Talk:Chain duality III (Ex)
(Difference between revisions)
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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 | ||
+ | $$\xymatrix{ | ||
+ | T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & | ||
+ | T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} & | ||
+ | T'^2F(M) \ar[d]_{e'_{F(M)}}\\ | ||
+ | FT(M)\ar[r]^{FT\varphi} & | ||
+ | FT^2M\ar[r]^{Fe_M}& | ||
+ | F(M) | ||
+ | }.$$ | ||
</wikitex> | </wikitex> |
Revision as of 11:32, 1 June 2012
We check this for objects .
Let be an element of .
We have to check the equality of
and
This follows from the commutative diagram