Associator, symmetry and unitors of the abelian sheaf tensor product

For presheaves $\mathcal{P},\mathcal{Q}$ the morphism ${\phi }_{\mathcal{P},\mathcal{Q}}:a(\mathcal{P}{\otimes }_{p,\mathbb{Z}}\mathcal{Q})\to a\mathcal{P}{\otimes }_{\mathbb{Z}}a\mathcal{Q}$ is defined by the universal property of sheafification [F10] from the presheaf map $\mathcal{P}{\otimes }_{p,\mathbb{Z}}\mathcal{Q}\to a\mathcal{P}{\otimes }_{\mathbb{Z}}a\mathcal{Q}$ given on $U$ by the sheafification maps ${\eta }_{\mathcal{P},U},{\eta }_{\mathcal{Q},U}$ composed with the presheaf-tensor map into the sheaf tensor [F1]. On stalks it is the canonical map which is an isomorphism because ${\otimes }_{\mathbb{Z}}$ preserves colimits in each variable [F6] and the stalks are the filtered colimits of [F7], while the right-hand side is the stalk of $a\mathcal{P}{\otimes }_{\mathbb{Z}}a\mathcal{Q}$ by [F3] and [F8]; hence ${\phi }_{\mathcal{P},\mathcal{Q}}$ is an isomorphism [F9]. It is natural in $\mathcal{P},\mathcal{Q}$ because each defining ingredient is.

For $x\otimes y\in {\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j$ the map $S$ sends $x\otimes y$ to $(-1{)}^{ij}y\otimes x$. It is a cochain map: by [F2] the differential of $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet })$ sends $x\otimes y$ to $d_{\mathcal{F}}x\otimes y+(-1{)}^{i}x\otimes d_{\mathcal{G}}y$, and applying $S$ gives $(-1{)}^{(i+1)j}y\otimes d_{\mathcal{F}}x+(-1{)}^{i+i(j+1)}{d}_{\mathcal{G}}y\otimes x$, while the differential of $\mathrm{Tot}({\mathcal{G}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{F}}^{\bullet })$ sends $S(x\otimes y)=(-1{)}^{ij}y\otimes x$ to $(-1{)}^{ij}(d_{\mathcal{G}}y\otimes x+(-1{)}^{j}y\otimes d_{\mathcal{F}}x)=(-1{)}^{ij}{d}_{\mathcal{G}}y\otimes x+(-1{)}^{ij+j}y\otimes d_{\mathcal{F}}x$, and the two expressions agree because $(-1{)}^{i+i(j+1)}=(-1{)}^{ij}$ and $(-1{)}^{ij+j}=(-1{)}^{(i+1)j}$. This is the Koszul sign rule; it is verified on pure tensors, and a morphism of total complexes is determined by its components on the summands [F13] and by its values on pure sections inside each summand [F10]. Composing $S$ with the symmetry in the opposite direction gives $(-1{)}^{ij}(-1{)}^{ji}=1$ on each summand, so $S$ is an isomorphism and is involutive, and it is natural in the two complex arguments because the exchange of pure tensors is.

The total complex $\mathrm{Tot}({\mathbb{Z}}_X[0]{\otimes }_{\mathbb{Z}}{\mathcal{F}}^{\bullet })$ has degree-$n$ term ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathcal{F}}^n$: the only nonzero term of ${\mathbb{Z}}_X[0]$ is in degree $0$ [F15], so the degree-$n$ diagonal has the single summand ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathcal{F}}^n$. Its differential is $d_{{\mathbb{Z}}_X}\otimes \mathrm{id}+(-1{)}^0\mathrm{id}\otimes d_{\mathcal{F}}^n=\mathrm{id}\otimes d_{\mathcal{F}}^n$ because the differential of the stalk complex is zero [F15] and by the formula of [F2]. The levelwise unitor $\lambda$ of [F4] is therefore an isomorphism of cochain complexes $\mathrm{\Lambda }$ [F14], and the same argument in the second variable, using the symmetry of [F4] or the unitor $\rho$, gives $\mathrm{P}$. Both are natural in ${\mathcal{F}}^{\bullet }$ because $\lambda$ and $\rho$ are natural [F4].

The componentwise module associator of [F5] is natural in $U$ because restriction maps are homomorphisms, so the isomorphisms $(\mathcal{F}(U){\otimes }_{\mathbb{Z}}\mathcal{G}(U)){\otimes }_{\mathbb{Z}}\mathcal{H}(U)\to \mathcal{F}(U){\otimes }_{\mathbb{Z}}(\mathcal{G}(U){\otimes }_{\mathbb{Z}}\mathcal{H}(U))$ assemble into an isomorphism of presheaves $(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G}){\otimes }_{p,\mathbb{Z}}\mathcal{H}\to \mathcal{F}{\otimes }_{p,\mathbb{Z}}(\mathcal{G}{\otimes }_{p,\mathbb{Z}}\mathcal{H})$, which remains an isomorphism after sheafification. Composing this with the isomorphisms of [step 1.1] for the pairs $(\mathcal{F}{\otimes }_{p,\mathbb{Z}}\mathcal{G},\mathcal{H})$ and $(\mathcal{F},\mathcal{G}{\otimes }_{p,\mathbb{Z}}\mathcal{H})$ gives the canonical isomorphism ${\alpha }_{\mathcal{F},\mathcal{G},\mathcal{H}}:(\mathcal{F}{\otimes }_{\mathbb{Z}}\mathcal{G}){\otimes }_{\mathbb{Z}}\mathcal{H}\to \mathcal{F}{\otimes }_{\mathbb{Z}}(\mathcal{G}{\otimes }_{\mathbb{Z}}\mathcal{H})$, natural in each variable, which on pure sections satisfies $(f\otimes g)\otimes h\mapsto f\otimes (g\otimes h)$; indeed this is the image under the sheafification maps of the module-level formula of [F5], and a morphism out of a tensor product is determined by its values on pure sections [F10].

The pentagon, triangle and hexagon identities of clause 2 hold. By [F9] it suffices to compare both sides on every stalk, where all morphisms are computed from module-level maps: the stalk maps of $\alpha$ are the associators of [F5], those of $\sigma$ and $\lambda ,\rho$ are the commutativity and unit constraints of the tensor product of abelian groups [F4]. On pure tensors these are the standard identities: in the pentagon both sides are the same reassociation of a pure tensor, in the triangle both sides multiply the tensor by the scalar through $\lambda (f\otimes s)=f\cdot s$ [F4], and in the hexagon both routes send a pure tensor $f\otimes (g\otimes h)$ to $g\otimes (h\otimes f)$ with all signs $+1$. The symmetry is involutive because ${\sigma }_{\mathcal{G},\mathcal{F}}{\sigma }_{\mathcal{F},\mathcal{G}}$ sends $m\otimes n$ to $n\otimes m$ and back to $m\otimes n$ [F4, F9].

On the summand ${\mathcal{F}}^i{\otimes }_{\mathbb{Z}}{\mathcal{G}}^j{\otimes }_{\mathbb{Z}}{\mathcal{H}}^k$ of the degree-$n$ term, the complex $\mathrm{Tot}(\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet }){\otimes }_{\mathbb{Z}}{\mathcal{H}}^{\bullet })$ has differential $d_{\mathcal{F}}\otimes \mathrm{id}\otimes \mathrm{id}+(-1{)}^i\mathrm{id}\otimes d_{\mathcal{G}}\otimes \mathrm{id}+(-1{)}^{i+j}\mathrm{id}\otimes \mathrm{id}\otimes d_{\mathcal{H}}$: apply the total-complex differential of [F2] to the outer tensor variable, whose inner factor is the total complex $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{G}}^{\bullet })$ with differential $d_{\mathcal{F}}\otimes \mathrm{id}+(-1{)}^i\mathrm{id}\otimes d_{\mathcal{G}}$ on its summand. The complex $\mathrm{Tot}({\mathcal{F}}^{\bullet }{\otimes }_{\mathbb{Z}}\mathrm{Tot}({\mathcal{G}}^{\bullet }{\otimes }_{\mathbb{Z}}{\mathcal{H}}^{\bullet }))$ carries the same differential on the same summand, obtained in the opposite order, so the reassociation map $A$, which on each summand is the sheaf-level associator $\alpha$ of [step 2.1] and hence an isomorphism, commutes with the differentials and is an isomorphism of cochain complexes [F14]. Its components are assembled from the coproducts of [F13], which exist by [F12], and it is natural because the copairings are unique [F11] and $\alpha$ is natural.

Every coherence identity of clause 3 compares two morphisms of cochain complexes between the same total complexes. A morphism of complexes is determined by its components on the summands [F13], and by [F10] each component out of a sheaf tensor product is determined by its values on pure sections, so it suffices to evaluate both sides on pure tensors, where the identities reduce to the sheaf-level ones of [step 3.1] combined with the sign computation of [step 1.2]: the reassociation identities hold because both sides reassociate a pure tensor in the same way, the triangle identities multiply by the scalar of the unit as in [F4], and the hexagon identities acquire the Koszul signs of [step 1.2], which are multiplicative and cancel in the same way on both routes. For the shift statement, ${\mathbb{Z}}_X[-p]$ has its single nonzero term ${\mathbb{Z}}_X$ in degree $p$ and ${\mathbb{Z}}_X[-q]$ its single nonzero term in degree $q$ [F15]; hence $\mathrm{Tot}({\mathbb{Z}}_X[-p]{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X[-q])$ has the single nonzero term ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X$ in degree $p+q$, where $S$ acts as $(-1{)}^{pq}$ times the exchange of the two factors, and under the unit identification ${\mathbb{Z}}_X{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_X\cong {\mathbb{Z}}_X$ of [F4] that is multiplication by $(-1{)}^{pq}$.

Clause 1 is [step 1.1]; clause 2 is [step 2.1] together with [step 3.1] and the symmetry and left unitor of [F4] together with $\rho :=\lambda \circ \sigma$; clause 3 is [step 3.2], [step 1.2], [step 1.3] and [step 4.1]. No choice principle is used anywhere. ∎