Chain duality II (Ex)
From Manifold Atlas
Let be an additive category with chain duality. Show that
induces an isomorphism (rather than just a chain equivalence) of chain complexes of abelian groups
![\displaystyle T_{M,N} \colon M \otimes_{\Aa} N \rightarrow N \otimes_{\Aa} M.](/images/math/b/f/2/bf291d6d20ed33cdcd2f791ababecd52.png)
Observe that is an involution. Define further
![\displaystyle T_{C,D} \colon C \otimes_{\Aa} D \rightarrow D \otimes_{\Aa} C \quad \textup{by} \quad (T_{C,D})_{p,q} = (-1)^{pq} T_{C_p,D_q}](/images/math/f/1/5/f1536cd409c07557de6cd30c3480ed20.png)
Show that it induces an isomorphism of chain complexes and is an involution.