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    • CommentRowNumber1.
    • CommentAuthorjin
    • CommentTimeMay 30th 2022
    • (edited May 30th 2022)

    I like the nPOV that regards ordinary cohomology H n(X;)H^n(X;\mathbb{Z}) as the derived hom space π 0Mor(X,B nZ)\pi_0 Mor(X, B^n Z) in the (,1)(\infty,1)-topos of simplicial sheaves over XX.

    How about a generalized (Eilenberg-Steenrod) cohomology EE, which includes topological K-theory, elliptic cohomology and cobordism cohomology theory.. etc? Can one find a suitable category CC such that all such EE satisfies

    E(X)Mor C(X,E¯),E(X) \sim Mor_{C}(X, \overline{E}),

    for some CC-object E¯\overline{E}?

    If not, is it true that any such EE is at least the limit of some spectral sequence of ordinary sheaf cohomology?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022
    • (edited May 30th 2022)

    That’s the Brown representability theorem: Whitehead-generalized cohomology theories are those whose classifying spaces are (stages of) spectra.

    So this doesn’t even need sheaves yet. If one does add sheaves to this picture, one gets Whitehead-generalized differential cohomology theories (differential K-theory, etc.)

    Some survey of this and related facts in the hom-space perspective on cohomology is in Section 2 (pp. 9) of The Character Map in Twisted Non-Abelian Cohomology

    • CommentRowNumber3.
    • CommentAuthorjin
    • CommentTimeJun 10th 2022

    Thanks for your comment, Urs. I understand that K()K(-) is a representable functor by, say, KUKU. But by “doesn’t even need sheaves yet” do you mean that it automatically gives a sheaf-theoretic description, perhaps by looking at the sheaf UKU(U)U \mapsto KU(U)?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2022
    • (edited Jun 10th 2022)

    Let’s introduce some notation to make this clearer:


    KU 0×BUʃFredGrpd KU_0 \,\simeq\, \mathbb{Z} \times B U \,\simeq\, ʃ Fred \;\in\; Grpd_\infty

    one means the bare homotopy type of the classifying space for complex K-theory.

    So for XX any topological space with ʃXʃ X \,\in\, Grpd Grpd_\infty its underlying homotopy type, we have that the K-theory of XX is (the homotopy type of) the mapping space (I am using notation as on pp. 37 of arXiv:2008.01101 and pp. 6 in arXiv:2112.13654):

    K(X)=KU 0(X)π 0Maps(ʃX,KU 0). K(X) \;=\; KU^0(X) \;\simeq\; \pi_0 Maps\big( ʃ X,\, KU_0 \big) \,.

    No sheaves so far. And this generalizes: For every Whitehead-generalized cohomology theory EE there is a spectrum E E_\bullet such that (assume XX is compact, if you like)

    E 0(X)π 0Maps(ʃX,E 0). E^0(X) \;\simeq\; \pi_0 Maps\big( ʃ X,\, E_0 \big) \,.

    No sheaves have been invoked anywhere yet. This all happens in the classical homotopy category.

    But now assume that XX is equipped with the structure of a smooth manifold. This canonically makes it an object in the cohesive \infty-topos

    SmthGrpd Sh (CartSp) SmthGrpd_\infty \;\coloneqq\; Sh_\infty\big( CartSp\big)

    of smooth \infty-groupoids.

    The shape modality

    ʃ:SmthGrpd ShpGrp DscSmthGrpd ʃ \;\colon\; SmthGrpd_\infty \xrightarrow{ \;\; Shp \;\; } Grp_\infty \xhookrightarrow{ \;\; Dsc \;\; } SmthGrpd_\infty

    still remembers the underlying bare homotopy type ʃXʃ X of XX, but XX is much richer now.

    In particular, there exists a sheaf of spectra KU^ Spectra(SmthGrp )\widehat {KU}_\bullet \,\in\, Spectra\Big(SmthGrp_\infty\Big) whose shape, in turn, is still the plain classifying homotopy type from above

    ʃKU^ KU , ʃ \widehat {KU}_\bullet \;\simeq\; KU_\bullet \,,

    but which represents differential K-theory K^(X)\widehat{K}(X) in that

    K^(X)=KU^ 0(X)π 0ʃMaps(X,KU^ 0)π 0SmthGrp (X,KU^). \widehat{K}(X) \;=\; \widehat{KU}{}^0(X) \;\simeq\; \pi_0 ʃ Maps\big( X ,\, \widehat{KU}_0 \big) \;\simeq\; \pi_0 SmthGrp_\infty\big( X ,\, \widehat{KU}\big) \,.

    Here on the right we have the shape of the mapping stack, equivalently the \infty-categorical hom-space of smooth \infty-groupoids.

    More generally, for every spectrum E E_\bullet one may find a sheaf of spectra E^ Spectra(SmthGrp )\widehat{E}_\bullet \,\in\, Spectra\big( SmthGrp_\infty \big) such that

    E^ 0(X)π 0ʃMaps(X,E^ 0)π 0SmthGrpd(X,E^ 0) \widehat{E}{}^0(X) \;\simeq\; \pi_0 ʃ Maps\big( X,\, \widehat{E}_0 \big) \;\simeq\; \pi_0 SmthGrpd\big( X,\, \widehat{E}_0 \big)

    is a corresponding differential cohomology.

    Finally, to actually answer your last question: The plain cohomology is still available in the differential context:

    E 0(X)π 0SmthGrpd (X,ʃE^ 0)π 0Grpd (ʃX,E 0). E^0(X) \;\simeq\; \pi_0 SmthGrpd_\infty\big(X,\, ʃ \widehat{E}_0 \big) \;\simeq\; \pi_0 Grpd_\infty\big( ʃX,\, E_0 \big) \,.

    Here the term in the middle may be understood as forming the sheaf hypercohomology of XX with coefficients in the locally constant \infty-stack ʃE^ 0ʃ \widehat{E}{}^0.

    For more on all this see also the differential cohomology hexagon.

    For a quick but complete account of the actual technical details see pp. 106 and then pp. 65 in arXiv:2009.11909.

    (Of course all this is also in dcct (schreiber), but the above references are more polished.)

    • CommentRowNumber5.
    • CommentAuthorjin
    • CommentTimeJun 18th 2022

    I appreciate for your thorough answer. I am one step closer to understand (the point of) cohesion.

    From the answer I could sense two possible generalizations, which you must know if true.

    1. Does any Whitehead-generalized cohomology theory EE has a differential version (i.e. a corresponding sheaf of spectra in Spectra(SmthGrp )Spectra(SmthGrp_{\infty})?

    2. The ordinary cohomology theory and the K-theory extend to their differential versions from Grpd\infty-Grpd to SmoothGrpd )SmoothGrpd_{\infty}). In the previous sentence, what are all of the other known variations of SmoothGrpd SmoothGrpd_{\infty} for the (generalized) cohomology theories to extend? It would be fantastic if it is true for any local model [1] that is reasonable enough.

    [1] By a local model I mean something like the topological n\mathbb{R}^n (corresponding to topological theories) and the smooth n\mathbb{R}^n (corresponding to differential theories).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2022
    • (edited Jun 18th 2022)
    1. Does any Whitehead-generalized cohomology theory EE has a differential version (i.e. a corresponding sheaf of spectra in Spectra(SmthGrp )Spectra(SmthGrp_{\infty})?

    Oh yes, all of them do. Under mild conditions there is a canonical such, in general there are many. Lecture notes include Section 4 “Differential extensions of generalized cohomology theories” (pp. 71) in Uli Bunke’s arXiv:1208.3961.

    In the previous sentence, what are all of the other known variations of SmoothGrpd ∞SmoothGrpd_{\infty} for the (generalized) cohomology theories to extend?

    Here I am not sure if I understand what you have in mind. Do you mean differential cohomology in \infty-toposes other than that of smooth \infty-groupoids? In that case, one answer is:

    Yes, this holds whenever there is an analog of the Poincaré-lemma/de Rham theorem. Namely, this enters in constructing a differential cohomology theory in “Hopkins-Singer style” (referring to Hopkins and Singer 2005, which started the topic) as a homotopy fiber product of a bare spectrum (over its Chern-Dold character) with the sheaf of de Rham complexes valued in a chain complex model for the real-ification of the spectrum (e.g. Example 4.29 on p. 68 here).

    For example, one can use complex analytic ∞-groupoids. The differential Whitehead-generalized cohomology theories in this case were discussed already in Quick & Hopkins arXiv:1212.2173.