nLab shape via cohesive path ∞-groupoid




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 ʃʃ \dashv \flat from the interpretation of the flat modality as sending any delooping B𝒢\mathbf{B} \mathcal{G} to the moduli stack for flat ∞-connections with structure ∞-group 𝒢\mathcal{G}:

ʃXB𝒢ʃXB𝒢higher parallel transporthigher flat bundles, \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 ʃXʃ X as higher parallel transport-functors for gauge group 𝒢\mathcal{G}.

Concretely, in the cohesive ∞-topos of smooth ∞-groupoids it is readily checked (dcct, see around here at smooth ∞-groupoid – structures) that B𝒢\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 ʃXʃ 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.



(1)H SmoothGroupoids Sh (CartesianSpaces)Sh (SmoothManifolds)PSh (SmoothManifolds) \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


(smooth extended simplices)

Δ smooth :ΔCartesianSpacesyH \Delta^\bullet_{smooth} \;\colon\; \Delta \longrightarrow CartesianSpaces \xhookrightarrow{\;\;\y\;\;} \mathbf{H}

for the cosimplicial object of smooth extended simplices, hence with

Δ smth n{(x 0,,x n) n+1| ix i=1} n+1 \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.


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

(2)Sing(A) lim[n]Δ opX(Δ smth n) lim[n]Δ opH(Δ smth n,X)Groupoids \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 AA (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 Δ n\Delta^n-shaped plots of the generalized smooth space XX.


For XDiffeologicalSpacesySmoothGroupoids 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


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

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

Sing:USing([U,A])=lim[n]Δ smoth n(A(Δ smth n×U)) \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 AA (Def. ),

  1. is in fact an (∞,1)-sheaf

    (3)Sing(A)Sh (SmthMfds)PSh (SmthMfds) \mathbf{Sing}(A) \;\in\; Sh_\infty(SmthMfds) \xhookrightarrow{\;\;\;} PSh_\infty(SmthMfds)
  2. as such is equivalent to the shape of AA:

    (4)ʃASing(A), ʃ A \;\simeq\; \mathbf{Sing}(A) \,,
  3. which means in particular (by ʃDiscShapeʃ \simeq Disc \circ Shape) that it is equivalent to the plain path ∞-groupoid from Def. , equipped with discrete smooth structure:

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

    hence, with (4), that

    (6)ʃADiscSing(A). 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.


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:


(smooth Oka principle) For

we have a natural equivalence

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

between the mapping stack from XX to the shape of AA and the shape of the mapping stack into AA itself.

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

The following proof (from June 2021, here) 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.


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

[X,ʃA](U) Sh (SmthMfds)(X×U,ʃA) PSh (SmthMfds)(X×U,(Ulim[n]Δ opA(U×Δ smth n))) lim[n]Δ opPSh (SmthMfds)(X×U,(UA(U×Δ smth n))) lim[n]Δ opPSh (SmthMfds)(X×U,[Δ smth n,A]) lim[n]Δ opPSh (SmthMfds)(Δ smth n×X×U,A) lim[n]Δ op([X,A](Δ smth n×U)) (ʃ[X,A])(U), \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}


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

Classification of principal \infty-bundles


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

B𝒢ʃB𝒢Bʃ𝒢 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 *B𝒢\ast \to \mathbf{B}\mathcal{G}) the delooping of its shape.


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

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

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

(𝒢PrinBund X) / equτ 0H(X,B𝒢)Sets. \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 𝒢\mathcal{G}-principal ∞-bundles is a discrete object in Groupoids DiscHGroupoids_\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):


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

P 0,P 1𝒢PrinBund(SmoothGroupoids|infty) X P_0, P_1 \;\in\; \mathcal{G}PrinBund(SmoothGroupoids|infty)_X

is a principal \infty-bundle on the cylinder over XX

P^𝒢PrinBund X× \widehat P \;\in\; \mathcal{G}PrinBund_{X \times \mathbb{R}}

whose restriction to the points 0,10,1 \,\in\, \mathbb{R} is equivalent to the given pair of bundles:

P^ |X×{0}P 0andP^ |X×{1}P 1. {\widehat P}_{\vert X \times \{0\}} \;\simeq\; P_0 \;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\; {\widehat P}_{\vert X \times \{1\}} \;\simeq\; P_1 \,.

We write

(9)(𝒢PrinBund X) / conc(𝒢PrinBund X)/(𝒢PrinBund X×) \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.


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

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

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

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

(𝒢PrinBund X) / concτ 0H(X,B𝒢). \big( \mathcal{G}PrinBund_X \big)_{/\sim_{conc}} \;\; \simeq \;\; \tau_0 \, \mathbf{H} \big( X,\, B \mathcal{G} \big) \,.


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

τ 0H(X,B𝒢) τ 0Pnts[X,B𝒢] τ 0Pnts[X,ʃB𝒢] τ 0Pntsʃ[X,B𝒢] τ 0PntsDiscSing[X,B𝒢] τ 0Sing[X,B𝒢] τ 0lim([X,B𝒢](Δ smth )) τ 0lim(H(X×Δ smth B𝒢)) τ 0lim(𝒢PrinBund X×Δ smth ) limτ 0(𝒢PrinBund X×Δ smth ) (τ 0(𝒢PrinBund X))/(τ 0(𝒢PrinBund X×Δ smth 1)) (𝒢PrinBund X) conc \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}



The identification in Prop. is due to

with a precursor observation in:

The particular conclusion (6) is also claimed in:

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

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

The statement of the smooth Oka principle is also

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

Last revised on August 9, 2022 at 07:01:08. See the history of this page for a list of all contributions to it.