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Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$ between two functors $b_! \circ a^*$ and

$d^* \circ c_!$. Usually this natural isomorphism is called base change.

Suppose we have another pull-back diagram, $d:Y\rightarrow Z$, $f:Z\rightarrow U$, $e:Y\rightarrow V$ and $g:V\rightarrow U$. Then we have another natural isomorphism $\kappa'$ between $e_! \circ d^*$ and

$g^*\circ f_!$.

By the universal property of pull-back, one can see that $a:W\rightarrow X$,$f \circ c:X\rightarrow U$, $e\circ b:W\rightarrow V$ and $g:V\rightarrow U$ is also a pull-back diagram. Denote the corresponding natural isomorphism by $\kappa''$.

Is it true that $\kappa''=\kappa'\circ \kappa$?

Probably the equality is a little confusing, but the formulation is clear if one thinks of it.

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1 Answer

Yes. Inject $f_ {!}$ into $f_ {\ast}$ to convert it into a claim concerning equality among "base change morphisms" (not generally isomorphisms) relating topological pushforward and pullback. Then it is a matter of just looking at the definitions (and how pullback is built as a sheafification of a certain presheaf).

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I don't quite like to think of the pull-back functor as certain construction. I prefer to think of it as the adjoint functor of push-forward $f_*$ . Can you kindly give some reference about proof? Thank you very much. –  xiyu Jun 22 '10 at 17:48
    
@xiyu: OK, so you want a compatibility involving adjunction morphisms and compositions of functors. Please state in such terms what "general nonsense" result you want, and post as an addition to the question (I can guess, but you should figure out what you want). I still encourage you to do the "bare hands" version with the definitions if you didn't; it is fairly straightforward, and I can tell you from experience that direct practice working closely with the explicit definitions is useful for computations arising in theoretical proofs. The adjoint viewpoint is nice, but not always best! –  Boyarsky Jun 22 '10 at 19:18
    
It is really straitforward by using the explicit construction of pull-back(Still I don't want to call it definition). We can construct the base change by adjunction morphisms $id\rightarrow f_*\circ f^*$ and $f^*\circ f_*\rightarrow id$, but it's not easy to check in this point of view. In general, of course I can formulate some general abstract nonsense statement, maybe it's too much and useless. –  xiyu Jun 23 '10 at 10:51
    
@xiyu: yes, I should have said "construction" rather than "definition". At your stage I reached the same conclusion that you have now reached (i.e., easy enough to check directly, so it's not worth getting confused by many commutative diagrams). For some other sites the pullback construction is more complicated than in topology, but the spirit is the same and so the direct verification easily carries over. In the land of abstract nonsense with adjunctions there is a useful little compatibility relating the pair of adjunction morphisms for which Hilton-Stammbach gives a nice proof. –  Boyarsky Jun 23 '10 at 18:39
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