A remark on singular cohomology and sheaf cohomology


  • Dan Petersen




We prove a comparison isomorphism between singular cohomology and sheaf cohomology.


Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. 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, Lecture Notes in Mathematics, 269, Springer-Verlag, Berlin-New York, 1972

Barwick, C., On left and right model categories and left and right Bousfield localizations, Homology Homotopy Appl. 12 (2010), no. 2, 245–320. http://projecteuclid.org/euclid.hha/1296223884

Bredon, G. E., Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967.

Brown, K. S., Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458. https://doi.org/10.2307/1996573

Brown, K. S. and Gersten, S. M., Algebraic $K$-theory as generalized sheaf cohomology, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266–292. Lecture Notes in Math., 341 Springer, Berlin, 1973.

Choudhury, U. and Gallauer Alves de Souza, M., Homotopy theory of dg sheaves, Comm. Algebra 47 (2019), no. 8, 3202–3228. https://doi.org/10.1080/00927872.2018.1554744

Dugger, D., Hollander, S., and Isaksen, D. C., Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51. https://doi.org/10.1017/S0305004103007175

Dugger, D. and Isaksen, D. C., Topological hypercovers and $mathbb A^1$-realizations, Math. Z. 246 (2004), no. 4, 667–689. https://doi.org/10.1007/s00209-003-0607-y

Elmendorf, A. D., Kriz, I., Mandell, M. A., and May, J. P., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, 47, American Mathematical Society, Providence, RI, 1997,


Hinich, V., Deformations of sheaves of algebras, Adv. Math. 195 (2005), no. 1, 102–164. https://doi.org/10.1016/j.aim.2004.07.007

Hinich, V. A. and Schechtman, V. V., On homotopy limit of homotopy algebras, in $K$-theory, arithmetic and geometry (Moscow, 1984–1986), pp. 240–264, Lecture Notes in Math., 1289, Springer, Berlin, 1987. https://doi.org/10.1007/BFb0078370

Jardine, J. F., Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87. https://doi.org/10.1016/0022-4049(87)90100-9

Jardine, J. F., Presheaves of chain complexes, $K$-Theory 30 (2003), no. 4, 365–420, https://doi.org/10.1023/B:KTHE.0000021707.27987.c9

Lurie, J., Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, Princeton, NJ, 2009. https://doi.org/10.1515/9781400830558

Sella, Y., Comparison of sheaf cohomology and singular cohomology, arXiv:1602.06674, 2016.

Shipley, B., $Hmathbb Z$-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007), no. 2, 351–379. https://doi.org/10.1353/ajm.2007.0014

Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.

Spanier, E. H., Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.

Thomason, R. W., Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552.



How to Cite

Petersen, D. (2022). A remark on singular cohomology and sheaf cohomology: Array. MATHEMATICA SCANDINAVICA, 128(2). https://doi.org/10.7146/math.scand.a-132191