Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Warning 2.5.6.20. Let $A_{\bullet }$ be a simplicial abelian group, and let $A_{\bullet }^{\operatorname{op}}$ be the opposite simplicial abelian group (obtained by precomposing the functor $A_{\bullet }: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{ Ab }$ with the order-reversal involution $\mathrm{Op}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}}$ of Notation 1.3.2.1). Then there is a canonical isomorphism of Moore complexes $\psi : \mathrm{C}_{\ast }( A^{\operatorname{op}} ) \simeq \mathrm{C}_{\ast }(A )$, given by $\psi (x) = (-1)^{n} x$ for $x \in A_ n$. This isomorphism carries the subcomplex $\mathrm{D}_{\ast }( A^{\operatorname{op}} )$ generated by the degenerate simplices of $A_{\bullet }^{\operatorname{op}}$ to the subcomplex $\mathrm{D}_{\ast }( A )$ generated by the degenerate simplices of $A_{\bullet }$, and therefore descends to an isomorphism of normalized Moore complexes $\mathrm{N}_{\ast }( A^{\operatorname{op}} ) \simeq \mathrm{N}_{\ast }( A )$, where we view $\mathrm{N}_{\ast }(A)$ and $\mathrm{N}_{\ast }(A^{\operatorname{op}})$ as quotients of $\mathrm{C}_{\ast }(A)$ and $\mathrm{C}_{\ast }(A^{\operatorname{op}})$ (as in Construction 2.5.5.7). Beware that the isomorphism $\psi $ does not carry the subcomplex $\widetilde{ \mathrm{N} }_{\ast }(A^{\operatorname{op}}) \subseteq \mathrm{C}_{\ast }(A^{\operatorname{op}})$ of Construction 2.5.6.16 to the subcomplex $\widetilde{ \mathrm{N} }_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A)$. Instead, it carries it carries $\widetilde{ \mathrm{N} }_{\ast }( A^{\operatorname{op}} )$ to a different subcomplex of $\mathrm{C}_{\ast }(A)$, given in degree $n$ by those elements $x \in \mathrm{C}_{n}(A) = A_ n$ satisfying $d_ i(x)$ for $0 \leq i < n$, and with differential given by $x \mapsto (-1)^{n} d_ n(x)$. This subcomplex is yet another incarnation of the normalized Moore complex of $A_{\bullet }$, which is canonically isomorphic to $\widetilde{ \mathrm{N} }_{\ast }(A)$ but not identical as a subcomplex of $\mathrm{C}_{\ast }(A)$.

More informally: the definition of the normalized Moore complex $\mathrm{N}_{\ast }(A)$ as a quotient of $\mathrm{C}_{\ast }(A)$ (via Construction 2.5.5.7) is compatible with passage from a simplicial abelian group $A_{\bullet }$ to its opposite $A_{\bullet }^{\operatorname{op}}$, but the realization as a subcomplex of $\mathrm{C}_{\ast }(A)$ (via Construction 2.5.6.16) is not.