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I like the nPOV that regards ordinary cohomology $H^n(X;\mathbb{Z})$ as the derived hom space $\pi_0 Mor(X, B^n Z)$ in the $(\infty,1)$-topos of simplicial sheaves over $X$.
How about a generalized (Eilenberg-Steenrod) cohomology $E$, which includes topological K-theory, elliptic cohomology and cobordism cohomology theory.. etc? Can one find a suitable category $C$ such that all such $E$ satisfies
$E(X) \sim Mor_{C}(X, \overline{E}),$for some $C$-object $\overline{E}$?
If not, is it true that any such $E$ is at least the limit of some spectral sequence of ordinary sheaf cohomology?
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
Thanks for your comment, Urs. I understand that $K(-)$ is a representable functor by, say, $KU$. 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 $U \mapsto KU(U)$?
Let’s introduce some notation to make this clearer:
By
$KU_0 \,\simeq\, \mathbb{Z} \times B U \,\simeq\, ʃ Fred \;\in\; Grpd_\infty$one means the bare homotopy type of the classifying space for complex K-theory.
So for $X$ any topological space with $ʃ X \,\in\,$ $Grpd_\infty$ its underlying homotopy type, we have that the K-theory of $X$ 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):
$K(X) \;=\; KU^0(X) \;\simeq\; \pi_0 Maps\big( ʃ X,\, KU_0 \big) \,.$No sheaves so far. And this generalizes: For every Whitehead-generalized cohomology theory $E$ there is a spectrum $E_\bullet$ such that (assume $X$ is compact, if you like)
$E^0(X) \;\simeq\; \pi_0 Maps\big( ʃ X,\, E_0 \big) \,.$No sheaves have been invoked anywhere yet. This all happens in the classical homotopy category.
But now assume that $X$ is equipped with the structure of a smooth manifold. This canonically makes it an object in the cohesive $\infty$-topos
$SmthGrpd_\infty \;\coloneqq\; Sh_\infty\big( CartSp\big)$The shape modality
$ʃ \;\colon\; SmthGrpd_\infty \xrightarrow{ \;\; Shp \;\; } Grp_\infty \xhookrightarrow{ \;\; Dsc \;\; } SmthGrpd_\infty$still remembers the underlying bare homotopy type $ʃ X$ of $X$, but $X$ is much richer now.
In particular, there exists a sheaf of spectra $\widehat {KU}_\bullet \,\in\, Spectra\Big(SmthGrp_\infty\Big)$ whose shape, in turn, is still the plain classifying homotopy type from above
$ʃ \widehat {KU}_\bullet \;\simeq\; KU_\bullet \,,$but which represents differential K-theory $\widehat{K}(X)$ in that
$\widehat{K}(X) \;=\; \widehat{KU}{}^0(X) \;\simeq\; \pi_0 ʃ Maps\big( X ,\, \widehat{KU}_0 \big) \;\simeq\; \pi_0 SmthGrp_\infty\big( X ,\, \widehat{KU}\big) \,.$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 $E_\bullet$ one may find a sheaf of spectra $\widehat{E}_\bullet \,\in\, Spectra\big( SmthGrp_\infty \big)$ such that
$\widehat{E}{}^0(X) \;\simeq\; \pi_0 ʃ Maps\big( X,\, \widehat{E}_0 \big) \;\simeq\; \pi_0 SmthGrpd\big( X,\, \widehat{E}_0 \big)$is a corresponding differential cohomology.
Finally, to actually answer your last question: The plain cohomology is still available in the differential context:
$E^0(X) \;\simeq\; \pi_0 SmthGrpd_\infty\big(X,\, ʃ \widehat{E}_0 \big) \;\simeq\; \pi_0 Grpd_\infty\big( ʃX,\, E_0 \big) \,.$Here the term in the middle may be understood as forming the sheaf hypercohomology of $X$ with coefficients in the locally constant $\infty$-stack $ʃ \widehat{E}{}^0$.
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.)
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.
Does any Whitehead-generalized cohomology theory $E$ has a differential version (i.e. a corresponding sheaf of spectra in $Spectra(SmthGrp_{\infty})$?
The ordinary cohomology theory and the K-theory extend to their differential versions from $\infty-Grpd$ to $SmoothGrpd_{\infty})$. In the previous sentence, what are all of the other known variations of $SmoothGrpd_{\infty}$ 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 $\mathbb{R}^n$ (corresponding to topological theories) and the smooth $\mathbb{R}^n$ (corresponding to differential theories).
- Does any Whitehead-generalized cohomology theory $E$ has a differential version (i.e. a corresponding sheaf of spectra in $Spectra(SmthGrp_{\infty})$?
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.
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