Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

384 colin mclartycomponents of this cover have non-empty intersectionU 12 = U 1 ∩ U 2 U 23 = U 2 ∩ U 3 U 31 = U 3 ∩ U 1while the triple intersection is empty:U 123 = U 1 ∩ U 2 ∩ U 3 =∅There is no point in the center where the components would all overlap. Inthe center is a hole. The precise relation between holes and covers is complex.For example the cover {U 1 , U 2 , U 3 } does not reveal the hole inside the torus,which is hidden inside each sleeve. Other covers reveal that one. CohomologysummarizescoversofaspaceM to produce cohomology groups H n (M) whichare close kin to the homology groups H n (M). The standard tool for it issheaves.On one definition, a sheaf on a space M is another space with a map S → Mwhere S is a union of partially overlapping partial covers of M, whichmay(or may not) patch together in many different ways to form many covers.¹⁴Another definition says a sheaf F on M assigns a set F(U) to each opensubset U ⊆ M, thought of heuristically as a set of ‘patches’ covering U. Thereare compatibility conditions among the sets over different open subsets of Mwhich we will not detail. If the sets F(U) are Abelian groups in a compatibleway then F is an Abelian sheaf.The cohomology of a space M became an infinite series H 1 , H 2 , ... offunctors from the category Ab M of Abelian sheaves on M to the categoryAb of ordinary Abelian groups. Then the entire series H 1 , H 2 , ... wasconceived as a single object called a δ-functor, which can be defined (up toisomorphism) by its place within a category of series of functors Ab M → Ab.Today’s tools were introduced in one of the most cited papers in mathematics(Grothendieck, 1957). In topology or complex analysis the functorialpatterns are always invoked although lower-level constructions are often takenas defining cohomology. Textbooks on cohomological number theory takethe functorial pattern as definition. Of course the theory of derived functorswas developed to get at geometry, arithmetic, etc. and not to call attentionto itself. So for example Washington on Galois cohomology never mentionsderived functors. But each time he writes ‘for proof see ... ’, the referenceuses them.¹⁵¹⁴ See the light introduction in Mac Lane (1986, pp. 252–256, 351–355).¹⁵ See Huybrechts (2005) on complex geometry, Hartshorne (1977) andMilne(1980) on numbertheory. Compare Washington (1997, pp. 102, 104, 105, 107, 108, 118, 120).