# nLab shape via cohesive path ∞-groupoid

Contents

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## 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

### 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

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

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 to the Oka principle in complex analytic geometry.

###### Proposition

(smooth Oka principle) For

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.

###### 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

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:

## References

The identification in Prop. is due to

with a precursor observation in:

The particular conclusion (6) is also claimed in:

• Severin Bunk, Section 3 of: The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces (arXiv:2007.06039)

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

Last revised on July 28, 2021 at 17:14:33. See the history of this page for a list of all contributions to it.