Talk:Pullback

Page merge?
Shouldn't this page be merged with Pullback (category theory)? The idea that a pullback is just precomposition is just an example of a pullback in the correct category. — Preceding unsigned comment added by ParanoidLepton (talk • contribs) 15:04, 1 October 2023 (UTC)

Cohomology
Algebraic topology is full of pullbacks in cohomology theories, but I haven't been able to find a good reference on wikipedia. Can someone help? Geometry guy 01:07, 10 February 2007 (UTC)
 * JP May, "A concise course in algebraic topology" although that book can be (very) hard to read if you don't already know algebraic topology. Its written at the genius-level, designed to wash-out first-year students. 67.198.37.16 (talk) 17:24, 7 May 2016 (UTC)


 * Here, Novikov, "Topology I General Survey" is extremely pleasant to read, it defines Pullback (cohomology) on page 48 (without actually giving it that name) so if Hom(C(K),G) are the homomorphisms from a chain complex C(K) to an abelian group G, and if f a map from C(K) to C(K') then $$f^*$$ is the pullback $$f^*:Hom(C(K'),G)\to Hom(C(K),G)$$ The standard notation for Hom(C(K),G) is C(K;G) and for the boundary operator $$\partial:C_n(K)\to C_{n-1}(K)$$ one has the pullback $$\partial^*:C_{n-1}(K;G)\to C_n(K;G)$$ and so from there one trivially defines the cohomology groups just like the homology groups. The above 2-sentence summary could be a foundation for the article Pullback (cohomology) as a stub, but I don't have the authority to create such an article from that red-link. 67.198.37.16 (talk) 18:08, 7 May 2016 (UTC)


 * Since I'm a registered Wikipedia editor, I took a liberty of starting a stub. I admit its exposition is not optimal but is a good start, I think. -- Taku (talk) 23:43, 11 June 2016 (UTC)

Designation of precomposition pullbacks
I have just double checked about the allegedly used term omitting pullback. The given reference says

"We have our usual Pavlovian response to seeing something equal to zero (and our usual abuse of notation omitting pullbacks in the notation here and in what follows), so we expand out ..."

In my opinion, this has nothing to do with designation, but is meant to be a descriptive explanation of how this precomposition pullback is handled. I hope it is okay to delete this. Thank you.