nLab shape via cohesive path ∞-groupoid

Idea

In any cohesive ∞-topos, the shape modality ʃ may be interpreted as sending any cohesive space to its path ∞-groupoid. This follows by the defining adjunction $ʃ \dashv \flat$ from the interpretation of the flat modality as sending any delooping $\mathbf{B} \mathcal{G}$ to the moduli stack for flat ∞-connections with structure ∞-group $\mathcal{G}$:

$$\frac{ ʃ X \xrightarrow{\;\nabla\;} \phantom{\flat} \mathbf{B}\mathcal{G} }{ \phantom{ʃ} X \xrightarrow{ \;\;\; } \flat \mathbf{B}\mathcal{G} } \;\;\;\; { { \text{higher parallel transport} } \atop { \text{higher flat bundles} }\,, }$$

because this identifies morphisms out of $ʃ X$ as higher parallel transport-functors for gauge group $\mathcal{G}$.

Concretely, in the cohesive ∞-topos of smooth ∞-groupoids it is readily checked (dcct, see around here at smooth ∞-groupoid – structures) that $\flat \mathbf{B}\mathcal{G}$ indeed classifies flat ∞-connections in the traditional sense.

It requires more work to show, even in this concrete smooth case, that $ʃ X$ is indeed modeled by a path ∞-groupoid in the traditional sense (say based on the idea of the smooth singular simplicial complex). That this is indeed the case is Prop. below, due to Dmitri Pavlov et al.

This fact has some remarkable consequences, which we develop further below.

Details

Throughout,

(1)$$\begin{aligned} \mathbf{H} &\,\coloneqq\, SmoothGroupoids_\infty \\ & \;\simeq\; Sh_\infty(CartesianSpaces) \;\simeq\; Sh_\infty(SmoothManifolds) \;\hookrightarrow\; PSh_\infty(SmoothManifolds) \end{aligned}$$

denotes the cohesive (∞,1)-topos of smooth ∞-groupoids, which is the hypercomplete (∞,1)-topos over the site of smooth manifolds or equivalently over the dense subsite of Cartesian spaces.

Smooth path $\infty$-groupoid

Definition

(smooth extended simplices)
Write

$$\Delta^\bullet_{smooth} \;\colon\; \Delta \longrightarrow CartesianSpaces \xhookrightarrow{\;\;\y\;\;} \mathbf{H}$$

for the cosimplicial object of smooth extended simplices, hence with

$$\Delta^n_{smth} \;\coloneqq\; \Big\{ (x_0, \cdots, x_n) \,\in\, \mathbb{R}^{n+1} \,\left\vert\, \sum_i x_i \,=\, 1 \right. \Big\} \;\subset\; \mathbb{R}^{n+1}$$

and with co-degeneracy and co-face maps given by addition of consecutive variables, and by insertion of zeros, respectively.

Definition

(path ∞-groupoid)
For $A \,\in\, SmoothGroupoids_\infty = Sh_\infty(CartesianSpaces)$, write

(2)$$\begin{aligned} Sing(A) & \;\coloneqq\; \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim} X \big( \Delta^n_{smth} \big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim} \mathbf{H} \big( \Delta^n_{smth}, X \big) \; \;\;\; \in Groupoids_\infty \end{aligned}$$

for the homotopy colimit (in ∞-groupoids) of the evaluation of $A$ (as an (∞,1)-presheaf) on the smooth extended simplices (Def. ).

Here in the second line we recall, just for emphasis, how, under the (∞,1)-Yoneda lemma, these values are the ∞-groupoids of the $\Delta^n$-shaped plots of the generalized smooth space $X$.

Example

For $X \,\in\, DiffeologicalSpaces \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$ a diffeological space (in particular: a smooth manifold) its path ∞-groupoid in the sense of Def. ) is its ordinary smooth singular simplicial complex (e.g. Christensen & Wu 13, Def. 4.3).

This follows by

1. the fully faithful embedding of diffeological spaces as the concrete 0-truncated objects (see here);

2. the fact that every simplicial set is the homotopy colimt over its cells.

Equivalence to smooth shape modality

Proposition

(smooth shape modality given by smooth path ∞-groupoid)

For $A \,\in\, SmoothGroupoids_\infty = Sh_\infty(SmthMfd)$, the smooth path $\infty$-groupoid, namely the (∞,1)-presheaf

$$\mathbf{Sing}(A) \;\colon\; U \;\mapsto\; Sing \big( [U,A] \big) \;=\; \underset{\underset{[n] \in \Delta^n_{smoth}}{\longrightarrow}}{lim} \Big( A \big( \Delta^n_{smth} \times U \big) \Big)$$

of smoothly parameterized path $\infty$-groupoids in $A$ (Def. ),

1. is in fact an (∞,1)-sheaf

   (3)$$\mathbf{Sing}(A) \;\in\; Sh_\infty(SmthMfds) \xhookrightarrow{\;\;\;} PSh_\infty(SmthMfds)$$

2. as such is equivalent to the shape of $A$:

   (4)$$ʃ A \;\simeq\; \mathbf{Sing}(A) \,,$$

3. which means in particular (by $ʃ \simeq Disc \circ Shape$) that it is equivalent to the plain path ∞-groupoid from Def. , equipped with discrete smooth structure:

   (5)$$\mathbf{Sing}(A) \;\simeq\; Disc \circ Sing(A) \,,$$

   hence, with (4), that

   (6)$$ʃ A \;\simeq\; Disc \circ Sing(A) \,.$$

This is due to Pavlov et al. 2019, Thm. 1.1 (see discussion here), announced in Pavlov 2014, Thm. 0.2. The analogous statement for sheaves of spectra (which follows essentially formally) was observed in Bunke-Nikolaus-Völkl 2013, Lem. 7.5 (see at differential cohomology hexagon for more on this case). The particular conclusion (6) is also claimed as Bunk 2020, Prop. 3.6 with Prop. 3.11.

Consequences

Smooth Oka principle

The following may be compared (SS21, p. 7) to the Oka principle in complex analytic geometry, now seen in higher differential topology:

Proposition

(smooth Oka principle) For

• $X \,\in\, SmoothManifolds \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$ a smooth manifold regarded as a smooth ∞-groupoid,

• $A \,\in\, SmoothGroupoids_\infty$ any smooth ∞-groupoid

we have a natural equivalence

(7)$$\big[ X, \, ʃ A \big] \;\; \simeq \;\; ʃ [X,A] \;\;\;\;\; \in \; SmoothGroupoids_\infty$$

between the mapping stack from $X$ to the shape of $A$ and the shape of the mapping stack into $A$ itself.

For precursor discussion see nForum comment 55210 (Nov 2015).

The following proof (from June 2021, here) as a direct consequence of Prop. is that appearing in SS21, §3.3.1. We later learned that the statement also appears as Clough 2021, Theorem B/Cor. 6.4.8, where it is proven independently, now implying a proof of Prop. .

Proof

For any $U \in SmoothManifolds$ consider the following sequence of natural equivalences of values (∞-groupoids) of (∞,1)-presheaves on SmoothManifolds (1):

$$\begin{aligned} [X, ʃA](U) & \;\simeq\; Sh_\infty(SmthMfds) \big( X \times U, ʃA \big) \\ & \;\simeq\; PSh_\infty(SmthMfds) \Big( X \times U, \, \big( U' \,\mapsto\, \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} A(U' \times \Delta^n_{\mathrm{smth}}) \big) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( X \times U, \, \big( U' \,\mapsto\, A(U' \times \Delta^n_{\mathrm{smth}}) \big) \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( X \times U, \, [\Delta^n_{\mathrm{smth}}, A] \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} PSh_\infty(SmthMfds) \Big( \Delta^n_{\mathrm{smth}} \times X \times U, \, A \Big) \\ & \;\simeq\; \underset{\underset{[n] \in \Delta^{\mathrm{op}}}{\longrightarrow}}{\mathrm{lim}} \big( [X,A]( \Delta^n_{\mathrm{smth}} \times U ) \big) \\ & \;\simeq\; \big( ʃ [X,A] \big)(U) \,. \end{aligned}$$

Here

• the first step uses the closed monoidal structure on presheaves;

• the second step is Prop. ;

• the third step uses that with $X \in SmoothManifolds$, by assumption, and with $\Delta^n_{smth} \in CartesianSpaces \hookrightarrow SmoothManifolds$ by Def. , also the Cartesian product $X \times \Delta^n_{smth} \,\in\, SmoothManifolds \xhookrightarrow{y} PSh_\infty(SmoothManifolds)$ is still representable, so that by the (∞,1)-Yoneda lemma, we may compute the colimit of presheaves objectwise;

• the fourth just makes explicit the internal hom in order to make manifest another appeal to the closed monoidal structure on presheaves in the fifth and sixth step;

• the last step applies again Prop. .

Since the composite of these equivalences is still natural in $U$, the statement (7) follows by the (∞,1)-Yoneda embedding.

Classification of principal $\infty$-bundles

Definition

For $\mathcal{G} \,\in\, Groups(SmoothGroupoids_\infty)$ any smooth ∞-group, write

$$B \mathcal{G} \;\; \coloneqq \;\; ʃ \mathbf{B} \mathcal{G} \;\; \simeq \mathbf{B} ʃ \mathcal{G}$$

for the shape of its delooping, equivalently (since shape preserves the simplicial (∞,1)-colimits and the finite products involved in defining the delooping as the realization of the Cech nerve of the effective epimorphism $\ast \to \mathbf{B}\mathcal{G}$) the delooping of its shape.

Remark

Notice that the delooping $\mathbf{B}\mathcal{G} \in \mathbf{H}$ of an ∞-group $\mathcal{G} \in Groups(\mathbf{H})$ is the moduli ∞-stack of $\mathcal{G}$-principal ∞-bundles, in that (NSS 12) for $X \in \mathbf{H}$ there is a natural equivalence (of ∞-groupoids)

(8)$$\mathcal{G}PrinBund_X \;\; \simeq \;\; \mathbf{H}(X,\mathbf{B}\mathcal{G}) \;\;\; \in \; Groupoids_\infty \,.$$

In particular, upon 0-truncation $\tau_0$, this means that the moduli stack $\mathbf{B}\mathcal{G}$ classifies equivalence classes of principal ∞-bundles:

$$\big(\mathcal{G}PrinBund_X\big)_{/\sim_{equ}} \;\; \simeq \;\; \tau_0\, \mathbf{H}(X,\mathbf{B}\mathcal{G}) \;\;\; \in \; Sets \,.$$

But for this reduced information much less than the full moduli stack may be necessary: A classifying space for $\mathcal{G}$-principal ∞-bundles is a discrete object in $Groupoids_\infty \xhookrightarrow{Disc} \mathbf{H}$, such that homotopy classes of maps into it still correspond to equivalence classes of principal ∞-bundles, at least over suitable domains (traditionally: paracompact topological spaces, hence in particular smooth manifolds).

Slightly coarser than plain equivalence classes of principal $\infty$-bundles are concordance classes of principal $\infty$-bundles (in fact, often the two notions coincide, see e.g. Roberts-Stevenson 16, Cor. 15):

Definition

(concordance classes of principal ∞-bundles)
For $\mathcal{G} \in Groups(SmoothGroupoids)_\infty$ and any $X \in SMoothGroupoids_\infty$,
we say that a concordance between two principal ∞-bundles

$$P_0, P_1 \;\in\; \mathcal{G}PrinBund(SmoothGroupoids|infty)_X$$

is a principal $\infty$-bundle on the cylinder over $X$

$$\widehat P \;\in\; \mathcal{G}PrinBund_{X \times \mathbb{R}}$$

whose restriction to the points $0,1 \,\in\, \mathbb{R}$ is equivalent to the given pair of bundles:

$${\widehat P}_{\vert X \times \{0\}} \;\simeq\; P_0 \;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\; {\widehat P}_{\vert X \times \{1\}} \;\simeq\; P_1 \,.$$

We write

(9)$$\big( \mathcal{G}PrinBund_X \big)_{/\sim_{conc}} \;\; \coloneqq \;\; \big( \mathcal{G}PrinBund_X \big) \big/ \big( \mathcal{G}PrinBund_{X \times \mathbb{R}} \big)$$

for the set of equivalence classes of principal ∞-bundles under this concordance relation.

Proposition

(classifying spaces for smooth principal ∞-bundles up to concordance) For

• any smooth ∞-group $\mathcal{G} \,\in\, Groups(SmoothGroupoids_\infty)$;

• any smooth manifold $X \,\in\, SmoothManifolds \xhookrightarrow{\;\;y\;\;} SmoothGroupoids_\infty$

the space $B \mathcal{G} \,\in\, Groupoids_\infty \xhookrightarrow{Disc} SmoothGroupoids_\infty$ (Def. ) is a classifying space for $\mathcal{G}$-principal ∞-bundles over $X$, up to concordance (Def. ), in that we have a natural bijection:

$$\big( \mathcal{G}PrinBund_X \big)_{/\sim_{conc}} \;\; \simeq \;\; \tau_0 \, \mathbf{H} \big( X,\, B \mathcal{G} \big) \,.$$

Proof

This follows as the composition of the following sequence of natural bijections:

$$\begin{aligned} \tau_0 \, \mathbf{H} \big( X, \, B \mathcal{G} \big) & \;\simeq\; \tau_0 \, Pnts \, \big[ X, \, B \mathcal{G} \big] \\ & \;\simeq\; \tau_0 \, Pnts \, \big[ X, \, ʃ \, \mathbf{B}\mathcal{G} \big] \\ & \;\simeq\; \tau_0 \, Pnts \, ʃ \, \big[ X, \, \mathbf{B}\mathcal{G} \big] \\ & \;\simeq\; \tau_0 \, Pnts \, Disc \, \mathrm{Sing} \big[ X, \, \mathbf{B}\mathcal{G} \big] \\ & \;\simeq\; \tau_0 \, \mathrm{Sing} \big[ X, \, \mathbf{B}\mathcal{G} \big] \\ & \;\simeq\; \tau_0 \, \underset{\longrightarrow}{\mathrm{lim}} \Big( \big[ X, \, \mathbf{B}\mathcal{G} \big] (\Delta^\bullet_{\mathrm{smth}}) \Big) \\ & \;\simeq\; \tau_0 \, \underset{\longrightarrow}{\mathrm{lim}} \Big( \mathbf{H} \big( X \times \Delta^\bullet_{\mathrm{smth}} \, \mathbf{B}\mathcal{G} \big) \Big) \\ & \;\simeq\; \tau_0 \, \underset{\longrightarrow}{\mathrm{lim}} \big( {\mathcal{G}}PrinBund_{X \times \Delta^\bullet_{\mathrm{smth}}} \big) \\ & \;\simeq\; \underset{\longrightarrow}{\mathrm{lim}} \, \tau_0 \big( {\mathcal{G}}PrinBund_{X \times \Delta^\bullet_{\mathrm{smth}}} \big) \\ & \;\simeq\; \Big( \tau_0 \big( {\mathcal{G}}PrinBund_{ X } \big) \Big) \big/ \Big( \tau_0 \big( {\mathcal{G}}PrinBund_{X \times \Delta^1_{\mathrm{smth}}} \big) \Big) \\ & \;\simeq\; \big( {\mathcal{G}}PrinBund_X \big)_{\sim_{\mathrm{conc}}} \end{aligned}$$

Here:

• the first step is the relation between the (∞,1)-categorical hom-space and the internal hom via the Points-functor $Pts(X) \coloneqq \mathbf{H}(\ast,X)$;

• the second step is the definition of the classifying space (Def. );

• the third step is the smooth Oka principle (Prop. );

• the fourth step is the relation $ʃ \;\simeq\; Disc \circ Sing$ (5) from Prop. ;

• the fifth step is the relation $Pnts \circ Disc \simeq id$, valied in any cohesive (∞,1)-topos;

• the sixth step is (2) from Def. ;

• the seventh step is the formula for the internal hom by the closed monoidal structure on presheaves;

• the eighth step is the modulation of principal $\infty$-bundles by their moduli stack (8);

• the ninth step uses that n-truncation is a left adjoint (∞,1)-functor and hence commutes with (∞,1)-colimits;

• the tenth step uses that the inclusion of the first two face maps into the opposite of the simplex category is a final functor (by this example, noting that this happens in the 1-category of 0-truncated ∞-groupoids, hence in Sets);

• the eleventh step invokes the definition (9) of concordance of principal $\infty$-bundles (Def. ).

Related concepts

• concordance

• higher parallel transport

References

The identification in Prop. is due to

• Dmitri Pavlov, Structured Brown representability via concordance, 2014 (pdf, pdf)

• Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves (arXiv:1912.10544)

with a precursor observation in:

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Lem. 7.5 of: Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures 11 1 (2016) (arXiv:1311.3188, doi:10.1007/s40062-014-0092-5)

The particular conclusion (6) is also claimed in:

• Severin Bunk, Section 3 of: The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces, J. Homotopy Relat. Struct. (2022) [arXiv:2007.06039, doi:10.1007/s40062-022-00318-7]

The “smooth Oka principle” is also proven (Theorem B) in:

• Adrian Clough, A Convenient Category for Geometric Topology, PhD thesis, UT Austin (2021) [pdf, pdf]

  The homotopy theory of differentiable sheaves [arXiv:2309.01757]

Talk notes:

• Adrian Clough, The smooth Oka principle, talk at CQTS (Oct 2022) [pdf]

The special case of the smooth path $\infty$-groupoid (Def. ) applied to diffeological spaces (among all smooth $\infty$-groupoids), was considered also in:

• J. Daniel Christensen, Enxin Wu, Def. 4.3 of: The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects, New York J. Math. 20 (2014), 1269-1303 (arXiv:1311.6394)

Application of the theorem to a (orbi-)smooth Oka principle and to the classification of equivariant principal bundles:

• Hisham Sati, Urs Schreiber, Equivariant principal $\infty$-bundles $[$arXiv:2112.13654$]$