal{H}\to \mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{V}$ the class of $Q({\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}})\circ {\mathrm{\Psi }}_1$ is $H^{p+q+r}(X,{\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}})((\alpha \cup \beta )\cup \gamma )$, and the class of ${\mathrm{\Psi }}_2$ is $\alpha \cup (\beta \cup \gamma )$; hence clause 4 holds.

Let $\mu :\mathcal{R}{\otimes }_{\mathbb{Z}}\mathcal{R}\to \mathcal{R}$ and $u:{\mathbb{Z}}_X\to \mathcal{R}$ satisfy the associativity and unit identities of clause 5, the unitors being ${\lambda }_{\mathcal{R}},{\rho }_{\mathcal{R}}$ of clause 2 [F10], and let $e\in H^0(X,\mathcal{R})$ be the class of $u$, so that the representing morphism of $e$ is $Q(u)$ [F2]. Unitality: by the definition of the cup product and naturality of the comparison [F4] applied to the pair $(u,{\mathrm{id}}_{\mathcal{R}})$ the morphism representing $e{\cup }_{\mu }\alpha$ is $\mu \circ c_{\mathcal{R},\mathcal{R}}\circ (Q(u){\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (0,p)=\mu \circ (u{\otimes }_{\mathbb{Z}}{\mathrm{id}}_{\mathcal{R}})\circ c_{{\mathbb{Z}}_X,\mathcal{R}}\circ ({\mathrm{id}}_{{\mathbb{Z}}_X}{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (0,p)=Q({\lambda }_{\mathcal{R}})\circ c_{{\mathbb{Z}}_X,\mathcal{R}}\circ ({\mathrm{id}}_{{\mathbb{Z}}_X}{\otimes }^{\mathbf{L}}\tilde{\alpha })\circ \kappa (0,p)$, where the last step is the unit identity $\mu \circ (u{\otimes }_{\mathbb{Z}}{\mathrm{id}}_{\mathcal{R}})={\lambda }_{\mathcal{R}}$; since ${\lambda }^{\mathbf{L}}(\mathcal{R})=Q({\lambda }_{\mathcal{R}})\circ c_{{\mathbb{Z}}_X,\mathcal{R}}$ [F14], the naturality of ${\lambda }^{\mathbf{L}}$ [F13] and the unit law ${\lambda }^{\mathbf{L}}({\mathbb{Z}}_X[-p])\circ \kappa (0,p)=\mathrm{id}$ [F12] give $e{\cup }_{\mu }\alpha =\alpha$, and the identity $\mu \circ ({\mathrm{id}}_{\mathcal{R}}{\otimes }_{\mathbb{Z}}u)={\rho }_{\mathcal{R}}$ gives $\alpha {\cup }_{\mu }e=\alpha$ in the same way. Associativity: let ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2$ be the representing morphisms of $(\alpha \cup \beta )\cup \gamma$ and $\alpha \cup (\beta \cup \gamma )$ with the identity pairings as in [step 4.1]; by naturality of the comparison [F4] applied to $(\mu ,{\mathrm{id}}_{\mathcal{R}})$ and to $({\mathrm{id}}_{\mathcal{R}},\mu )$ the representing morphisms of $(\alpha {\cup }_{\mu }\beta ){\cup }_{\mu }\gamma$ and $\alpha {\cup }_{\mu }(\beta {\cup }_{\mu }\gamma )$ are $Q(\mu \circ (\mu {\otimes }_{\mathbb{Z}}{\mathrm{id}}_{\mathcal{R}}))\circ {\mathrm{\Psi }}_1$ and $Q(\mu \circ ({\mathrm{id}}_{\mathcal{R}}{\otimes }_{\mathbb{Z}}\mu ))\circ {\mathrm{\Psi }}_2$. By [step 4.1] $Q({\alpha }_{\mathcal{R},\mathcal{R},\mathcal{R}})\circ {\mathrm{\Psi }}_1={\mathrm{\Psi }}_2$, so with $Q({\alpha }_{\mathcal{R},\mathcal{R},\mathcal{R}}{)}^{-1}=Q({\alpha }_{\mathcal{R},\mathcal{R},\mathcal{R}}^{-1})$ [F5] the first representative is $Q(\mu \circ (\mu {\otimes }_{\mathbb{Z}}{\mathrm{id}}_{\mathcal{R}}))\circ Q({\alpha }_{\mathcal{R},\mathcal{R},\mathcal{R}}^{-1})\circ {\mathrm{\Psi }}_2=Q(\mu \circ ({\mathrm{id}}_{\mathcal{R}}{\otimes }_{\mathbb{Z}}\mu ))\circ {\mathrm{\Psi }}_2$ by the associativity identity $\mu \circ (\mu {\otimes }_{\mathbb{Z}}{\mathrm{id}}_{\mathcal{R}})=\mu \circ ({\mathrm{id}}_{\mathcal{R}}{\otimes }_{\mathbb{Z}}\mu )\circ {\alpha }_{\mathcal{R},\mathcal{R},\mathcal{R}}$ of clause 5; the two representing morphisms therefore agree and $(\alpha {\cup }_{\mu }\beta ){\cup }_{\mu }\gamma =\alpha {\cup }_{\mu }(\beta {\cup }_{\mu }\gamma )$.

Clause 1 is [step 1.1], clause 2 is [step 1.2], clause 3 is [step 1.3], clause 4 is [step 4.1], and clause 5 is [step 5.1] together with [step 1.5] for the commutative case. The Axiom of Choice is used only as the standing hypothesis of the sheaf cohomology groups: the representing morphisms of the classes come from the supplied functorial injective resolution datum, which exists under AC (Enough injective abelian sheaves) and whose comparison maps use the Axiom of Dependent Choice implied by AC [F25], and the identifications $1_X\leftrightarrow {\mathrm{id}}_{{\mathbb{Z}}_X}$ and the class of $u$ are those of [F2]; no other selection occurs in the proof, every morphism used being one of the canonical maps $\kappa ,{\kappa }_{p,q},c,C,{\sigma }^{\mathbf{L}},{\lambda }^{\mathbf{L}},{\rho }^{\mathbf{L}},{\alpha }^{\mathbf{L}},A,\alpha ,\sigma ,\lambda ,\rho$ or the given pairing. ∎