# Chain duality III (Ex)

Let $F \colon \Aa \rightarrow \Aa'$$; Let F \colon \Aa \rightarrow \Aa' be a functor of additive categories with chain duality. Show that the assignment M \otimes_{\Aa} N \rightarrow F(M) \otimes_{\Aa'} F(N) given by \varphi \colon TM \rightarrow N \quad \mapsto \quad F (\varphi) \circ G(M) \colon T' F (M) \rightarrow FT(M) \rightarrow F(N) induces a \Zz_2-equivariant chain map C \otimes_{\Aa} C \rightarrow F(C) \otimes_{\Aa'} F(C) for any C \in \Bb(\Aa). == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]F \colon \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment

$\displaystyle M \otimes_{\Aa} N \rightarrow F(M) \otimes_{\Aa'} F(N)$

given by

$\displaystyle \varphi \colon TM \rightarrow N \quad \mapsto \quad F (\varphi) \circ G(M) \colon T' F (M) \rightarrow FT(M) \rightarrow F(N)$

induces a $\Zz_2$$\Zz_2$-equivariant chain map

$\displaystyle C \otimes_{\Aa} C \rightarrow F(C) \otimes_{\Aa'} F(C)$

for any $C \in \Bb(\Aa)$$C \in \Bb(\Aa)$.