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In the entry history of cohomology with local coefficients mention is made of Reidemeister’s use of cochains on the universal cover taken with the action of the fundamental group. A bit later on it is written:
Steenrod in 1942 independently generalized the concept of ordinary cohomology to cohomology with coefficients in a local system of groups, later known as cohomology with local coefficients. In 1943, he acknowledged Reidemeister’s precedence showing that (after a suitable interpretation) the theory includes Reidemeister’s concept of Überdeckung.
I do not want to make any contentious point here, but would observe the Steenrod wrote the review of Reidemeister’s book for the BAMS in 1939. Presumably the idea germinated and adapted unseen to be applied in the slightly different context 3 years later. This increases my wariness as to attribution of great and fundamental ideas on the basis of their ’invention’/ discovery by X rather than Y. Unfortunately the citations index culture is probably here to stay! (Perhaps we should delete ‘’independently’. I am not sure.)
On another tack under the same heading, I am reading Isaksen’s treatment of local systems on pro-simplicial sets (in his paper: D. C. Isaksen, A model structure on the category of pro-simplicial sets , Trans. Amer. Math. Soc., 353, (2001), 2805–2841) If anyone has the time I would appreciate a view on his definition of a local system (on a pro-space) as being an object of the colimit of the ind-category of local systems on the individual parts of the pro-space. I think this is alright (and am fairly certain that it is equivalent to a lax-colimit of them) but the lack of use of isomorphisms rather than identities in this seems a bit risky to me.
This problem is useful/important because of the link with étale homotopy theory. The local systems seem to be over some sort of virtual fundamental groupoid that need not have any points/ objects. It looks in need of an overhaul but I do not see what overhaul and thus how t present it in what I am writing in a fairly elementary fashion.
I checked up. In Steenrod’s paper of 1942, he does not cite Reidemeister, but in a foot note says: It has come to the author’s attention that “local coefficients” and Reidemeister’s ’U”berdeckungen” (Topologie der Polyeder, Leipzig, 1938) are equivalent concepts.
That is strange except (that like me) perhaps when reviewing a book he did not read every page!:-(
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