nLab shape modality

(shape does not preserve homotopy fibers of points over positive-dimensional smooth spheres)
For $n \in \mathbb{N}_+$ consider the $n$ -sphere in its standard incarnation as a smooth manifold (or just as a D-topological space) and as such as a 0-truncated smooth $\infty$ -groupoid

S^n_{smth} \,\in\, SmthMfd \hookrightarrow SmthGrpd_\infty \,.

This is such that the shape is the $n$ -sphere as a geometrically discrete higher homotopy type

\esh S^n_{smth} \,\simeq\, S^n \,\in\, Grpd_\infty \xhookrightarrow{Disc} SmthGrpd_{\infty}.

Now for $x \colon \ast \to S^n_{smth}$ any point (and using that $\esh \ast \,=\, \ast$ by the condition that $\esh$ preserves finite products):

its fiber product over $S^n_{smth}$ may be computed in SmthMfd and hence is that same point:
its homotopy fiber product over $S^n = \esh S^n_{smth}$ is the loop space $\Omega S^n$ , which (under our assumption that $n$ is a positive number) is far from equivalent to the point (for example: $\Omega S^1 \,\simeq\, \mathbb{Z}$ ).

Hence

n \in \mathbb{N}_+ \;\;\;\; \vdash \;\;\;\; \ast \;\simeq\; \esh \big( \ast \underset {S^n_{smth}} {\times} \ast \big) \;\neq\; \big( (\esh\ast) \underset {\esh S^n_{smth}} {\times} (\esh\ast) \big) \;\simeq\; \Omega S^{n} \,.

However, the shape modality does preserve some useful classes of homotopy fiber products after all:

General

Proposition

(cohesive shape preserves looping and delooping)

For $\mathbf{H}$ a cohesive $\infty$ -topos and $\mathbf{B}\mathcal{G} \,\in\, \mathbf{H}$ a pointed connected object, hence the delooping of an $\infty$ -group $\mathcal{G} \,\in\, Grp(\mathbf{H})$ , the cohesive shape modality preserves the looping homotopy fiber product:

\esh \big( \ast \underset {\mathbf{B}\mathcal{G}}{\times} \ast \big) \;\; \simeq \;\; \ast \underset {\esh \mathbf{B}\mathcal{G}}{\times} \ast \,.

Proof

Since groupoid objects in an (∞,1)-topos are effective, the delooping of an $\infty$ -group is the simplicial $\infty$ -colimit of the simplicial object $\mathcal{G}^{\times^\bullet}$ of finite products of $\mathcal{G}$ . Since the shape operation preserves finite products and general colimits, it preserves the delooping:

\esh \mathbf{B}\mathcal{G} \;\simeq\; \esh \underset{\underset{n \in \mathbb{N}}{\longrightarrow}} {\lim} \mathcal{G}^{\times^n} \;\simeq\; \underset{\underset{n \in \mathbb{N}}{\longrightarrow}} {\lim} \big( \esh \mathcal{G} \big)^{\times^n} \;\simeq\; \mathbf{B} \esh \mathcal{G} \,.

Finally, using again that groupoid objects in an (∞,1)-topos are effective, the claim follows:

\esh \big( \ast \underset{\mathbf{B}\mathcal{G}}{\times} \ast \big) \;\simeq\; \esh \mathcal{G} \;\simeq\; \esh \big( \ast \underset{\esh\mathbf{B}\mathcal{G}}{\times} \ast \big) \,.

Proposition

(cohesive shape preserves homotopy fiber products over discrete objects)
If $\mathbf{H}$ is a cohesive $\infty$ -topos over $Grpd_\infty$ , then its shape modality preserves homotopy fiber products over discrete objects.

The following argument follows dcct, Thm. 3.8.19, which in turn follows discussion with Mike Shulman. The diagram is taken from the writeup of the proof given in SS21, Prop. 3.38.

Proof

By a version of the $\infty$ -Grothendieck construction (Beardsley & Péroux 2022, Lem. 3.10), a slice $\infty$ -topos over the inverse image of a plain $\infty$ -groupoid $B$ is equivalent to the $\infty$ -category of $\infty$ -functors $B \to \mathbf{H}$ . This way, the $B$ -sliced adjunction $Shp \dashv Dscr$ is seen to be of the following form:

Now $Fnctr(B,Shp)$ preserves binary products as soon as $Shp$ does (because products of presheaves are computed objectwise). But binary products in the equivalent slice category $\mathbf{H}_{/B}$ are the homotopy fiber products in question, and hence the claim follows.

Proposition

(shape preserves homotopy fibers of maps from discrete to delooped objects)
For $\mathbf{H}$ is a cohesive $\infty$ -topos over $Grpd_\infty$ the shape modality preserves the homotopy fibers of morphisms

f \,\colon\, X \longrightarrow \mathbf{B}\mathcal{G}

from any discrete object $X \,\in\, Grpd_\infty \xhookrightarrow{Dscr} \mathbf{H}$
to any delooped object $\mathbf{B}\mathcal{G} \,\in\, \mathbf{H}$

in that

\esh fib(f) \;\simeq\; fib\big( \esh f \big) \,.

Proof

Since a discrete $\infty$ -groupoid $X$ is the coproduct of its connected components, and since coproducts are preserved by homotopy pullback (due to universal colimits in an $\infty$ -topos) and by the shape modality (being a left adjoint), we may assume without restriction of generaliity that also $X$ is connected, hence $X \,\simeq\,B \mathcal{K}$ with $\ast \twoheadrightarrow B\mathcal{K}$ an effective epimorphism.

First, observe that the pasting law for homotopy pullbacks gives the pasting diagram of homotopy pullbacks shown on the left below:

\array{ \mathcal{G} &\longrightarrow& fib(f) &\longrightarrow& \ast \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \ast &\twoheadrightarrow& B\mathcal{K} &\underset{f}{\longrightarrow}& \mathbf{B}\mathcal{G} } \;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\; \array{ \esh \mathcal{G} &\longrightarrow& \esh fib(f) &\longrightarrow& \ast \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \ast &\twoheadrightarrow& B\mathcal{K} &\underset{\esh f}{\longrightarrow}& B\mathcal{G} \mathrlap{\,.} }

Now, as shown on the right, the shape modality

preserves the left pullback square and its bottom effective epimorphism, by Prop. ;
preserves the outer pullback rectangle by Prop. .

Therefore also the square on the far right is homotopy Cartesian, by the “reverse pasting law” (this Prop.). But this is equivalently the claim to be shown.

More specific cases

The following discussion is about the topological/smooth shape modality commuting with looping (homotopy fiber products of points) of mapping stacks between delooping stacks of certain topological groups. This is a technical point of crucial importance in the construction and discussion of equivariant classifying spaces for equivariant principal bundles – though not traditionally presented as being about behaviour under a shape modality. The following perspective is from SS21, though the hard results themselves are either classical (essentially Conner & Floyd 1964, Lemma 31.8 for the case of compact $\Gamma$ ) or, in more generality, due to Uribe & Lück 2014. (Beware that the following differs from their notation by the exchange “ $G$ ” $\leftrightarrow$ “ $\Gamma$ ”! For us here, $G$ denotes the equivariance group and $\Gamma$ the structure group.)

Definition

(Condition H on families of topological group homomorphisms)
Let $G, \Gamma \,\in\, Grp(kTop)$ be topological groups and let

\mathcal{R} \,\subset\, \underset{ H \subset G }{\coprod} Hom(H,\Gamma)

be a set of pairs $(H,\rho)$ consisting of a topological subgroup $H$ of $G$ and a continuous homomorphism $H \to \Gamma$ such that (Uribe & Lück 2014, Def. 3.4) $\mathcal{R}$ is closed under:

fiber products, i.e. under products in $Grp(kTop)_{/\Gamma}$ ,
conjugation by $\Gamma$ (under postcomposition),
conjugation by $G$ (under precomposition).

We say (with Uribe & Lück 2014, Def. 6.1) that such data satisfies Condition H if for all $(H, \rho) \,\in\, \mathcal{R}$ we have:

the path component of $\rho$ in $HomMap(H,\Gamma)$ is contained in its orbit under the $\Gamma$ -conjugation action:

$HomMap(H,\Gamma)_{\rho} \;\; \subset \;\; \Gamma(\rho) \;\coloneqq\; \big\{ Ad_\gamma \circ \rho \;\big\vert\; \gamma \in \Gamma \big\}$
the coset space-coprojections
1. $G \longrightarrow G/H$
2. $\Gamma \longrightarrow \Gamma/Stab_\Gamma(\rho)$
  
  (by the the stabilizer subgroup $Stab_{\Gamma}(\rho) \subset \Gamma$ under the conjugation action)
admit local sections.
the map

$\array{ \Gamma/Stab_\Gamma(\rho) &\longrightarrow& HomMap(H,\Gamma) \\ [\gamma] &\mapsto& Ad_\gamma \circ \rho }$

is a homeomorphism onto its image.

Example

Examples of families of topological group homomorphisms satsifying Condition H (Def. ) include the following:

$\Gamma$ and $G$ any Lie groups with $\Gamma$ almost connected

with $\mathcal{R}$ consisting of all compact subsgroups $H \subset G$ and all their homomorphisms $H \to \Gamma$

(established in Uribe & Lück 2014, Thm. 6.3)
$\Gamma =$ PU(ℋ) and $G$ any finite group

with $\mathcal{R}$ consisting of all subgroups $H \subset G$ and all their homomorphisms $H \to PU(\mathcal{H})$

(established in Uribe & Lück 2014, Sec. 15)

For the following formulation of the main result of Uribe & Lück 2014 in terms of cohesive homotopy theory we use the continuous diffeology

kTop \xrightarrow{CDfflg} DiffSp \hookrightarrow SmthGrp_\infty

and the fact that this preserves shapes and shapes of mapping spaces (see the introduction of SS21).

Proposition

(shape commutes with looping of mapping stacks between delooping stacks of certain pairs of topological groups)
For

$G$ a finite group
$\Gamma$ a compact Lie group or PU(ℋ),

we have for all homomorphisms $\rho \,\colon\, H \to \Gamma$ from subgroups $H \subset G$ that the shape modality commutes with the looping-operation on the mapping stack between the deloopings of $H$ and $\Gamma$ in $SmoothGrpd_{\infty}$ :

\esh \Omega_\rho Map \big( \mathbf{B}H ,\, \mathbf{B}\Gamma \big) \;\; \simeq \;\; \Omega_\rho \esh Map \big( \mathbf{B}H ,\, \mathbf{B}\Gamma \big) \,.

Proof

First notice (e.g. from the discussion at stabilizer subgroup, here) that

(1)

Stab_\Gamma(\rho) \;\simeq\; \Omega_\rho Map \big( \mathbf{B}H ,\, \mathbf{B}\Gamma \big)

From the discussion at equivariant principal bundles we have, under the given assumptions, that

B_G \Gamma \;\;\; \colon \;\;\; G/H \;\mapsto\; \esh \, Map \big( \mathbf{B}H ,\, \mathbf{B}\Gamma \big)

is the $G$ -equivariant homotopy type of the $G$ -equivariant classifying space of $\Gamma$ .

(For $\Gamma$ compact Lie this is the Murayama-Shimakawa result as reviewed in Guillou, May & Merling 2017, Thm. 3.1, while for $\Gamma = PU(\mathcal{H})$ this is due to Bárcenas, Espinoza, Joachim & Uribe 2012, with a re-derivation in a more explicitly cohesive context in SS21).

Moreover, by Exp. both cases satisfy Condition H (Def. ), whence the main result Uribe & Lück 2014, Sec. 13 applies, which, using the above identification of the equivariant classifying space, gives a weak homotopy equivalence of the form

\underset{ { [\rho] \in } \atop { Hom(H,\Gamma)/\Gamma } }{\coprod} B Stab_\Gamma(\rho) \;\;\; \simeq \;\;\; \esh Map \big( \mathbf{B}H ,\, \mathbf{B}\Gamma \big) \,.

(For $\Gamma$ a Lie group this is the classical fact that nearby homomorphisms from compact Lie groups are conjugate.)

Observing that $B Stab_\Gamma(\rho) \,\simeq\, \mathbf{B} \esh Stab_\Gamma(\rho)$ , the looping of this equivalence at the basepoint given by $\rho$ yields the claim.

Relative shape and factorization system

Generally, given an (∞,1)-topos $\mathbf{H}$ (or just a 1-topos) equipped with an idempotent monad $\esh \colon \mathbf{H} \to \mathbf{H}$ (a (higher) modality/closure operator) which preserves (∞,1)-pullbacks over objects in its essential image, one may call a morphism $f \colon X \to Y$ in $\mathbf{H}$ $\esh$ -closed if the unit-diagram

\array{ X &\overset{\eta_X}{\longrightarrow}& \esh(X) \\ \big\downarrow {\mathrlap{{}^f}} && \big\downarrow{\mathrlap{{}^{\esh(f)}}} \\ Y & \overset{\eta_Y}{\longrightarrow} & \esh(Y) }

is an (∞,1)-pullback diagram. These $\esh$ -closed morphisms form the right half of an orthogonal factorization system, the left half being the morphisms that are sent to equivalences in $\mathbf{H}$ .

Definition

Let $(\Pi\dashv \Disc\dashv \Gamma):H\to\infty\Grpd$ be an infinity-connected (infinity,1)-topos, let $\esh:=\Disc \Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$ -morphism, let $c_{\esh} f$ denote the ∞-pullback

\array{ c_{\esh} f &\longrightarrow& {\esh} X \\ \big\downarrow && \big\downarrow^{\mathrlap{{\esh}_f}} \\ Y &\xrightarrow{1_{(\Pi\dashv \Disc)}} & {\esh}Y }

$c_{\esh} f$ is called $\esh$ -closure of $f$ .

$f$ is called $\esh$ -closed if $X\simeq c_{\esh}f$ .

If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $\esh$ -equivalence then $g$ is $\esh$ -closed; this is seen by using that $\esh$ is idempotent.

$\Pi$ -closed morphisms are a right class of an orthogonal factorization system (in an (∞,1)-category) and hence, as discussed there, are closed under limits, composition, retracts and satisfy the left cancellation property.

Remark

(shape-modal maps as open maps)
A consequence of the previous property is that the class of $\esh$ -closed morphisms gives rise to an admissible structure in the sense of structured spaces on an (∞,1)-connected (∞,1)-topos, hence they serve as a class of a kind of open maps.

Shape via cohesive path ∞-groupoid

See at shape via cohesive path ∞-groupoid.

Examples

Internal locally constant $\infty$ -stacks

In a cohesive (∞,1)-topos $\mathbf{H}$ with an ∞-cohesive site of definition, the fundamental ∞-groupoid-functor $\esh$ satisfies the above assumptions (this is the example gives this entry its name). The $\esh$ -closed morphisms into some $X \in \mathbf{H}$ are canonically identified with the locally constant ∞-stacks over $X$ . The correspondence is effectively what is called categorical Galois theory.

Proposition

Let $H$ be a cohesive (∞,1)-topos possessing a ∞-cohesive site of definition. Then for $X\in H$ the locally constant ∞-stacks $E\in \L\Const(X)$ , regarded as ∞-bundle morphisms $p:E\to X$ are precisely the $\esh$ -closed morphisms into $X$

Formally étale morphisms

If a differential cohesive (∞,1)-topos $\mathbf{H}_{th}$ , the de Rham space functor $\Im$ satisfies the above assumptions. The $\Im$ -closed morphisms are precisely the formally étale morphisms.