Chain duality III (Ex)

Let $F : \Aa \rightarrow \Aa'$$<wikitex>; Let $F : \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment $$ 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 $C \in \Bb(\Aa)$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]F : \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment

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

given by

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

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

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

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