aa r X i v : . [ m a t h . A T ] F e b A REMARK ON SINGULAR COHOMOLOGY AND SHEAF COHOMOLOGY
DAN PETERSENA
BSTRACT . We prove a comparison isomorphism between singular cohomology and sheaf cohomology.
Let X be a topological space, A an abelian group. We say that X is cohomologically locally connected (with respect to A ) if for all x ∈ X and all k ∈ Z , lim −→ U ∋ x H k sing (U , x ; A) = 0 . The goal of this note is to give a homotopy-theoretic proof of the following result:
Suppose that the topological space X is cohomologically locally connected with respect to A .Then sheaf cohomology and singular cohomology of X with coefficients in A are canonically isomorphic. It is of course well known that singular cohomology coincides with sheaf cohomology for ‘rea-sonable’ spaces. I will offer two excuses for having written this paper:(i) The proof given here seems to me to be the ‘right’ one. Although the argument is not simple inthe sense of being elementary or self-contained, it is simple in that it only uses general principlesand makes the result seem inevitable; there is no cleverness involved.(ii) In the classical literature one finds a proof of this result under the assumption that X is coho-mologically locally connected and paracompact Hausdorff . It is a surprisingly recent theorem ofSella [Sel16] that the result is true also without the assumption that X is paracompact Haus-dorff. However, Sella’s argument is quite intricate, and reading it makes one wonder what is‘really’ going on.We also give a quick argument for the more refined statement that if A is a commutative ring thenthere is a canonical isomorphism of E ∞ -algebras RΓ(X , A) ≃ C • sing (X , A) in the ∞ -category of cochaincomplexes. Let us first of all outline the standard proof of the comparison isomorphism between sheaf andsingular cohomology, which can be found with some variations in many textbooks. Denote by S thecochain complex of presheaves on X which in degree k is given by U C k sing (U , A) , and for a presheaf F denote by F its sheafification. The proof proceeds as follows:(i) Let A be the constant presheaf. Argue that the unit map A → S is a quasi-isomorphism on stalks.(ii) Apply the sheafification functor to get a quasi-isomorphism A → S .(iii) Argue that the sheaves S k are acyclic, so that A → S is an acyclic resolution of the constant sheaf.(iv) Argue that the map Γ(X , S ) → Γ(X , S ) is a quasi-isomorphism. Each of the steps (i), (iii) and (iv) of §0.4 require some point-set hypotheses on X . Step (i) isof course equivalent to X being cohomologically locally connected. Steps (iii) and (iv) require X to be paracompact Hausdorff. For step (iii) a natural idea is to observe that the presheaves S k are The author acknowledges support by ERC-2017-STG 759082 and a Wallenberg Academy Fellowship. The notion of being cohomologically locally connected is classically defined by the stronger condition that for all k ∈ Z ,all x ∈ X and all neighborhoods x ∈ U , there exists a smaller neighborhood x ∈ V ⊂ U such that H k sing (U , x ; A) → H k sing (V , x ; A) is the zero map. The definition used here seems more natural. Sella makes the stronger assumption that X is semi-locally contractible , but his proof only requires X to be cohomologicallylocally connected. asque . Unfortunately, the sheafification of a flasque presheaf is not in general flasque. But one maymoreover note that S k is an epipresheaf , i.e. a presheaf which satisfies gluing, but not unique gluing.Now if F is any epipresheaf and X is paracompact Hausdorff, then one can show that the natural map F(X) → F(X) is surjective [Bre67, Ch. I, Theorem 6.2]. It follows in particular the sheafification of aflasque epipresheaf on a hereditarily paracompact
Hausdorff space is flasque. If X is not hereditarilyparacompact Hausdorff it may not be true that S is flasque, and a slightly more elaborate argument isneeded: namely, it will always be true that S = S is a flasque sheaf of rings, and S k is a module over S . On a paracompact Hausdorff space one knows that flasque sheaves are soft, modules over softsheaves of rings are fine, and fine sheaves are acyclic (see [Bre67, Ch. II, §9]; hence again S is an acyclicresolution. For (iv), one may identify the image of Γ(X , S ) → Γ(X , S ) with the filtered colimit overall open coverings U of X of the complexes of ‘ U -small cochains’. Since X is paracompact Hausdorff, Γ(X , S ) → Γ(X , S ) is onto, and then the result follows from the ‘theorem of small simplices’ [Spa66,Ch. 4, Sect. 4, Theorem 14]. As mentioned in §0.3, Sella proved a comparison isomorphism without assuming X paracompactHausdorff. His approach is to turn S into a sheaf not by means of sheafification, but instead by a verycarefully constructed procedure of taking a colimit over smaller and smaller singular simplices. Itturns out that the result is a resolution of the constant sheaf by products of skyscraper sheaves, whoseglobal sections can be compared to the singular cochains on X . A homotopy theorist, presented with the proof outline of §0.4, might proclaim that step (ii) is‘morally wrong’, and therefore also (iii) and (iv). Indeed one could object that we should not have tosheafify S , since it is already a sheaf . As this statement is obviously false as stated — the presheaves S k are certainly not sheaves — I should explain what it means. The point is that we may think of S asa presheaf taking values in the ∞ -category of cochain complexes (a higher categorical enhancement ofthe classical derived category of abelian groups). There is a well defined notion of what it means fora presheaf taking values in an ∞ -category to be a sheaf, and it turns out that S is always a sheaf inthis higher categorical sense. Let M be a model category (or a complete ∞ -category), and X a topological space. Apresheaf F : Open(X) op → M is called a hypersheaf if for any open hypercover V • → U , the map F(U) −→ holim F(V • ) is a weak equivalence. A typical example of a hypersheaf is given by the functor U RΓ(U , F) , for a fixed sheaf ofabelian groups F . Said differently, Godement’s canonical flabby resolution Gode(F) is a hypersheaf.This fact is well known, but let us give the proof for the reader’s convenience:
Let R be a commutative ring, and let K be a bounded below cochain complex of flabbysheaves of R -modules on X . Then K is a hypersheaf.Proof. Since flabbiness is preserved when restricting to an open subset, it is enough to verify thehypersheaf axiom for an open hypercover ǫ : V • → X . For every such hypercover there is a complexof sheaves ˇC ǫ (K) on X such that holim K(V • ) = Γ(X , ˇC ǫ (K)) , with a map K → ˇC ǫ (K) inducing themap K(X) → holim K(V • ) . Explicitly, ˇC ǫ (K) is the homotopy limit of the cosimplicial object which indegree n is given by K restricted to V n , pushed forward to X . We note in passing that the homotopylimit may be computed as the totalization since every cosimplicial chain complex is Reedy fibrant.We claim now that:(1) K → ˇC ǫ (K) is a quasi-isomorphism, One can also define the weaker notion of an ∞ -sheaf , by imposing the descent condition only for ˇCech covers, ratherthan general hypercovers. It is an insight of Lurie [Lur09] that ∞ -sheaves are for several reasons more fundamental thanhypersheaves; in particular, the ∞ -category of hypersheaves can be recovered from the ∞ -category of ∞ -sheaves by theprocess of hypercompletion. We will not need to consider ∞ -sheaves here, and for presheaves of bounded below cochaincomplexes the two notions are in fact equivalent. ˇC ǫ (K) is a bounded below complex of flabby sheaves,from which the result follows, since bounded below flabby resolutions can be used to compute thederived functor RΓ(X , − ) . For the first point it is enough to check on stalks, which reduces the claim tothe case that X is a point. But a hypercover of a point is a contractible Kan complex and in particularits cohomology is trivial. The second point is straightforward, using that flabbiness of a sheaf ispreserved by taking products, restricting to an open subset, and pushing forward. (cid:3) The presheaf S is a hypersheaf.Proof. Let V • → U be an open hypercover. Then hocolim V • → U is a weak equivalence [DI04,Theorem 1.2], and in particular for any spectrum (or space) E we have that holim E V • ≃ E U . Applythis to the Eilenberg-MacLane spectrum HA . (cid:3) Note in the proof of Proposition 0.11 that by E U we mean the ‘derived’ function spec-trum, which is a weak homotopy invariant of U . If the expression E U were interpreted more naively,as a topological sequential spectrum whose n th component is the space of based maps from U + to E n , then it would not be true that HA U is equivalent to the singular cochains on U without point-sethypotheses on U . A consequence of Proposition 0.11 is a comparison isomorphism between singular and sheaf co-homology. It seems natural to formulate this in terms of the local projective model structure on presheavesof cochain complexes , described e.g. in [Jar03; Hin05; CS19]. It is the differential graded analogue of themore familiar local model structure on simplicial presheaves. The homotopy category of this modelcategory is the classical unbounded derived category of sheaves of R -modules. Let X be a topological space, R a commutative ring. There is a model structure on thecategory of presheaves of unbounded cochain complexes of R -modules on X , such that the weak equivalencesare the maps inducing quasi-isomorphisms on all stalks, and the fibrations are the pointwise surjections whosekernel is a hypersheaf. One can understand the model structure of Proposition 0.14 as follows. The category of un-bounded cochain complexes Ch R is a left proper combinatorial model category, and therefore thesame is true for the category of presheaves on X with values in Ch R , equipped with the projectivemodel structure (which in this context is often called the global projective model structure). It followsfrom Smith’s theory of combinatorial model categories that left proper combinatorial model cate-gories admit left Bousfield localizations at arbitrary sets of maps, and we may in particular form theBousfield localization at a dense set of hypercovers; the result is precisely the local projective modelstructure on presheaves. See Barwick [Bar10], in particular Application IV of Section 4, where theanalogous procedure is carried out for ˇCech covers instead of hypercovers, giving a model categoryof ∞ -sheaves rather than hypersheaves. The fact that the equivalences in this Bousfield localizationare precisely the stalkwise equivalences is not immediate, and is due to Dugger–Hollander–Isaksen[DHI04] in the setting of simplicial presheaves. See [CS19] for the analogous result for cochain com-plexes. Let X be a topological space, cohomologically locally connected with respect to A . There is acanonical isomorphism H • sheaf (X , A) ∼ = H • sing (X , A) .Proof. Consider the constant presheaf A on X . By Proposition 0.10, Proposition 0.11 and Proposi-tion 0.14 we know that both presheaves S and Gode(A) are fibrant replacements of A for the localprojective model structure. The global sections functor is right Quillen, since it is the composite of theright adjoint of the localization functor and the global sections functor on presheaves. So the resultfollows by Ken Brown’s lemma. (cid:3) A reader may wonder where the point-set topology happened in this proof. Surely onecan not prove a theorem in general topology merely by mumbling about hypersheaves. The answer isthat the result of Dugger–Isaksen used in Proposition 0.11 is a genuinely nontrivial point-set theorem: ndeed, their result says that if V • → U is any open hypercover of an arbitrary topological space U , then hocolim V • ≃ U . The conclusion of their result says that the homotopy colimit of a certain diagramcoincides with its usual colimit, which we usually only expect for suitably cofibrant diagrams, whichis far from the case here. The theory of sheaf cohomology has traditionally been approached using the tools ofhomological algebra and derived categories. The homotopy-theoretic approach to sheaf theory usedhere was developed to treat situations where the methods of homological algebra break down, suchas sheaves of simplicial sets or spectra. A full history of the subject would be out of place here butcontributions of [SGA4; Bro73; BG73; Tho85; Jar87; DHI04; Lur09] should be mentioned. This noteillustrates that these methods can be useful and clarifying also in very classical and linear situations.
Suppose that A is a commutative ring. The argument can be jazzed up to prove that RΓ(X , A) and C • sing (X , A) are quasi-isomorphic as E ∞ -algebras in Ch A . In fact they are canonically isomorphicin the associated ∞ -category. One way to prove this fact is model categorical: our argument in anutshell was to consider the zig-zag of weak equivalences of presheaves on X , S ∼ ←− A ∼ −→ Gode(A) , and argue that both S and Gode(A) are fibrant, and can therefore be used to compute cohomology.All that is needed to prove the stronger result is to argue that this is a zig-zag of presheaves of E ∞ -algebras on X , and that both S and Gode(A) are fibrant for a suitable model structure on presheavesof E ∞ -algebras on X . This is indeed possible, but since not all of the details are in an easily quotableform in the literature, we will give a slightly different argument. Recall that the totalization functor from cosimplicial cochain complexes to cochain complexes ismonoidal, but not symmetric monoidal. It is however symmetric monoidal in a homotopical sense:it induces a symmetric monoidal functor between the associated ∞ -categories. A more classicalway of expressing this homotopy symmetricity property uses the Eilenberg–Zilber operad Z of Hinich–Schechtman [HS87], which has the following properties:(i) If O is an operad in cochain complexes, then the totalization of a cosimplicial O -algebra is an O ⊗ Z -algebra.(ii) There is a canonical quasi-isomorphism of operads Z → Com .We remark first of all that these facts explain why S and Gode(A) are presheaves of E ∞ -algebras.Indeed, the singular cochains of any space are manifestly obtained as the totalization of a cosimplicalcommutative ring, hence form a Z -algebra by (i), and by (ii), any Z -algebra may be considered as an E ∞ -algebra. The Godement construction is the composition of a symmetric monoidal functor frompresheaves on X to cosimplicial presheaves on X , and the functor of totalization, which in particularmeans that the Godement construction applied to a presheaf of O -algebras is a presheaf of O ⊗ Z -algebras.But this means now that we may start from the quasi-isomorphism A → S of presheaves of Z -algebras, and apply the Godement functor to obtain a commuting square A Gode(A) S Gode( S ) ∼∼ ∼∼ of quasi-isomorphisms of presheaves of Z ⊗ Z -algebras. All entries in this diagram except the top leftcorner are hypersheaves. It follows that taking global sections gives a zig-zag of quasi-isomorphismsof Z ⊗ Z -algebras, hence of E ∞ -algebras, C • sing (X , A) = Γ(X , S ) ∼ −→ Γ(X , Gode( S )) ∼ ←− Γ(X , Gode(A)) = RΓ(X , A) , which gives the result. EFERENCES [Bar10] Clark Barwick. “On left and right model categories and left and right Bousfield localizations”.
Ho-mology Homotopy Appl. K -theory as generalized sheaf cohomology”.In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) . 1973,266–292. Lecture Notes in Math., Vol. 341.[Bre67] Glen E. Bredon.
Sheaf theory . McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967, pp. xi+272.[Bro73] Kenneth S. Brown. “Abstract homotopy theory and generalized sheaf cohomology”.
Trans. Amer.Math. Soc.
186 (1973), pp. 419–458.[CS19] Utsav Choudhury and Martin Gallauer Alves de Souza. “Homotopy theory of dg sheaves”.
Comm.Algebra
Math. Proc. Cambridge Philos. Soc. A -realizations”. Math. Z.
Adv. Math. K -theory, arith-metic and geometry (Moscow, 1984–1986) . Vol. 1289. Lecture Notes in Math. Springer, Berlin, 1987,pp. 240–264.[Jar03] J. F. Jardine. “Presheaves of chain complexes”. K -Theory J. Pure Appl. Algebra
Higher topos theory . Vol. 170. Annals of Mathematics Studies. Princeton University Press,Princeton, NJ, 2009, pp. xviii+925.[Sel16] Yehonatan Sella. “Comparison of sheaf cohomology and singular cohomology”. Preprint available atarXiv:1602.06674. 2016.[SGA4]
Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos . Lecture Notes in Mathematics,Vol. 269. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin,A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.Springer-Verlag, Berlin-New York, 1972, pp. xix+525.[Spa66] Edwin H. Spanier.
Algebraic topology . McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966,pp. xiv+528.[Tho85] Robert W. Thomason. “Algebraic K -theory and étale cohomology”. Ann. Sci. École Norm. Sup. (4)
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