Talk:Chain duality III (Ex)
From Manifold Atlas
(Difference between revisions)
(2 intermediate revisions by one user not shown) | |||
Line 1: | Line 1: | ||
<wikitex>; | <wikitex>; | ||
− | + | The most interesting part is to check equivariance, say for objects $M\in\mathbb A$. | |
Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$. | Let $\varphi:TM\to M$ be an element of $M\otimes_{\mathbb A}M$. | ||
We have to check the equality of | We have to check the equality of | ||
− | $T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$ | + | $$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$$ |
− | and $F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M) | + | and $$F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M).$$ |
+ | This follows from the commutative diagram | ||
+ | $$\xymatrix{ | ||
+ | T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & | ||
+ | T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} & | ||
+ | T'^2F(M) \ar[d]_{e'_{F(M)}}\\ | ||
+ | FT(M)\ar[r]^{FT\varphi} & | ||
+ | FT^2M\ar[r]^{Fe_M}& | ||
+ | F(M) | ||
+ | }$$ | ||
+ | as the first square commutes by naturality of $G$ and the second one by definition of a | ||
+ | functor of categories with chain duality. | ||
</wikitex> | </wikitex> |
Latest revision as of 12:35, 1 June 2012
The most interesting part is to check equivariance, say for objects .
Let be an element of .
We have to check the equality of
and
This follows from the commutative diagram
as the first square commutes by naturality of and the second one by definition of a functor of categories with chain duality.