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1. • CommentRowNumber1.
   • CommentAuthorjin
   • CommentTimeMay 30th 2022
   • (edited May 30th 2022)
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   Author: jin
   Format: MarkdownItexI 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2022
   • (edited May 30th 2022)
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   Author: Urs
   Format: MarkdownItexThat'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](https://arxiv.org/pdf/2009.11909.pdf#page=9)) of *[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Twisted Non-Abelian 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

3. • CommentRowNumber3.
   • CommentAuthorjin
   • CommentTimeJun 10th 2022
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   Author: jin
   Format: MarkdownItexThanks 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 $U \mapsto KU(U)$?

   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 $U \mapsto KU(U)$?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 10th 2022
   • (edited Jun 10th 2022)
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   Author: Urs
   Format: MarkdownItexLet'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 $ʃ X \,\in\,$ [[infinity-groupoids|$Grpd_\infty$]] 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](https://arxiv.org/pdf/2008.01101.pdf#page=37) of [arXiv:2008.01101](https://arxiv.org/abs/2008.01101) and [pp. 6](https://arxiv.org/pdf/2112.13654.pdf#page=6) in [arXiv:2112.13654](https://arxiv.org/abs/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 $$ SmthGrpd_\infty \;\coloneqq\; Sh_\infty\big( CartSp\big) $$ of [[smooth infinity-groupoid|smooth $\infty$-groupoids]]. 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 [[(infinity,1)-categorical hom-space|$\infty$-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](https://arxiv.org/pdf/2009.11909.pdf#page=106) and then [pp. 65](https://arxiv.org/pdf/2009.11909.pdf#page=65) in [arXiv:2009.11909](https://arxiv.org/abs/2009.11909). (Of course all this is also in [[schreiber:dcct]], but the above references are more polished.)

   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

   $$SmthGrpd_\infty \;\coloneqq\; Sh_\infty\big( CartSp\big)$$

   of smooth $\infty$-groupoids.

   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.)

5. • CommentRowNumber5.
   • CommentAuthorjin
   • CommentTimeJun 18th 2022
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   Author: jin
   Format: MarkdownItexI 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 $E$ has a differential version (i.e. a corresponding sheaf of spectra in $Spectra(SmthGrp_{\infty})$? 2. 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).

   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 $E$ has a differential version (i.e. a corresponding sheaf of spectra in $Spectra(SmthGrp_{\infty})$?

   2. 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).

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 18th 2022
   • (edited Jun 18th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 1. 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](https://arxiv.org/pdf/1208.3961.pdf#page=71)) in Uli Bunke's [arXiv:1208.3961](https://arxiv.org/abs/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 [[Quadratic Functions in Geometry, Topology, and M-Theory|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](https://arxiv.org/pdf/2009.11909.pdf#page=68)). 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](https://arxiv.org/abs/1212.2173).

   1. 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.