Contents

Idea

A Euclidean-topological $\infty$-groupoid is an ∞-groupoid equipped with cohesion in the form of Euclidean topology.

Examples of 1-truncated type are topological groupoids/topological stacks whose topologies are detectable by maps out of Euclidean topologies, for instance internal groupoids in topological manifolds.

More generally, every simplicial topological space whose topology is degreewise detectable by Euclidean topologies canonically identifies with a Euclidean-topological $\infty$-groupoid. Various constructions with simplicial toppological spaces find their natural home in this (∞,1)-topos. For instance

Definition

Definition

Let CartSp${}_{top}$ be the site whose underlying category has as objects the Cartesian spaces $\mathbb{R}^n$, $n \in \mathbb{N}$ equipped with the Euclidean topology and as morphisms the continuous maps between them; and whose coverage is given by good open covers.

Definition

Define

$ETop \infty Grpd := (\infty,1)Sh(CartSp_{top})$

to be the (∞,1)-category of (∞,1)-sheaves on $CartSp_{top}$.

Properties

General

Proposition

The (∞,1)-topos $ETop \infty Grpf$ is a cohesive (∞,1)-topos.

Proof

The site CartSp${}_{top}$ an ∞-cohesive site. See there for details.

For completeness we record general properties of cohesive (∞,1)-toposes implied by this.

Corollary

$ETop\infty Grpd$ is

Definition

We say that $ETop \infty Grpd$ defines Euclidean-topological cohesion. An object in $ETop \infty Grpd$ we call a Euclidean-topological $\infty$-groupoid.

Proposition

Write TopMfd for the category of topological manifolds. This becomes a large site with the open cover coverage. We have an equivalence of (∞,1)-categories

$ETop\infty Grpd \simeq \hat Sh_{(\infty,1)}(TopMfd)$

with the hypercompletion of the (∞,1)-category of (∞,1)-sheaves on TopMfd.

Proof

Since every topological manifold admits an open cover by open balls homeomorphic to a Cartesian space it follows that CartSp${}_{top}$ is a dense sub-site of $TopMfd$. Accordingly the categories of sheaves are equivalent

$Sh(CartSp_{top}) \simeq Sh(TopMfd) \,.$

By the discussion at model structure on simplicial sheaves it follows that the hypercomplete (∞,1)-toposes over these sites are equivalent

$\hat Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(TopMfd) \,.$

But by the above proposition we have that before hypercompletion $Sh_{(\infty,1)}(CartSp_{top})$ is cohesive. This means that it is in particular a local (∞,1)-topos. By the discussion there, this means that it already coincides with its hypercompletion, $Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(CartSp_{top})$.

Definition

Write $Top_1$ for the 1-category of Hausdorff topological spaces and continuous maps. There is a canonical functor

$j : Top_1 \to \tau_{\leq 0}ETop\infty Grpd \hookrightarrow ETop\infty Grpd$

given by sending a topological space $X$ to the 0-truncated (∞,1)-sheaf (= sheaf) on CartSp${}_{top}$ externally represented by $X$ under the embedding $CartSp_{top} \hookrightarrow Top$:

$j(X) : (U \in CartSp_{top}) \mapsto Hom_{Top}(U,X) \in Set \hookrightarrow \infty Grpd \,.$
Proposition

The functor $j$ exhibits TopMfd as a full sub-(∞,1)-category of $ETop\infty Grpd$

$TopMfd \hookrightarrow ETop\infty Grpd \,.$
Proof

With the above proposition this follows directly by the (∞,1)-Yoneda lemma.

Model category presentation

We dicuss some aspects of the presentation of $ETop \infty Grpd$ by model category structures.

Proposition

Let $[CartSp_{top}^{op}, sSet]_{proj,loc}$ be the Cech-local projective model structure on simplicial presheaves. This is a presentation of $ETop \infty Grpd$

$([CartSp_{top}^{op}, sSet]_{proj,loc})^\circ \simeq ETop \infty Grpd \,.$

Also the model structure on simplicial sheaves sSh(CartSp_{{top})_{loc} is a presentation

$(sSh(CartSp_{top})_{loc})^\circ \simeq ETop \infty Grpd \,.$
Proof

The first statement is a special case of the general discussion at model structure on simplicial presheaves. Similarly, by the general discussion at model structure on simplicial sheaves we have that this presents the hypercompletion of the (∞,1)-category of (∞,1)-sheaves. But by the above $ETop\infty Grpd$ already is hypercomplete.

Moreover:

Proposition

$ETop\infty Grpd$ is also the hypercompletion of the (∞,1)-topos presented by the local model structure on simplicial presheaves over all of Mfd (or over any small dense sub-site such as for instance the full sub-category of manifolds bounded in size by some regular cardinal).

$\hat{}([Mfd^{op}, sSet]_{proj,loc})^\circ \simeq ETop \infty Grpd \,.$
Proof

By the above proposition.

While the model structures on simplicial presheaves over both sites present the same (∞,1)-category, they lend themselves to different computations:

the model structure over $CartSp_{top}$ has more fibrant objects and hence fewer cofibrant objects, while the model structure over $Mfd$ has more cofibrant objects and fewer fibrant objects. More specifically:

Proposition

Let $X \in [Mfd^{op}, sSet]$ be an object that is globally fibrant , separated and locally trivial, meaning that

1. $X(U)$ is an inhabited Kan complex for all $U \in Mfd$;

2. for every covering $\{U_i \to U\}$ in Mfd the descent comparison morphism $X(U) \to [Mfd^{op}, sSet](C(\{U_i\}), X)$ is a full and faithful (∞,1)-functor;

3. for contractible $U$ we have $\pi_0[Mfd^{op}, sSet](C(\{U_i\}), X) \simeq *$.

Then the restriction of $X$ along $CartSp_{top} \hookrightarrow Mfd$ is a fibrant object in the local model structure $[CartSp_{top}^{op}, sSet]_{proj,loc}$.

Proof

The fibrant objects in the local model structure are precisely those that are Kan complexes over every object and for which the descent morphism is an equivalence for all covers.

The first condition is given by the first assumption. The second and third assumptions imply the second condition over contractible manifolds, such as the Cartesian spaces.

Example

Let $G$ be a topological group, regarded as the presheaf over Mfd that it represents. Write $\bar W G$ (see the notation at simplicial group) for the simplicial presheaf on $Mfd$ given by the nerve of the topological groupoid $(G \stackrel{\to}{\to} *)$. (This is a presentation of the delooping of the 0-truncated ∞-group $G \in ETop\infty Grpd$, see the discussion below. )

The fibrant resolution of $\bar W G$ in $[Mfd^{op}, sSet]_{proj,loc}$ is (the rectification of) its stackification: the stack $G Bund$ of topological $G$-principal bundles. But the canonical morphism

$\bar W G \to G Bund$

is a full and faithful functor (over each object $U \in Mfd$): it includes the single object of $\bar W G$ as the trivial $G$-principal bundle. The automorphism of the single object in $\bar W G$ over $U$ are $G$-valued continuous functions on $U$, which are precisely the automorphisms of the trivial $G$-bundle. Therefore this inclusion is full and faithful, the presheaf $\bar W G$ is a separated prestack.

Moreover, it is locally trivial: every Cech cocycle for a $G$-bundle over a Cartesian space is equivalent to the trivial one. Equivalently, also $\pi_0 G Bund(\mathbb{R}^n) \simeq *$.

Therefore $\bar W G$, when restricted $CartSp_{top}$, does become a fibrant object in $[CartSp_{top}^{op}, sSet]_{proj,loc}$.

On the other hand, let $X \in Mfd$ be any non-contractible manifold. Since in the projective model structure on simplicial presheaves every representable is cofibrant, this is a cofibrant object in $[Mfd^{op}, sSet]_{proj,loc}$. However, it fails to be cofibrant in $[CartSp_{top}^{op}, sSet]_{proj,loc}$. Instead, there a cofibrant replacement is given by the Cech nerve $C(\{U_i\})$ of any good open cover $\{U_i \to X\}$.

This yields two different ways to compute the first nonabelian cohomology

$H^1_{ETop}(X,G) := \pi_0 ETop\infty Grpd (X, \mathbf{B}G)$

in $ETop\infty Grpd$ on $X$ with coefficients in $G$, as

1. $\cdots \simeq \pi_0 [Mfd^{op}, sSet](X, G Bund) \simeq \pi_0 G Bund(X)$;

2. $\cdots \simeq \pi_0 [CartSp_{top}^{op}, sSet](C(\{U_i\}), \bar W G) \simeq H^1_{Ch}(X,G)$.

In the first case we need to construct the fibrant replacement $G Bund$. This amounts to computing $G$-cocycles = $G$-bundles over all manifolds and then evaluate on the given one, $X$, by the 2-Yoneda lemma.

In the second case however we cofibrantly replace $X$ by a good open cover, and then find the Cech cocycles with coefficients in $G$ on that.

For ordinary $G$-bundles the difference between the two computations may be irrelevant in practice, because ordinary $G$-bundles are very well understood. However for more general coefficient objects, for instance general topological simplicial groups $G$, the first approach requires to find the full ∞-stackification to the ∞-stack of all principal ∞-bundles, while the second approach requires only to compute specific coycles over one specific base object. In practice the latter is often all that one needs.

Structures in the cohesive $(\infty,1)$-topos $ETop \infty Grpd$

We discuss what some of the general abstract Structures in a cohesive (∞,1)-topos look like in the model $ETop \infty Grpd$.

As usual, write

$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : ETop \infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd$

for the defining quadruple of adjoint (∞,1)-functors that refine the global section (∞,1)-geometric morphism to ∞Grpd.

Cohesive $\infty$-groups

By the general properties of cohesive (∞,1)-toposes with an ∞-cohesive site of definition, every ∞-group object is presented by a presheaf of simplicial groups. For $ETop\infty Grpd$ among these are the simplicial topological groups. See there for more details.

Geometric homotopy and Galois theory

We discuss the realization of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos in $ETop \inft Grpd$.

Proposition

Let $X$ be a paracompact topological space such that that $X$ admits a good open cover by open balls (for instance a paracompact manifold).

Then $\Pi(X) := \Pi(i(X)) \in \infty Grpd$ is equivalent to the standard fundamental ∞-groupoid of a topological space that is presented by the singular simplicial complex $Sing X$

$\Pi(X) \simeq Sing X \,.$

Equivalently, under geometric realization $\mathbb{L}|-| : \infty Grpd \to Top$ we have that there is a weak homotopy equivalence

$X \simeq |\Pi(X)| \,.$
Proof

By the discussion at ∞-cohesive site we have an equivalence $\Pi(-) \simeq \mathbb{L} \lim_\to$ to the derived functor of the sSet-colimit functor $\lim_\to : [CartSp^{op}, sSet]_{proj,loc} \to sSet_{Quillen}$.

To compute this derived functor, let $\{U_i \to X\}$ be a good open cover by open balls, hence homeomorphically by Cartesian space. By goodness of the cover the Cech nerve $C(\coprod_i U_i \to X) \in [CartSp^{op}, sSet]$ is degreewise a coproduct of representables, hence a split hypercover. By the discussion at model structure on simplicial presheaves we have that in this case the canonical morphism

$C(\coprod_i U_i \to X) \to X$

is a cofibrant resolution of $X$ in $[CartSp^{op}, sSet]_{proj,loc}$. Accordingly we have

$\Pi(X) \simeq (\mathbb{L} \lim_\to) (X) \simeq \lim_\to C(\coprod_i U_i \to X) \,.$

Using the equivalence of categories $[CartSp^{op}, sSet] \simeq [\Delta^{op}, [CartSp^{op}, Set]]$ and that colimits in presheaf categories are computed objectwise and finally using that the colimit of a representable functor is the point (an incarnation of the Yoneda lemma) we have that $\Pi(X)$ is presented by the Kan complex that is obtained by contracting in the Cech nerve $C(\coprod_i U_i)$ each open subset to a point.

The classical nerve theorem asserts that this implies the claim.

Remark

We may regard Top itself as a cohesive (∞,1)-topos. $(\Pi_{Top}\dashv Disc_{Top} \dashv \Gamma_{Top} \dashv coDisc_{Top}) Top \stackrel{\simeq}{\to} \infty Grpd$. This is discussed at discrete ∞-groupoid.

Using this the above proposition may be stated as saying that for $X$ a paracompact topological space that admits a good open cover we have

$\Pi_{ETop\infty Grpd}(X) \simeq \Pi_{Top}(X) \,.$
Proposition

Let $X_\bullet$ be a good simplicial topological space that is degreewise paracompact and degreewise admits a good open cover, regarded naturally as an object $X_\bullet \in Top^{\Delta^{op}} \to ETop \infty Grpd$.

We have that the intrinsic $\Pi(X_\bullet) \in \infty Grpd$ coincides under geometric realization $\mathbb{L}|-| : \infty Grpd \stackrel{\simeq}{\to} Top$ with the ordinary geometric realization of simplicial topological spaces $|X_\bullet|_{Top^{\Delta^{op}}}$

$|\Pi(X_\bullet)| \simeq |X_\bullet|_{Top^{\Delta^{op}}} \,.$
Proof

Write $Q$ for Dugger’s cofibrant replacement functor on $[CartSp^{op}, sSet]_{proj,loc}$ (discussed at model structure on simplicial presheaves). On a simplicially constant simplicial presheaf $X$ it is given by

$Q X := \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{U_0 \to \cdots \to U_n \to X} U_0 \right) \,,$

where the coproduct in the integrand of the coend is over all sequences of morphisms from representables $U_i$ to $X$ as indicated. On a general simplicial presheaf $X_\bullet$ it is given by

$Q X_\bullet := \int^{[k] \in \Delta} \Delta[k] \cdot Q X_k \,,$

which is the simplicial presheaf that over any $\mathbb{R}^n \in CartSp$ takes as value the diagonal of the bisimplicial set whose $(n,r)$-entry is $\coprod_{U_0 \to \cdots \to U_n \to X_k} CartSp_{top}(\mathbb{R}^n,U_0)$.

Since coends are special colimits, the colimit functor itself commutes with them and we find

\begin{aligned} \Pi(X_\bullet) & \simeq (\mathbb{L} \lim_\to) X_\bullet \\ & \simeq \lim_\to Q X_\bullet \\ & \simeq \int^{[n] \in \Delta} \Delta[k] \cdot \lim_\to (Q X_k) \,. \end{aligned}

By the discussion at Reedy model structure this coend is a homotopy colimit over the simplicial diagram $\lim_\to Q X_\bullet : \Delta \to sSet_{Quillen}$

$\cdots \simeq hocolim_\Delta \lim_\to Q X_\bullet \,.$

By the above proposition we have for each $k \in \mathbb{N}$ weak equivalences $\lim_\to Q X_k \simeq (\mathbb{L} \lim_\to) X_k \simeq Sing X_k$, so that

\begin{aligned} \cdots &\simeq hocolim_\Delta Sing X_k \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot Sing X_k \\ & \simeq diag Sing(X_\bullet)_\bullet \end{aligned} \,.

By the discussion at geometric realization of simplicial topological spaces, this maps to the homotopy colimit of the simplicial topological space $X_\bullet$, which is just its geometric realizaiton if it is proper.

Paths and geometric Postnikov towers

We discuss the notion of geometric path ∞-groupoids realized in $ETop\infty Grpd$.

In the above constructions of $\Pi(X)$ the actual paths are not explicit. We discuss here presentations of $\mathbf{\Pi}(X)$ in terms of actual paths.

By prop. we have

Proposition

Let $X$ be a a paracompact topological space, regarded as an object of $ETop\infty Grpd$. Then $\mathbf{\Pi}(X)$ is presented by the constant simplicial presheaf

$Disc Sing(X) \,:\, (U,[k]) \mapsto Hom_{Top}(\Delta^k, X) \,.$

Possibly more natural would seem to look at the topological Kan complex that remembers the topology on the spaces of paths:

Definition

For $X$ a paracompact topological space, define the simplicial presheaf

$\mathbf{Sing} X : (U,[k]) \mapsto Hom_{Top}(U \times \Delta^k, X) \,.$
Observation

Also $\mathbf{Sing} X$ is a presentation of $\mathbf{\Pi}(X)$

Proof

For each fixed $U \in CartSp$ the inclusion of simplicial sets

$Sing X \to \mathbf{Sing}(X)(U)$

is a weak homotopy equivalence, since $U \in CartSp$ is contractible.

Therefore the inclusion of simplicial presheaves

$Disc Sing X \to \mathbf{Sing} X$

is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$. This implies the claim with prop. .

Remark

Typically one is interested in mapping out of $\mathbf{\Pi}(X)$. While it is clear that $Disc Sing X$ is cofibrant in $[CartSp^{op}, sSet]_{proj,loc}$, it is harder to determine the necessary resolutions of $\mathbf{Sing}X$.

Cohomology and principal $\infty$-bundles

We dicuss aspects of the intrinsic cohomology of $E Top \infty Grpd$ and of the principal ∞-bundles that it classifies.

Definition

Let $A \in$ ∞Grpd be any discrete ∞-groupoid. Write $|A| \in$ Top for its geometric realization. For $X$ any topological space, the nonabelian cohomology of $X$ with coefficients in $A$ is the set of homotopy classes of maps $X \to |A|$

$H_{Top}(X,A) := \pi_0 Top(X,|A|) \,.$

We say $Top(X,|A|)$ itself is the cocycle ∞-groupoid for $A$-valued nonabelian cohomology on $X$.

Similarly, for $X, \mathbf{A} \in ETop \infty Grpd$ two e-topological $\infty$-groupoids, write

$H_{ETop}(X,\mathbf{A}) := \pi_0 ETop\infty Grpd(X,\mathbf{A})$

for the intrinsic cohomology of $ETop \infty Grpd$ on $X$ with coefficients in $\mathbf{A}$.

Proposition

Let $A \in$ ∞Grpd, write $Disc A \in ETop \infty Grpd$ for the corresponding discrete topological ∞-groupoid. Let $X \in Top_1 \stackrel{i}{\hookrightarrow} ETop \infty Grpd$ be a paracompact topological space regarded as a 0-truncated Euclidean-topological $\infty$-groupoid.

We have an isomorphism of cohomology sets

$H_{Top}(X,A) \simeq H_{ETop}(X,Disc A)$

and in fact an equivalence of cocycle ∞-groupoids

$Top(X,|A|) \simeq ETop\infty Grpd(X, Disc A) \,.$
Proof

By the $(\Pi \dashv Disc)$-adjunction of the locally ∞-connected (∞,1)-topos $ETop \infty Grpd$ we have

$ETop\infty Grpd(X, Disc A) \simeq \infty Grpd(\Pi(X), A) \underoverset{\simeq}{|-|}{\to} Top(|\Pi X|, |A|) \,.$

From this the claim follows by the above proposition.

Proposition

Let $G$ be a well-pointed simplicial topological group degreewise in TopMfd. Then the $(\infty,1)$-functor $\Pi : \mathrm{ETop}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd}$ preserves homotopy fibers of all morphisms of the form $X \to \mathbf{B}G$ that are presented in $[\mathrm{CartSp}_{\mathrm{top}}^{\mathrm{op}}, \mathrm{sSet}]_{proj}$ by morphism of the form $X \to \bar W G$ with $X$ fibrant.

Proof

Notice that since (∞,1)-sheafification preserves finite (∞,1)-limits we may indeed discuss the homotopy fiber in the global model structure on simplicial presheaves.

Write $Q X \stackrel{\simeq}{\to} X$ for the global cofibrant resolution given by $Q X : [n] \mapsto \coprod_{\{U_{i_0} \to \cdots \to U_{i_n} \to X_n\}} U_{i_0}$, where the $U_{i_k}$ range over $\mathrm{CartSp}_{\mathrm{top}}$ . (Discussed at model structure on simplicial presheaves – cofibrant replacement. ) This has degeneracies splitting off as direct summands, and hence is a good simplicial topological space that is degreewise in TopMfd. Consider then the pasting of two pullback diagrams of simplicial presheaves

$\array{ P' &\stackrel{\simeq}{\to}& P &\to& W G \\ \downarrow && \downarrow && \downarrow \\ Q X &\stackrel{\simeq}{\to}& X &\to & \bar W G } \,.$

By the discussion at geometric realization of simplicial topological spaces we have that the rightmost vertical morphism is a fibration in $[CartSp_{top}^{op}, sSet]_{proj}$. Since fibrations are stable under pullback, the middle vertical morphism is also a fibration (as is the leftmost one). Since the global model structure is a right proper model category it follows then that also the top left horizontal morphism is a weak

Since the square on the right is a pullback of fibrant objects with one morphism being a fibration, $P$ is a presentation of the homotopy fiber of $X \to \bar W G$. Hence so is $P'$, which is moreover the pullback of a diagram of good simplicial spaces.

By prop. we have that on the outer diagram $\Pi$ is presented by geometric realization of simplicial topological spaces $|-|$. By the discussion of realization of simplicial principal bundles there, we have a pullback in $\mathrm{Top}_{\mathrm{Quillen}}$

$\array{ {|P|} &\to& {|W G|} \\ \downarrow && \downarrow \\ {|Q X|} & \to & {|\bar W G|} }$

which exhibits $|P|$ as the homotopy fiber of $|Q X| \to |\bar W G|$. But this is a model for $|\Pi(X \to \bar W G)|$.

Twisted cohomology

See twisted bundle .

Universal coverings and geometric Whitehead towers

We discuss geometric Whitehead towers in $ETop\infty Grpd$.

Proposition

Let $X$ be a pointed] [[paracompact topological space that admits a good open cover. Then its ordinary Whitehead tower $* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X$ in Top coincides with the image under the intrinsic fundamental ∞-groupoid functor $|\Pi(-)|$ of its geometric Whitehead tower $X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X$ in $ETop \infty Grpd$:

\begin{aligned} |\Pi(-)| & : (X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X) \in ETop\infty Grpd \\ & \mapsto (* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X) \in Top \end{aligned} \,.
Proof

By the general discussion at Whitehead tower in an (∞,1)-topos the geometric Whitehead tower is characterized for each $n$ by the fiber sequence

$X^{\mathbf{(n)}} \to X^{\mathbf{(n-1)}} \to \mathbf{B}^n \mathbf{\pi}_n(X) \to \mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{(n-1)}(X) \,.$

By the above proposition on the fundamental ∞-groupoid we have that $\mathbf{\Pi}_n(X) \simeq Disc Sing X$. Since $Disc$ is right adjoint and hence preserves homotopy fibers this implies that $\mathbf{B} \mathbf{\pi}_n(X) \simeq \mathbf{B}^n Disc \pi_n(X)$, where $\pi_n(X)$ is the ordinary $n$th homotopy group of the pointed topological space $X$.

Then by the above proposition on geometric realization of homotopy fibers we have that under $|\Pi(-)|$ the space $X^{\mathbf{(n)}}$ maps to the homotopy fiber of $|\Pi(X^{\mathbf{(n-1)}})| \to B^n |Disc \pi_n(X)| = B^n \pi_n(X)$.

By induction over $n$ this implies the claim.

Path $\infty$-groupoid and geometric Postnikov towers

Let $C$ be an ∞-connected site. We give an explicit presentation of the constant path inclusion $X \to \mathbf{\Pi}(X)$ in the locally ∞-connected (∞,1)-topos over $C$ such that the component maps are cofibrations.

Remark

The projective model structure on simplicial presheaves $[C^{op}, sSet]_{proj}$ has a set of generating cofibrations

$I = \{ U \cdot \partial \Delta[n] \hookrightarrow U \cdot \Delta[n] | U \in C, n \in \mathbb{N}) \} \,.$

See model structure on functors for details.

Definition

Write

$\mathbf{Sing} : C \to [C^{op}, sSet]$

for the functor given by applying the small object argument to this set $I$ to obtain a functorial factorization of the terminal morphisms $U \to *$ into a cofibration followed by an acyclic fibration

$U \hookrightarrow \mathbf{Sing} U \stackrel{\simeq}{\to} * \,.$

Let

$\mathbf{Sing} : [C^{op}, sSet] \to [C^{op}, sSet]$

be the Yoneda extension (left Kan extension through the Yoneda embedding) of this functor to all of $[C^{op}, sSet]$.

Remark

For $U \in C$ the simplicial presheaf $\mathbf{Sing}U$ is a resolution of the (nerve of the) fundamental groupoid $\Pi_1(U)$:

the non-degenerate components of $\mathbf{Sing}U$ at the first stage of the small object argument are such that a map out of them into a simplicial presheaf $A$ are given by commuting diagrams

$\array{ U_0 \coprod U_0 &\stackrel{(s,t)}{\to}& U \\ \downarrow && \downarrow \\ U_0 \times \Delta &\to& A } \,.$

This is a $U$-parameterized family of objects of $A$ together with a $U_0$-parameterized family of morphisms of $A$ associated to the pairs of points $(s,t) \in U$, hence to the “straight paths” from $s$ to $t$. At the next stage for every triangle of such straight path a 2-morphism is thrown in, and so on. So $\mathbf{Sing}U$ indeed is an $\infty$-groupoid of paths in $U$.

Proposition

The functor $\mathbf{Sing}$ is the left adjoint of a Quillen adjunction

$(\mathbf{Sing} \dashv R) : [C^{op}, sSet]_{proj, loc} \to [C^{op}, sSet]_{proj, loc} \,.$

Its left derived functor is equivalent to the intrinsic fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

$\mathbb{L}\mathbf{Sing}(-) \simeq \Pi(-)$

and the constant path inclusion $Id \to \Pi$ is presented by the canonical natural transformation $Id \to \mathbf{Sing}$.

Proof

On an arbitrary simplicial presheaf $X$ the functor $\mathbf{Sing}$ is given by the coend

$\mathbf{Sing} : X \mapsto \int^{U \in C} X(U) \cdot \mathbf{Sing}U \,.$

By construction this preserves all colimits. Hence by the adjoint functor theorem (using that domain and codomain are presheaf categories) we have that $\mathbf{Sing}$ is a left adjoint. Explicitly, the right adjoint is given by

$R X : U \mapsto [C^{op}, sSet](\mathbf{Sing}U, X) \,.$

We check that $\mathbf{Sing}$ is also a left Quillen functor first for the global projective model structure. For that, notice that the above expression is the evaluation of the left Quillen bifunctor (see the examples-section there for details)

$\int^C (-) \cdot (-) : [C^{op}, sSet]_{proj} \times [C, [C^{op}, sSet]_{proj}]_{inj} \to [C^{op}, sSet]_{proj} \,.$

Since every representable $U$ is cofibrant in $[C^{op}, sSet]_{proj}$ and since $U \to \mathbf{Sing}U$ is a cofibration by the small object argument, we have that $\mathbf{Sing}U$ is cofibrant in $[C^{op}, sSet]_{proj}$ for all $U$. This means that also $\mathbf{Sing}(-)$ is cofibrant in $[C, [C^{op}, sSet]_{pro}]_{inj}$. Since $\int^C (-) \cdot (-)$ is a left Quillen bifunctor it follows that $\int^C (-)\cdot \mathbf{Sing}$ is a left Quillen functor. Hence it preserves cofibrations and acyclic cofibrations.

This establishes that $\mathbf{Sing}$ is a left simplicial Quillen functor on $[C^{op}, sSet]_{proj}$.

Since this is a left proper model category we have by the discussion at simplicial Quillen adjunction that for showing that this does descend to the local model structure it is sufficient to check that the right adjoint preserves local fibrant objects. Which, in turn, is implied if $\mathbf{Sing}$ send covering Cech nerves to weak equivalences.

Let therefore $C(\coprod_i U_i \to U)$ be the Cech nerve of a covering family in the site $C$. We may write this as the coend

$C(\coprod_i U_i) = \int^{[k] \in \Delta} \Delta[k] \cdot \left( \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \right) \,,$

where by assumption on the ∞-connected site $C$ all the $U_{i_0, \cdots, i_n}$ are representable. By precomposing the projection $C(\coprod_i U_i) \to X$ with the objectwise Bousfield-Kan map that replaces the simplices with the fat simplex $\mathbf{\Delta} : \Delta \to sSet$, we get the morphisms

$C(\coprod_i U_i) = \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \right) \stackrel{\simeq}{\to} C(\coprod_i U_i) \to U \,.$

Here the first map is an objectwise weak equivalence by Bousfield-Kan (see the examples at Reedy model structure for details). Hence by 2-out-of-3 we may equivalently check that $\mathbf{Sing}$ sends these morphisms to weak equivalences in $[C^{op}, sSet]_{proj}$.

Since $\mathbf{Sing}$ commutes with all colimits and hence coends the result of applying it to this morphism is

$\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n} \right) \to \mathbf{Sing}U \,.$

Since the fat simplex is cofibrant in $[\Delta, sSet_{Quillen}]_{proj}$ and since the above is an evaluation of the left Quillen bifunctor

$\int^\Delta (-) \cdot (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, [C^{op}, sSet]_{proj}]_{inj} \to [C^{op}, sSet]_{proj}$

the functor $\int^\Delta \mathbf{\Delta} \cdot (-)$ is left Quillen and hence preserves weak equivalences between cofibrant objects (by the factorization lemma), such as the morphisms $\mathbf{Sing}U \stackrel{\simeq}{\to} *$. Therefore we have a commuting diagram

$\array{ \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n} \right) &\stackrel{\simeq}{\to}& \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} * \right) \\ \downarrow && \downarrow^{\simeq} \\ \mathbf{Sing}U &\stackrel{\simeq}{\to}& * } \,,$

with weak equivalences in $[C^{op}, sSet]_{proj}$ as indicated: the top morphism is a weak equivalence by the argument just given, the bottom one by the small object argument-construction of $\mathbf{Sing}$ and the right vertical morphism is a weak equivalence by the assumption on an ∞-connected site. It follows by 2-out-of-3 that also the left vertical morphism is a weak equivalence.

This establishes the fact that $\mathbf{Sing}$ is left Quillen on the local model structure on simplicial presheaves. By the discussion at simplicial Quillen adjunction this implies that its left derived functor is a left adjoint (∞,1)-functor. Hence it preserves (∞,1)-colimits and so is determined on representatives. There $\mathbf{Sing} U \simeq *$ does coindice with $\Pi(U) \simeq *$, hence both (∞,1)-functors are equivalent.

Corollary

For all cofibrant $X \in [C^{op}, sSet]_{proj,loc}$, the de Rham coefficient object $\mathbf{\Pi}_{dR} X$ is presented by the ordinary pushout

$\array{ X &\to& * \\ \downarrow && \downarrow \\ \mathbf{Sing}X &\to& \mathbf{\Pi}_{dR} X }$

in $[C^{op}, sSet]$.

Proof

By definition we have that $\mathbf{\Pi}_{dR}$ is the (∞,1)-pushout $\mathbf{\Pi}(X) \coprod_X *$ in $Sh_{(\infty,1)}(C)$. By the above proposition we have a cofibrant presentation of the pushout diagram as indicated (all three objects cofibrant, at least one of the two morphisms a cofibration). By the general discussion at homotopy colimit the ordinary pushout of that diagram does compute the (∞,1)-colimit.

Homotopy localization

We discuss that the homotopy localization of topological $\infty$-groupoids reproduces Top $\simeq$ ∞Grpd, following (Dugger).

Idea

A central result about the (∞,1)-topos $Sh_{(\infty,1)}(Top)$ of ∞-stacks on Top is that the homotopy localization is equivalent to Top itself

$Sh_{(\infty,1)}(Top)^I \simeq Top \,.$

A discussion of this is in (the nice but not quite finished) (Dugger).

In fact, this is true even for Lie ∞-groupoids, i.e. ∞-stacks on Diff: the homotopy invariant ones model plain topological spaces.

This provides a useful resolution of topological spaces that often helps to disentangle the two different roles played by a topological space: on the one hand as a model for an ∞-groupoid, in the other as a locale.

Details

Let $SPSh(Diff)^{loc}$ be the local model structure on simplicial presheaves obtained by left Bousfield localization at the Cech nerves of Cech covers with respect to the standard Grothendieck topology on Diff. This is a model for ∞-stacks on Diff.

Let $SPSh(Diff)^{loc}_I$ be furthermore the left Bousfield localization at the set of projection morphisms out of products of the form $X \times \mathbb{R} \to X$ for all $X \in Diff$. The $\infty$-stacks that are local objects with respect to these morphisms are the homotopy invariant $\infty$-stacks, so this localization models the (∞,1)-topos of homotopy invariant $\infty$-stacks on $Diff$.

$L : SSet \stackrel{\leftarrow}{\to} SPSh(Diff) : R$

where $L$ sends a simplicial set to the simplicial presheaf constant on that simplicial set, and where evaluates a simplicial presheaf on the manifold that is the point.

Theorem (Dugger)

This adjunction $(L \dashv R)$ is a Quillen equivalence with respect to the standard model structure on simplicial sets on the left and the above model structure $SPSh(Diff)_{loc}^I$ on the right.

Section 3.2 in

Some discussion of the $(\infty,1)$-category of $(\infty,1)$-sheaves on the category of manifolds and its restriction to open balls and a discussion of its homotopy localization is in:

Discussion of geometric realization of simplicial topological principal bundles and of their classifying spaces is in