Schreiber
path ∞-groupoid

Idea

The fundamental ∞-groupoid $\Pi (X)$ of a topological space $X$ is the ∞-groupoid given by the Kan complex $\mathrm{Sing}X$ whose k-morphisms are continuous $k$-dimensional paths in $X$.

The notion of homotopy $\mathrm{\infty }$-groupoid is the generalization of this as we generalize from the (∞,1)-topos Top of nice topological spaces to an (∞,1)-topos $H$ of “structured” or “parameterized” spaces, namely of ∞-stacks. It encapsulates the notion of geometric homotopy groups in the (∞,1)-topos $H$.

A notion of homotopy $\mathrm{\infty }$-groupoid in an $(\mathrm{\infty }\mathrm{,1})$-topos $H$ is present notably when $H$ is a locally contractible (∞,1)-topos, a condition analogous to the condition on an ordinary topos to be locally connected. Various further structures and results in $H$ are induced in this case, such as

• a notion of geometric realization,

• a notion of parallel transport,

• a notion of structured singular cohomology,

• and an intrinsic de Rham theorem

on objects $X\in H$.

Unstructured geometric homotopy $\mathrm{\infty }$-groupoid

Let $C$ be some site and let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ be the (∞,1)-sheaf/∞-stack (∞,1)-topos over $C$. The canonical morphism of sites $p:C\to *$ induces the terminal geometric morphism

$$(p^{*}⊣p_{*})=:(\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$(p^* \dashv p_*) =: (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

consisting of

• the direct image $p_{*}=\Gamma$ – the global sections functor,

which is right adjoint to

• the inverse image $p^{*}=\mathrm{LConst}$, the constant ∞-stack functor.

We say $H$ is a locally contractible (∞,1)-topos if we also have a left adjoint $\Pi$ of $\mathrm{LConst}$ which we shall see is the operation of forming the bare fundamental ∞-groupoid of an ∞-stack.

$$(p_{!}⊣p^{*}⊣p_{*})=:(\Pi ⊣\mathrm{LConst}⊣\Gamma ):H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(*)=\mathrm{\infty }\mathrm{Grpd}.$$(p_! \dashv p^* \dashv p_*) =: (\Pi \dashv LConst \dashv \Gamma) : \;\;\; \mathbf{H} = Sh_{(\infty,1)}(C) \stackrel{ \overset{\Pi}{\to} }{ \stackrel{ \overset{LConst}{\leftarrow} } { \underset{\Gamma}{\to} } } Sh_{(\infty,1)}(*) = \infty Grpd \,.

A decent amount of geometric information is encoded in this essential geometric morphism, such as notably the geometric Galois theory of objects in $H$.

Recall that for an ordinary topological space $X$ with fundamental groupoid ${\Pi }_1(X)$ the representation category $\mathrm{Func}({\Pi }_1(X),\mathrm{Core}(\mathrm{Set}))$ of functors into the core or (the maximal groupoid in) the category Set of sets is equivalent to the category of locally constant sheaves/covering spaces on $X$.

Analogously, the constant ∞-stack $\mathrm{\infty }\mathrm{Cov}:=\mathrm{LConst}(\mathrm{Core}(\mathrm{\infty }\mathrm{Grpd}))\in H$ on the core of ∞Grpd is the classifying $\mathrm{\infty }$-stack for locally constant ∞-stacks on objects $X\in H$ hence for $\mathrm{\infty }$-covering spaces on $X$. We write

$$\mathrm{\infty }\mathrm{Cov}(X):=H(X,\mathrm{\infty }\mathrm{Cov})$$\infty Cov(X) := \mathbf{H}(X,\infty Cov)

for the $\mathrm{\infty }$-groupoid of locally constant ∞-stacks on $X$.

By adjunction the locally constant $\mathrm{\infty }$-stacks on $X$ – the local systems – are equivalently given by (∞,1)-functors from the ∞-groupoid $\Pi (X)$ to $\mathrm{\infty }\mathrm{Grpd}$:

$$\mathrm{\infty }\mathrm{Cov}(X)\simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),\mathrm{\infty }\mathrm{Grpd}).$$\infty Cov(X) \simeq \infty Grpd(\Pi(X),\infty Grpd) \,.

In $H=$ Top, this is the relation satisfied by the fundamental ∞-groupoid $\Pi (X)=\mathrm{Sing}X$ of a topological space $X$. Accordingly here in a general $(\mathrm{\infty }\mathrm{,1})$-topos $H$ we may think of the functor $\Pi :H\to \mathrm{\infty }\mathrm{Grpd}$ as giving for each generalized space its geometric homotopy ∞-groupoid of geometric paths in it.

Alternatively, regarding this from the perspective of Top under the equivalence $\mid -\mid :$ ∞Grpd $\stackrel{\simeq }{\to }$ Top induced by ordinary geometric realization, we may think for $X\in H$ of

$$\mid X\mid :=\mid \Pi (X)\mid \in \mathrm{Top}$$|X| := |\Pi(X)| \in Top

as the geometric (topological) realization of the structured object $X$.

Structured geometric homotopy $\mathrm{\infty }$-groupoid

Using the adjunction, the homotopy $\mathrm{\infty }$-groupoid in $\mathrm{\infty }\mathrm{Grpd}$ may be reflected back into $H$, where it provides internal notion of the homotopy $\mathrm{\infty }$-groupoid $\Pi (X):=\mathrm{LConst}\Pi (X)$. This constitutes an endo-adjunction

$$(\Pi ⊣♭):=(\mathrm{LConst}\circ \pi ⊣\mathrm{LConst}\circ \Gamma ):H\overleftarrow{\to }H.$$(\mathbf{\Pi} \dashv \flat) := (LConst \circ \pi \dashv LConst \circ \Gamma) : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} \,.

The unit of this provides us with the constant path inclusion, a natural morphism

$$X\to \Pi (X)$$X \to \mathbf{\Pi}(X)

We can see that differential cohomology in an (∞,1)-topos is effectively the obstruction theory to extensions through this morphism.

Geometric path $\mathrm{\infty }$-groupoid

So far the notion of geometric path in $X$ that underlies the notion of morphisms in $\Pi (X)$ is entirely implicit. In applications it is useful to have a model for the structured path $\mathrm{\infty }$-groupoid that is explicitly built from an interval object

$$\begin{array}{ccc}& & R\\ & {}^0\nearrow & & {\nwarrow }^1\\ *& & & & *\end{array}$$\array{ && R \\ & {}^0 \nearrow && \nwarrow^{1} \\ {*} &&&& {*} }

in $C$. This canonically induces a cosimplicial object ${\Delta }_R:\Delta \to C$ of geometric $k$-simplices built from $R$ and thereby a singular simplex functor $X\mapsto X^{{\Delta }_R^{•}}$. Its left derived functor we call the geometric path $\mathrm{\infty }$-groupoid functor ${\Pi }_R:H\to H$.

We show below that if all representables $U\in C$ are contractible with respect to the interval object $R$ in that the canonical morphism ${\Pi }_R(U)\to *$ is an equivalence, the path $\mathrm{\infty }$-groupoid functor ${\Pi }_R$ is equivalent to the canonical structured path $\mathrm{\infty }$-groupoid,

$${\Pi }_R\simeq \Pi .$$\mathbf{\Pi}_R \simeq \mathbf{\Pi} \,.

Relative and infinitesimal homotopy $\mathrm{\infty }$-groupoid

The entire situation so far may be discussed also in a relative version, where we have an (∞,1)-topos $H$ sitting by an essential geometric morphism $f:H\to B$ over another one $B$

$$H\stackrel{\stackrel{f_{!}}{\to }}{\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}}B.$$\mathbf{H} \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{B} \,.

Of particular interest is the case where $H$ is an infinitesimal thickening of $B=H_{\mathrm{red}}$, with $f$ induced from a morphism of sites that takes a test space with infinitesimal thickening to its reduced ordinary underlying space. In this case the functor $f_{!}$ identifies infinitesimal neighbour points in the same way as the functor $\Pi$ identifies points connected by a finite path. We therefore write

$$({\Pi }_{\inf }⊣{\mathrm{LConst}}_{\inf }⊣{\Gamma }_{\inf }):H\stackrel{\stackrel{{\Pi }_{\inf }}{\to }}{\stackrel{\stackrel{{\mathrm{LConst}}_{\inf }}{\leftarrow }}{\underset{{\Gamma }_{\inf }}{\to }}}{H}_{\mathrm{red}}$$(\Pi_{inf} \dashv LConst_{inf} \dashv \Gamma_{inf}) : \mathbf{H} \stackrel{\overset{\Pi_{inf}}{\to}} {\stackrel{\overset{LConst_{inf}}{\leftarrow}}{\underset{\Gamma_{inf}}{\to}}} \mathbf{H}_{red}

for the relative geometric morphism and as before reflect back ${\Pi }_{\inf }$ into $H$ to obtain the endo-adjunction

$$({\Pi }_{\inf }⊣{♭}_{\inf }):=({\mathrm{LConst}}_{\inf }\circ {\Pi }_{\inf }⊣{\mathrm{LConst}}_{\inf }\circ {\Gamma }_{\inf }):H\stackrel{\stackrel{{\Pi }_{\inf }}{\to }}{\underset{{♭}_{\inf }}{\leftarrow }}H.$$(\mathbf{\Pi}_{inf} \dashv \flat_{inf}) := (LConst_{inf} \circ \Pi_{inf} \dashv LConst_{inf} \circ \Gamma_{inf}) : \mathbf{H} \stackrel{\overset{\mathbf{\Pi}_{inf}}{\to}}{\underset{\flat_{inf}}{\leftarrow}} \mathbf{H} \,.

Here ${\Pi }_{\inf }$ we call the infinitsimal homotopy $\mathrm{\infty }$-groupoid.

There are then a canonical natural morphisms

$$X\to {\Pi }_{\inf }(X)\to \Pi (X)$$X \to \mathbf{\Pi}_{inf}(X) \to \mathbf{\Pi}(X)

whose composite equals the canonical natural morphism $X\to \Pi (X)$ from above.

Morphisms out of ${\Pi }_{\inf }$ may be interpreted as ∞-Lie algebroid valued differential forms. The obstruction theory to extensions through $X\to {\Pi }_{\inf }(X)$ is that of infinitesimal differential cohomology. Extensions through ${\Pi }_{\inf }(X)\to \Pi (X)$ corresponds to integration of ∞-Lie algebroid valued differential forms.

Infinitesimal path $\mathrm{\infty }$-groupoid

As before, there is realization of the infinitesimal homotopy $\mathrm{\infty }$-groupoid by infinitesimal geometric paths in the case that we have an infinitesimal line object $D\hookrightarrow R$. This induces for representables an infinitesimal singular simplicial complex functor $X\mapsto X^{({\Delta }_{\inf }^{•})}$ and its left derived functor ${\Pi }_D:H\to K$ we call the infinitesmal path $\mathrm{\infty }$-groupoid functor.

We show that if the objects in the site for $H$ are indeed infinitesimal thickenings of the objects of the site for $B$, then this is equivalent to the infinitesimal homotopy $\mathrm{\infty }$-groupoid functor

$${\Pi }_D\simeq {\Pi }_{\inf }.$$\mathbf{\Pi}_{D} \simeq \mathbf{\Pi}_{inf} \,.

This is used to show that the equivalent morphism ${\Pi }_{\inf }(X)\to \Pi (X)$ induces isomorphisms in $R$-cohomology. Together with the identification $H(\Pi (X),B^{n}R)\simeq \mathrm{Top}(\mid X\mid ,{ℬ}^n{ℝ}_{\mathrm{disc}})$ induced from the adjunction $(\Pi ⊣\mathrm{LConst}⊣\Gamma )$ this is the de Rham theorem

$$H^n({\Pi }_{\inf }(X,)R)\simeq H^n(\mid X\mid ,ℝ)$$H^n(\mathbf{\Pi}_{inf}(X,)R) \simeq H^n(|X|, \mathbb{R})

in our $(\mathrm{\infty }\mathrm{,1})$-topos.

Unstructured geometric homotopy $\mathrm{\infty }$-groupoid

We begin by disucssing the situation of a locally contractible $(\mathrm{\infty }\mathrm{,1})$-topos.

Definition

An ordinary Grothendieck topos $𝒯$ is called locally connected if the terminal global sections geometric morphism $(\mathrm{LConst}⊣\Gamma ):𝒯\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{Set}$ is an essential geometric morphism in that there is a further left adjoint $({\Pi }_0⊣\mathrm{LConst}):𝒯\stackrel{\stackrel{{\Pi }_0}{\to }}{\underset{\mathrm{LConst}}{\leftarrow }}\mathrm{Set}$. The functor ${\Pi }_0:𝒯\to \mathrm{Set}$ sends each object $X\in 𝒯$ to its set of connected components as seen by the geometric interpretation of objects in $𝒯$. Notably if $𝒯$ is the category of sheaves on the category of open subsets of a topological space, then ${\Pi }_0$ sends each sheaf to the set of ordinary connected components of its corresponding etale space.

A careful look at known results about geometric homotopy groups in an (∞,1)-topos shows that the following natural definition captures the correct (∞,1)-topos-theoretic analog of this situation.

Definition

We say that an (∞,1)-sheaf/∞-stack (∞,1)-topos $H$ is a locally contractible (∞,1)-topos if the canonical global section geometric morphism is an essential geometric morphism in that we have a pair of adjoint (∞,1)-functors

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}.$$(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,.

The left adjoint $\Pi$ to the constant ∞-stack functor $\mathrm{LConst}$ we call the homotopy $\mathrm{\infty }$-groupoid-functor or fundamental $\mathrm{\infty }$-grupoid-functor.

It defines a notion of geometric geometric homotopy groups in $H$: for $X\in H$ and $n\in \mathbb{N}$ we set

$${\pi }_n^{\mathrm{geom}}(X):={\pi }_n\Pi (X),$$\pi_n^{geom}(X) := \pi_n \Pi(X) \,,

where on the right we have the ordinary homotopy groups in ∞Grpd $\simeq$ Top.

Properties

The following definition captures a large source of examples for locally contractible (∞,1)-toposes.

Definition

Say a site $C$ has geometrically contractible objects if the constant $(\mathrm{\infty }\mathrm{,1})$-presheaf functor

$$\mathrm{Const}:\mathrm{\infty }\mathrm{Grpd}\to \mathrm{\infty }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$$Const : \infty Grpd \to \infty PSh_{(\infty,1)}(C)

factors through ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$. Or in terms of models: if the constant simplicial presheaf functor

$$\mathrm{Const}:{\mathrm{sSet}}_{\mathrm{Quillen}}\to \mathrm{sPSh}(C)$$Const : sSet_{Quillen} \to sPSh(C)

sends fibrant objects of ${\mathrm{sSet}}_{\mathrm{Quillen}}$ to fibrant objects in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Examples/Proposition

The following sites have geometrically contractibel objects, in the above sense:

• CartSp;

• the site ${\mathrm{CartSp}}_{\mathrm{th}}\subset 𝕃$ of smooth loci consisting smoth loci of the form $R^n\times D_{(k)}^n$ with the second factor infinitesimal.

Proposition

The (∞,1)-topos of a site with geometrically contractible objects is a locally contractible (∞,1)-topos in that the constant ∞-stack-functor has a left adjoint

$$(\Pi ⊣\mathrm{LConst}):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\overleftarrow{\to }\mathrm{\infty }\mathrm{Grpd}.$$(\Pi \dashv LConst) : Sh_{(\infty,1)}(C) \stackrel{\leftarrow}{\to} \infty Grpd \,.

Proof

The sSet-functor $\mathrm{LConst}:\mathrm{sSet}\to \mathrm{sPSh}(C)$ given on $S\in \mathrm{sSet}$ by ${\mathrm{LConst}}_S:U\mapsto S$ for all $U\in C$ has an sSet-left adjoint

$$\Pi :X\mapsto {\int }^{U}X(U)=\underrightarrow{\lim }X$$\Pi : X \mapsto \int^U X(U) = \lim_\to X

because for $X\in \mathrm{sPSh}(C)$ and $S\in \mathrm{sSet}$ we have

$$\begin{array}{rl}\mathrm{sPSh}(X,{\mathrm{Const}}_S)& ={\int }_U\mathrm{sSet}(X(U),{\mathrm{Const}}_S(U))\\ & ={\int }_U\mathrm{sSet}(X(U),S)\\ & =\mathrm{sSet}({\int }^{U}X(U),S)\end{array}$$\begin{aligned} sPSh(X,Const_S) &= \int_U sSet(X(U), Const_S(U)) \\ & = \int_U sSet(X(U), S) \\ & = sSet( \int^U X(U) , S) \end{aligned}

naturally in $X$ and $S$. Regarded as a functor ${\mathrm{sSet}}_{\mathrm{Quillen}}\to \mathrm{sPSh}(C{)}_{\mathrm{proj}}$ the functor $\mathrm{LConst}$ manifestly preserves fibrations and acyclic fibrations and hence

$$(\Pi ⊣\mathrm{LConst}):\mathrm{sPSh}(C{)}_{\mathrm{proj}}\overleftarrow{\to }{\mathrm{sSet}}_{\mathrm{Quillen}}$$(\Pi \dashv LConst) : sPSh(C)_{proj} \stackrel{\leftarrow}{\to} sSet_{Quillen}

is a Quillen adjunction, in particular $\Pi :\mathrm{sPSh}(C{)}_{\mathrm{proj}}\to {\mathrm{sSet}}_{\mathrm{Quillen}}$ preserves cofibrations. Since the cofibrations of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ are the same, also $\Pi :\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}\to {\mathrm{sSet}}_{\mathrm{Quillen}}$ preserves cofibrations. And by assumption on $C$ we have that $\mathrm{LConst}:{\mathrm{sSet}}_{\mathrm{Quillen}}\to \mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ preserves fibrant objects. Since ${\mathrm{sSet}}_{\mathrm{Quillen}}$ is a left proper model category it follows with HTT, corollary A.3.7.2 that also

$$(\Pi ⊣\mathrm{LConst}):\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}\overleftarrow{\to }{\mathrm{sSet}}_{\mathrm{Quillen}}$$(\Pi \dashv LConst) : sPSh(C)_{proj}^{loc} \stackrel{\leftarrow}{\to} sSet_{Quillen}

is a Quillen adjunction.

Remark

By the rules of Yoneda reduction we have for $X={\coprod }_i{U}_i$ a coproduct of representables that $\Pi (X)={\coprod }_i*$.

By Dugger’s cofibrant replacement theorem we have that every object $X$ in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$, hence also in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ has a cofibrant replacement by a simplicial presheaf

$$\hat{X}={\int }^{[n]\in \Delta }\Delta [n]\cdot \left(\coprod _{i_n}{U}_{i_n}\right)$$\hat X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_{i_n} \right)

that is degreewise a coproduct of representables. The image of this under $\Pi$ is

$$\Pi (\hat{X})={\int }^{[n]\in \Delta }\Delta [n]\cdot (\coprod _{i_n}*).$$\Pi(\hat X) = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} * \right) \,.

This reproduces the familiar computation of the fundamental $\mathrm{\infty }$-groupoid of a space as discussed at homotopy groups in an (∞,1)-topos.

Corollary

On a site $C$ with geometrically contractible objects, the two adjunctions constituting the essential geometric morphism

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\overrightarrow{\overleftarrow{\to }}\mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv LConst \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\to}{\stackrel{\leftarrow}{\to}} \infty Grpd

are such that the composite

$$(\Pi \mathrm{LConst}⊣\Gamma \mathrm{LConst}):\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\mathrm{LConst}}{\to }}{\underset{\Gamma }{\leftarrow }}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{\Pi }{\to }}{\underset{\mathrm{LConst}}{\leftarrow }}\mathrm{\infty }\mathrm{Grpd}$$(\Pi LConst \dashv \Gamma LConst) \; : \; \infty Grpd \stackrel{\overset{LConst}{\to}}{\underset{\Gamma}{\leftarrow}} Sh_{(\infty,1)}(C) \stackrel{\overset{\Pi}{\to}}{\underset{LConst}{\leftarrow}} \infty Grpd

is (equivalent to) the identity adjunction $(Id⊣\mathrm{Id})$.

Remark

This implies that such $(\mathrm{\infty }\mathrm{,1})$-toposes $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ have the same shape – in the sense of shape theory for (∞,1)-toposes – as the point.

Geometric realization

Let $X=$ CartSp.

In

• Daniel Dugger, Universal Homotopy Theories

is defined a geometric realization functor

$$\mid -\mid :{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})\to \mathrm{Top}.$$|-| : Sh_{(\infty,1)}(CartSp) \to Top \,.

Proposition Up to the equivalence between $\mathrm{\infty }\mathrm{Grpd}$ and $\mathrm{Top}$ this “geometric realization” is just $\Pi (-)$.

Proof By prop 2.8 of Universal Homotopy Theories for every $X\in \mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ there is a cofibrant replacement of the form

$$\hat{X}=d(X_{•,•})\simeq {\int }^{[n]}\Delta [n]\times \widehat{X_n},$$\hat X = d(X_{\bullet, \bullet}) \simeq \int^{[n]} \Delta[n] \times \widehat {X_n} \,,

where $\widehat{X_n}$ is in turn a good cover of $X_n$

$$\widehat{(}{X}_n)={\int }^{[k]}{\Delta }^k\cdot \left(\coprod _{i_k}{U}_{i_k}\right).$$\widehat (X_n) = \int^{[k]} \Delta^k \cdot \left( \coprod_{i_k} U_{i_k} \right) \,.

$\Pi$ sends $\widehat{X_i}$ to

$$\Pi (\widehat{X_n})={\int }^{[k]}{\Delta }^k\cdot (\coprod _{i_k}*)$$\Pi(\widehat {X_n}) = \int^{[k]} \Delta^k \cdot \left( \coprod_{i_k} * \right)

which is the $\mathrm{\infty }$-groupoid incarnation of the topological space $X_n^{\mathrm{Top}}$ underlying $X_n$. So

$$\Pi (\hat{X})={\int }^{[n]}\Delta [n]\times \mathrm{Sing}(X_n^{\mathrm{Top}}),$$\Pi(\hat X) = \int^{[n]} \Delta[n] \times Sing(X_n^{Top}) \,,

Applying $\mid -\mid :\mathrm{sSet}\to \mathrm{Top}$ yields

$$\mapsto \simeq {\int }^{[n]}{\Delta }_{\mathrm{Top}}^n\times X_n^{\mathrm{Top}},$$\mapsto \simeq \int^{[n]} \Delta^n_{Top} \times X_n^{Top} \,,

the standard geometric realization.

Structured geometric homotopy $\mathrm{\infty }$-groupoid

Definition

We obtain yet another endo-adjunction by composing the pair of adjunctions $(\Pi ⊣\mathrm{LConst})$ and $(\mathrm{LConst}⊣\Gamma )$ in the other direction. This is reflects the unstructured homtopy $\mathrm{\infty }$-groupoid back into $H$.

Definition

Write

$$(\Pi ⊣♭):=(\mathrm{LConst}\circ \Pi ⊣\mathrm{LConst}\circ \Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\stackrel{\stackrel{\Pi }{\to }}{\underset{\mathrm{LConst}}{\leftarrow }}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\mathrm{LConst}}{\to }}{\underset{\Gamma }{\leftarrow }}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C).$$(\mathbf{\Pi} \dashv \mathbf{\flat}) := (LConst \circ \Pi \dashv LConst \circ \Gamma ) \; : \; Sh_{(\infty,1)}(C) \stackrel{\overset{\Pi}{\to}}{\underset{LConst}{\leftarrow}} \infty Grpd \stackrel{\overset{LConst}{\to}}{\underset{\Gamma}{\leftarrow}} Sh_{(\infty,1)}(C) \,.

We say

• $\Pi (-)$ the structured or internal homotopy ∞-groupoid functor;

• for $A\in H$ the intrinsic cohomology with coefficients in $♭(A)$ is flat differential cohomology;

  $$H_{\mathrm{flat}}(X,A):={\pi }_{0}H(X,♭A).$$H_{flat}(X,A) := \pi_0 \mathbf{H}(X,\flat{A}) \,.

The unit of the adjunction $(\Pi ⊣\mathrm{LConst})$ with components

$$X\to \Pi (X)$$X \to \mathbf{\Pi}(X)

we call the constant path inclusion .

Remark

The notion of extension along the constant path inclusion, hence the notion of localization that identifies $X$ with $\Pi (X)$ encodes crucial information about the internal geometry of $H$. We may think of differential cohomology in an (∞,1)-topos as the obstruction theory to such extensions.

Properties

The following lemma is a simple formal consequence of the definitions so far, but plays an central conceptual role. Its main impact arises from applying it to the geometric path $\mathrm{\infty }$-groupoid construction ${\Pi }_R$ discussed below that is, if it exists, equivalent to the structured homotopy $\mathrm{\infty }$-groupoid functor $\Pi$.

Lemma

Let $k\in H$ be an abelian group object in $H$ and let

$$k:=\Gamma (k)\in \mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$$k := \Gamma(\mathbf{k}) \in \infty Grpd \simeq Top

be the unstructured group object underlying it.

Write $\mid X\mid \in \mathrm{Top}$ for the image of $\Pi (X)\in \mathrm{\infty }\mathrm{Grpd}$ under the canonical equivalence $\mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$.

Then the internal $k$-cohomology of $\Pi (X)$ is isomorphic to the ordinary cohomology of $\mid X\mid$ in Top with coefficients in $k$.

$${\pi }_{0}H(\Pi (X),k)\simeq H^n(\mid X\mid ,k).$$\pi_0\mathbf{H}(\mathbf{\Pi}(X),\mathbf{k}) \simeq H^n(|X|,k) \,.

In fact, even the cocycle categories are equivalent:

$$H(\Pi (X),B^{n}k)\simeq \mathrm{Top}(\mid X\mid ,K(k,n)),$$\mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n \mathbf{k}) \simeq Top(|X|, K(k,n)) \,,

where $K(k,n)$ is the corresponding Eilenberg-MacLane space.

Proof

This is just the defining adjunctions at work:

$$\begin{array}{rl}H(\Pi (X),B^{n}k)& :=H(\mathrm{LConst}\Pi (X),B^{n}k)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),\Gamma B^{n}k)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),{ℬ}^n\Gamma k)\\ & =:\mathrm{\infty }\mathrm{Grpd}(\Pi (X),{ℬ}^{n}k)\\ & \simeq \mathrm{Top}(\mid X\mid ,K(k,n))\end{array}.$$\begin{aligned} \mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n\mathbf{k}) &:= \mathbf{H}(LConst \Pi (X),\mathbf{B}^n \mathbf{k}) \\ &\simeq \infty Grpd(\Pi(X), \Gamma \mathbf{B}^n \mathbf{k}) \\ &\simeq \infty Grpd(\Pi(X), \mathcal{B}^n \Gamma \mathbf{k}) \\ &=: \infty Grpd(\Pi(X), \mathcal{B}^n k) \\ &\simeq Top(|X|,K(k,n)) \end{aligned} \,.

Here ${ℬ}^{n}k$ denotes the $k$-fold delooping in ∞Grpd and we use that the right adjoint $\Gamma$ preserves loop space objects and hence also deloopings.

Remark

It is useful to reflect this statement back into $H$, where it has an even simpler appearance:

$$H(\Pi (X),B^{n}k)\simeq H(X,\mathrm{LConst}{ℬ}^{n}k).$$\mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}^n\mathbf{k}) \simeq \mathbf{H}(X, LConst \mathcal{B}^n k) \,.

We will see below that $\Pi (X)$ has a natural model $\cdots \simeq {\mathrm{Pi}}_R(X)$ by geometric singular simplices in $X$, which identifies $H(\Pi (X),B^{n}k)$ effectively with the intrinsic singular cohomology of $X\in H$. Moreover, the inclusion of the infinitesimal path ∞-groupoid ${\Pi }_{\inf }(X)\hookrightarrow {\Pi }_R(X)\simeq \Pi (X)$ identifies this naturally with the intrinsic de Rham cohomology of $X$. As a result, we will find that the above equivalence is effectively the statement of the intrinsic de Rham theorem in $H$.

Example

Let the underlying site be $C=$ CartSp and write $R\in H$ for the internal incarnation of the canonical line object $ℝ$: the ∞-stack that is just the sheaf represented by $ℝ\in \mathrm{CartSp}$. Then of course

$$\Gamma R=ℝ\in \mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$$\Gamma R = \mathbb{R} \in \infty Grpd \simeq Top

is the real line regarded as a topological space, but – crucially – equipped with the discrete topology : this is just the set of global sections $\Gamma R={\mathrm{Sh}}_{\mathrm{CartSp}}(*,R)={\mathrm{Hom}}_{\mathrm{CartSp}}({ℝ}^0,{ℝ}^1)=ℝ\in \mathrm{Set}\hookrightarrow \mathrm{Top}$ of the smooth incarnation $R$ of $ℝ$.

So we have for every $X\in H$ that

$$H(\Pi (X),B^{n}R)\simeq \mathrm{Top}(\mid X\mid ,K(ℝ,n))$$\mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n R) \simeq Top(|X|, K(\mathbb{R},n))

and hence that

$${\pi }_{0}H(\Pi (X),B^{n}R)\simeq H^n(\mid X\mid ,ℝ)$$\pi_0 \mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n R) \simeq H^n(|X|, \mathbb{R})

is the ordinary real cohomology of the geometric realization $\mid X\mid$ of $X$.

Geometric path $\mathrm{\infty }$-groupoid

We wish to show that, at least under suitable conditions, in a locally contractible (∞,1)-topos $H$ one can find an interval object $R$ such that the structured homotopy $\mathrm{\infty }$-groupoid functor $\Pi :H\to H$ is equivalently given by a functor ${\Pi }_R$ that is locally given by forming $k$-dimensional geometric paths in an object $X$, modeled on the interval object $R$ and hence akin to a structured singular complex of $X$.

This explicit realization of the abstractly defined $\Pi$ in terms of a path $\mathrm{\infty }$-groupoid ${\Pi }_R$ connects the cohomology of $\Pi (X)$ manifestly with the notion of parallel transport and local systems on $X$ and induces in a smooth (∞,1)-topos the canonical morphism ${\Pi }_{\inf }(X)\hookrightarrow {\Pi }_R(X)\simeq \Pi (X)$ from the infinitesimal path ∞-groupoid.

Definition

Let $C$ be a site and let $(𝒯=\mathrm{Sh}(C),R)$ be a lined topos with $R\in C\subset \mathrm{Sh}(C)$ a representable line object. Thought of as a cartesian interval object this induces a cosimplicial object

$${\Delta }_R:\Delta \to C$$\Delta_R : \Delta \to C

of standard $k$-simplices in $C$ modeled on $R$.

Write $\mathrm{sPSh}(C)$ or ${\mathrm{sPSh}}_C$ for the SSet enriched category of simplicial presheaves on $C$, write $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ or $\mathrm{sPSh}(C{)}_{\mathrm{inj}}$ for the projective or injective, respectively, global model structure on simplicial presheaves and $\mathrm{sPSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}$ and $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ for the corresponding Cech model structure on simplicial presheaves.

For $U\in C\subset \mathrm{Sh}(C)$ a representable we can form the simplicial object

$$U^{{\Delta }_R^{•}}\in [{\Delta }^{\mathrm{op}},\mathrm{PSh}(C)]\simeq \mathrm{sPSh}(C)$$U^{\Delta^\bullet_{R}} \in [\Delta^{op}, PSh(C)] \simeq sPSh(C)

by forming degreewise the internal hom of presheaves. This is a naive model for the geometric path $\mathrm{\infty }$-groupoid of $U$.

Definition

(geometric path $\mathrm{\infty }$-groupoid)

To extend this construction from representables $U$ to general objects $X\in \mathrm{sPSh}(C)$ use the small object argument to choose a functorial factorization

$$\begin{array}{ccc}U& \hookrightarrow & {\Pi }_R(U)\\ & \searrow & {\downarrow }^{\simeq }\\ & & U^{{\Delta }_R^{•}}\end{array}$$\array{ U &\hookrightarrow& \mathbf{\Pi}_R(U) \\ & \searrow & \downarrow^\simeq \\ && U^{\Delta^\bullet_{R}} }

into a cofibration $U\hookrightarrow {\Pi }_R(U)$ and a weak equivalence ${\Pi }_R(U)\stackrel{\simeq }{\to }{U}^{{\Delta }_R^{•}}$ in the global projective model structure $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ and hence also in the local projective model structure $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$. Since all representables $U$ are cofibrant in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ it follows that also ${\Pi }_R(U)$ is cofibrant in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ and hence also in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Then for general $X\in \mathrm{sPSh}(C)$ set

$${\Pi }_R(X):={\int }^{U\in C}{\Pi }_R(U)\cdot X(U).$$\mathbf{\Pi}_R(X) := \int^{U \in C} \mathbf{\Pi}_R(U) \cdot X(U) \,.

This defines an sSet-enriched functor

$${\Pi }_R:\mathrm{sPSh}(C{)}_{\mathrm{proj}}\overleftarrow{\to }\mathrm{sPSh}(C{)}_{\mathrm{proj}}:{♭}_R$$\mathbf{\Pi}_R : sPSh(C)_{proj} \stackrel{\leftarrow}{\to} sPSh(C)_{proj} : \flat_R

which by general nonsense has a right adjoint ${♭}_R$.

Lemma

The functor ${\Pi }_R$ preserves cofibrations and global acyclic cofibrations in $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$. Moreover, there is a canonical morphism

$$X\to {\Pi }_R(X)$$X \to \mathbf{\Pi}_R(X)

natural in $X\mathrm{in}\mathrm{sPSh}(C)$ which is a cofibration when $X$ is cofibrant in $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$.

Proof

We use that the coend over the tensoring of the simplicial model category $\mathrm{sPSh}(C)$ over ${\mathrm{sSet}}_{\mathrm{Quillen}}$

$$\int (-)\cdot (-):[C,\mathrm{sPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}\times [C^{\mathrm{op}},{\mathrm{SSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj}}\to \mathrm{sPSh}(C{)}_{\mathrm{proj}}$$\int (-)\cdot (-) : [C,sPSh(C)_{proj}]_{inj} \times [C^{op},SSet_{Quillen}]_{proj} \to sPSh(C)_{proj}

in the definition of ${\Pi }_R$ is a left Quillen bifunctor (as discussed there) on the injective global model structure of functors from $C$ to the projective global model structure on simplicial presheaves and the projective global model structure on simplicial presheaves itself.

This implies that with one of the arguments fixed and cofibrant, the functor respects cofibrations and acyclic cofibrations in the other argument.

So for $X\in [C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ fixed and cofibrant applying the functor to the natural cofibration $((-)\to {\Pi }_R(-))\in [C,\mathrm{sPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}$ yields a cofibration

$$X={\int }^{U\in C}U\cdot X(U)\to {\int }^{U\in C}{\Pi }_R(U)\cdot X(U)={\Pi }_R(X).$$X = \int^{U \in C} U \cdot X(U) \to \int^{U \in C} \mathbf{\Pi}_R(U) \cdot X(U) = \mathbf{\Pi}_R(X) \,.

Similarly, since ${\Pi }_R(-)\in [C,\mathrm{sPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}$ is degreewise cofibrant and hence cofibrant,for $X\to Y$ a cofibration or acyclic cofibration in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ the induced morphism

$${\Pi }_R(X)={\int }^{U\in C}{\Pi }_R(U)\cdot X(U)\to {\int }^{U\in C}{\Pi }_R(U)\cdot Y(U)={\Pi }_R(Y)$$\mathbf{\Pi}_R(X) = \int^{U \in C} \mathbf{\Pi}_R(U)\cdot X(U) \to \int^{U \in C} \mathbf{\Pi}_R(U) \cdot Y(U) = \mathbf{\Pi}_R(Y)

is a cofibration or acyclic cofibration in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$, respectively.

Proposition

If in the underlying lined topos $(𝒯=\mathrm{Sh}(C),R)$ all representable objects are contractible with respect to R in that the canonical morphism

$${\Pi }_R(U)\to *$$\mathbf{\Pi}_R(U) \to *

is a global weak equivalence, and if the localization is at good covers (the Cech nerve is degreewise a coproduct of representables), then the adjunction ${\Pi }_R⊣{♭}_R$ above is a Quillen adjunction with respect to the Cech model structure on simplicial presheaves

$${\Pi }_R:\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}\overleftarrow{\to }\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}:{♭}_R.$$\mathbf{\Pi}_R : sPSh(C)_{proj}^{loc} \stackrel{\leftarrow}{\to} sPSh(C)_{proj}^{loc} : \flat_R \,.

Proof

We first notice two lemmas.

Lemma 1 ${\Pi }_R$ sends Cech nerves $(C(\{U_i\})\to U)$ of good covers to global weak equivalences.

Proof of lemma 1: By the assumption that the cover is good, we have a weak equivalence of simplicial sets

$${\Pi }_R((\coprod _i{U}_i{)}^{{\times }_U^{•+1}})(V)\stackrel{\simeq }{\to }*.$$\mathbf{\Pi}_R( (\coprod_i U_i)^{\times_U^{\bullet+1}})(V) \stackrel{\simeq}{\to} * \,.

Moreover by Bousfield-Kan we have a weak equivalence $\Delta \stackrel{\simeq }{\to }*$ of cosimplicial simplicial sets.

The coend

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

is a Quillen bifunctor, so that we have a weak equivalence

$${\int }^n{\Pi }_R((\coprod _i{U}_i{)}^{{\times }_U^{(n+1)}})(V)\times {\Delta }^n\stackrel{\simeq }{\to }{\int }^n{\Delta }^n\stackrel{\simeq }{\to }*$$\int^n \mathbf{\Pi}_R((\coprod_i U_i)^{\times_U^{(n+1)}})(V) \times \mathbf{\Delta}^n \stackrel{\simeq}{\to} \int^n \mathbf{\Delta}^n \stackrel{\simeq}{\to} *

Lemma 2: The right adjoint ${♭}_R$ preserves the fibrant objects of $\mathrm{SsPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Proof of lemma 2. By the general mechanism of left Bousfield localization, the fibrant objects of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ are the fibrant objects of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ that are local objects with respect to the set of good Cech covers.

Being a right Quillen functor on $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$, the functor ${♭}_R$ preserves the global fibrancy of objects. To show moreover that ${♭}_R(A)$ is a Cech cover local object for $A$ globally fibrant, we need to show that for all $C({\coprod }_i{U}_i)\to U$ we have that $\mathrm{sPSh}(U,{♭}_R(A))\to \mathrm{SPSh}(C({\coprod }_i{U}_i),{♭}_R(A))$ is a weak equivalence.

By the adjunction ${\Pi }_R⊣{♭}_R$ this is the same as $\mathrm{sPSh}({\Pi }_R(U),A)\to \mathrm{sPSh}({\Pi }_R(C({\coprod }_i{U}_i)),A)$ being a weak equivalence. Since both ${\Pi }_R(U)$ and ${\Pi }_R(C({\coprod }_i{U}_i))$ are cofibrant (the original objects are cofibrant by the discussion of cofibrant objects at model structure on simplicial presheaves and ${\Pi }_R$ preserves global cofibrations, which, by the definition of left Bousfield llocalization are the same as local cofibrations) this is a map of derived hom-spaces $R\mathrm{Hom}({\Pi }_R(U),A)\to \mathrm{R}\mathrm{Hom}(\Pi (C(\coprod (U_i))),A)$. And since by lemma 1 the map ${\Pi }_R(C({\coprod }_i{U}_i))\to \Pi (U)$ is a global weak equivalence, hence a local weak equivalence, this is indeed a weak equivalence.

Proof of the proposition

By the properties of left Bousfield localization, the cofibrations of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ are the same as those of $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ and these are preserved by ${\Pi }_R$, due to it being left Quillen with respect to the global model structure.

So it is sufficient to show that ${\Pi }_R$ sends cofibrations $Y\to X$ that are local equivalences to cofibrations that are local equivalences.

By the properties of local acyclic cofibrations these are characterized by the fact that for every fibrant local object $A$ the morphism $\mathrm{SPSh}(Y,A)\to \mathrm{SPSh}(X,A)$ is an acyclic Kan fibrations (where the crucial point is that we do not have to pass to a cofibrant replacement of $X$ and $Y$).

So we need to show that for $Y\to X$ such a local acyclic cofibration, the morphism $\mathrm{sPSh}({\Pi }_R(Y),A)\to \mathrm{sPSh}({\Pi }_R(X),A)$ is an acyclic Kan fibration for all fibrant and local $A$. By the adjunction ${\Pi }_R⊣{♭}_R$ this is equivalent to $\mathrm{sPSh}(Y,{♭}_R(A))\to \mathrm{sPSh}(\hat{X},{♭}_R(A))$ being a weak equivalence. But by lemma 2 ${♭}_R(A)$ is still locally fibrant. Therefore this is indeed a local weak equivalence, by their above mentioned characterization.

Properties

On a site of geometrically contractible objects with localization at good Cech covers, the structured homotopy $\mathrm{\infty }$-groupoid functor $\Pi$ and the geometric path $\mathrm{\infty }$-groupoid functor ${\Pi }_R$ are equivalent as left derived functors/(∞,1)-functors

$$\Pi \simeq {\Pi }_R.$$\mathbf{\Pi} \simeq \mathbf{\Pi}_R \,.

Proof

Every object $X\in \mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ has a cofibrant replacement $\tilde{X}$ that is degeewise a coproduct of representables (as described here).

$$\tilde{X}={\int }^{[n]}{\Delta }^n\cdot \left(\coprod _{i_n}{U}_{i_n}\right).$$\tilde X = \int^{[n]} \Delta^n \cdot \left( \coprod_{i_n} U_{i_n} \right) \,.

By coend manipulations as above we have ${\Pi }_R(\tilde{X})\simeq {\int }^n{\Delta }^n\cdot {\Pi }_R({\tilde{X}}_n)$. Hence with the above

$${\Pi }_R(\tilde{X})\simeq {\int }^{[n]}{\Delta }^n\cdot (\coprod _{i_n}{\Pi }_R(U_{i_n})).$$\mathbf{\Pi}_R(\tilde X) \simeq \int^{[n]} \mathbf{\Delta}^n \cdot \left( \coprod_{i_n} \mathbf{\Pi}_R(U_{i_n}) \right) \,.

By the assumptions that for all representables $U$ we have a weak equivalence ${\Pi }_R(U)\stackrel{\simeq }{\to }*$ this is weakly equivalent to

$${\Pi }_R(\tilde{X})\simeq {\int }^{[n]}{\Delta }^n\cdot (\coprod _{i_n}*).$$\mathbf{\Pi}_R(\tilde X) \simeq \int^{[n]} \mathbf{\Delta}^n \cdot \left( \coprod_{i_n} {*} \right) \,.

This is in the image of $\mathrm{LConst}:\mathrm{\infty }\mathrm{Grpd}\to \mathrm{SPSh}(C)$. We claim that its preimage in $\mathrm{\infty }\mathrm{Grpd}$ is the bare path $\mathrm{\infty }$-groupoid of $X$.

Similarly one sees that

$$(\mathrm{LConst}{♭}_R(𝒜)\simeq \mathrm{LConst}𝒜.$$(LConst \flat_R(\mathcal{A}) \simeq LConst \mathcal{A} \,.

Using all this, we find that $\Gamma \Pi$ is indeed left adjoint to $\mathrm{LConst}$:

$$\begin{array}{rl}H(X,\mathrm{LConst}𝒜)& \simeq H(X,(\mathrm{LConst}𝒜{)}_{\mathrm{flat}})\\ & \simeq H(\Pi (X),\mathrm{LConst}𝒜)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(\Gamma \Pi (X),\Gamma \mathrm{LConst}𝒜)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(\Gamma \Pi (X),𝒜)\end{array}$$\begin{aligned} \mathbf{H}(X, LConst \mathcal{A}) & \simeq \mathbf{H}(X, (LConst \mathcal{A})_{flat}) \\ &\simeq \mathbf{H}(\mathbf{\Pi}(X), LConst \mathcal{A}) \\ &\simeq \infty Grpd(\Gamma \mathbf{\Pi}(X), \Gamma LConst \mathcal{A}) \\ &\simeq \infty Grpd(\Gamma \mathbf{\Pi}(X), \mathcal{A}) \end{aligned}

So that we may indeed identify it with $\Pi (X)\in \mathrm{\infty }\mathrm{Grpd}$.

Remark

This discussion is just a slight variation of the discussion on pages 17 and 29

• Daniel Dugger, Sheaves and homotopy theory (web, dvi, pdf)

For $I$ the interval object, in that text the Bousfield localization at the morphisms $\{X\times I\to X\mid X\}$ is considered. That makes all representables there equivalent to the point. Then one already has $\tilde{X}\simeq {\int }^n{\Delta }^n({\coprod }_{\mathrm{in}}{U}_{i_n})\simeq {\int }^n{\Delta }^n({\coprod }_{\mathrm{in}}*)$ because of the localization. Instead of localizing the whole category, here we apply $\Pi$ to a given object, which there has the same effect.

Smooth singular cohomology

For $X\in H$ a cocycle on ${\Pi }_R(X)$ with values in $B^{n}R$ is a structured (smooth) singular cochain on $X$. By the adjunction property we have that the induced smooth singular cohomology is isomorphic to the real cohomology of the geometric realization $\mid X\mid$ of $X$

Observation

We have a natural isomorphism

$$H^n({\Pi }_R(X),R)\simeq H^n(\mid X\mid ,ℝ).$$H^n(\mathbf{\Pi}_R(X),R) \simeq H^n(|X|, \mathbb{R}) \,.

Proof

By the adjunction equivalences we even have an equivalence of cocycle categories

$$\begin{array}{rl}{H}^n(\Pi (X),R)& \simeq {\pi }_{0}H(\Pi (X),B^{n}R)\\ & \simeq {\pi }_0\mathrm{Top}(\mid X\mid ,{ℬ}^nℝ)\\ & \simeq H^n(\mid X\mid ,ℝ)\end{array}.$$\begin{aligned} H^n(\mathbf{\Pi}(X),R) & \simeq \pi_0 \mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n R) \\ & \simeq \pi_0 Top(|X|, \mathcal{B}^n \mathbb{R}) \\ & \simeq H^n(|X|, \mathbb{R}) \end{aligned} \,.

Infinitesimal homotopy $\mathrm{\infty }$-groupoid

We now look at the analogous discussion with the canonical morphism of sites $C\to *$ replaced by a morphism $\mathrm{Red}:C\to C_{\mathrm{red}}$, where $C$ is a site of infinitesimally thickenings of the objects of $C_{\mathrm{red}}$ and $\mathrm{Red}$ the operation that discards the thickening.

The discussion is similar to the discussion around page 7 in

• Carlos Simpson, Constantin Teleman, de Rham’s theorem for $\mathrm{\infty }$-stacks (pdf)

Definition

Restrict attention now to the site $C={\mathrm{CartSp}}_{\mathrm{th}}$ of smooth loci of the form ${ℝ}^n\times D_k$. This are cartesian spaces times an infinitesimal space, the formal duals of smooth algebras of the form $C^{\mathrm{\infty }}({ℝ}^n)\otimes W$ for $W$ a Weil algebra in the sense if synthetic differential geometry.

There is then the evident morphism of sites $\mathrm{Red}:{\mathrm{CartSp}}_{\mathrm{th}}\to \mathrm{CartSp}$ that discards the infinitesimal thickening.

Proposition

The induced geometric morphism of (∞,1)-toposes is essential in that we have three adjoint (∞,1)-functors

$$({\Pi }_{\inf }⊣{\mathrm{LConst}}_{\inf }⊣{\Gamma }_{\inf }):H:={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}({\mathrm{CartSp}}_{\mathrm{th}})\overrightarrow{\overleftarrow{\underset{}{\to }}}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})=:H_{\mathrm{red}}$$(\Pi_{inf} \dashv LConst_{inf} \dashv \Gamma_{inf} ) : \mathbf{H} := Sh_{(\infty,1)}(CartSp_{th}) \stackrel{\overset{}{\to}}{\stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}} Sh_{(\infty,1)}(CartSp) =: \mathbf{H}_{red}

Proof

It is sufficient to observe that the image ${\mathrm{LConst}}_{\inf }X=X(\mathrm{Red}(-))$ of fibrant objects $X\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}^{\mathrm{loc}}$ is fibrant in $[{\mathrm{CartSp}}_{\mathrm{th}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}^{\mathrm{loc}}$. The claim then follows by exactly the same argument as above, that showed that $\Gamma :[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}^{\mathrm{loc}}\to {\mathrm{sSet}}_{\mathrm{Quillen}}$ is an essential geometric morphism if $\mathrm{LConst}$ sends fibrant objects to objects that satisfy descent.

As before we form the structured infinitesimal homotopy $\mathrm{\infty }$-groupoid by reflecting back into $H$, and we write

$$({\Pi }_{\inf }⊣{♭}_{\inf }):=({\mathrm{LConst}}_{\inf }\circ {\Pi }_{\inf }⊣{\mathrm{LConst}}_{\inf }\circ {\Gamma }_{\inf }):H\stackrel{\stackrel{{\Pi }_{\inf }}{\leftarrow }}{\underset{{♭}_{\inf }}{\to }}H.$$(\mathbf{\Pi}_{inf} \dashv \flat_{inf}) := (LConst_{inf} \circ \Pi_{inf} \dashv LConst_{inf}\circ \Gamma_{inf}) : \mathbf{H} \stackrel{\overset{\Pi_{inf}}{\leftarrow}}{\underset{\flat_{inf}}{\to}} \mathbf{H} \,.

The image of the unit of the $(\Pi ⊣\mathrm{LConst})$ precomposed with ${\Pi }_{\inf }$ under ${\mathrm{LConst}}_{\inf }$ yields a canonical natural morphism

$${\Pi }_{\inf }(X)\to \Pi (X).$$\mathbf{\Pi}_{inf}(X) \to \mathbf{\Pi}(X) \,.

This we may think of as the inclusion of infintiesimal paths into all paths in $X$. We justify this below by realizing ${\Pi }_{\inf }$ concretely in terms of infinitesimal paths and realizing this morphism as the derived natural transformation ${\Pi }_D(X)\to {\Pi }_R(X)$ induced by the inclusion $D\hookrightarrow R$ of the infiniesimal line object into the line object $R$.

Properties

Proposition

The natural morphism ${\Pi }_{\inf }(X)\to \Pi (X)$ induces an isomorphism on $R$-cohomology

$${\pi }_{0}H(\Pi (X),B^{n}R)\stackrel{\simeq }{\to }{\pi }_{0}H({\Pi }_{\inf }(X),B^{n}R).$$\pi_0 \mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}^n R) \stackrel{\simeq}{\to} \pi_0 \mathbf{H}(\mathbf{\Pi}_{inf}(X), \mathbf{B}^n R) \,.

Proof

We prove this below by first producing an equivalence ${\Pi }_D\simeq {\Pi }_{\inf }$ analogous to the equivalence ${\Pi }_R\simeq \Pi$ and then showing that ${\Pi }_D(X)\to {\Pi }_R(X)$ induces an equivalence in $R$-cohomology.

Infinitesimal path $\mathrm{\infty }$-groupoid

We still consider the site $C={\mathrm{CartSp}}_{\mathrm{th}}$ of infinitesimally thickened cartesian space. Now we make use of the infinitesimal line object

$$D=\{x\in R\mid x^2=0\}\hookrightarrow R$$D = \{x \in R | x^2 = 0\} \hookrightarrow R

inside there to exhibit a realization of the infinitesimal homotopy $\mathrm{\infty }$-groupoid in terms of infinitesimal paths.

Definition

For $U\in C$ we may then form the infinitesimal singular simplicial complex $U^{({\Delta }_D^{•})}$, regarded as a simplicial presheaf on $C$. (Notice that this is not an internal hom, which we indicate by the parenthesis in the exponent.) This comes with a canonical natural inclusion

$$U^{({\Delta }_D^{•})}\mathrm{to}{U}^{{\Delta }_R^{•}}.$$U^{(\Delta^\bullet_D)} to U^{\Delta^\bullet_R} \,.

We form a functorial cofibrant replacement of this that is compatible with the one we have chosen for $U^{{\Delta }_R^{•}}$ above by forming the pullback $Q$ in

$$\begin{array}{ccccc}& & & & U\\ & & & \swarrow & \downarrow \\ {\Pi }_D(U)& \to & Q& \to & {\Pi }_R(U)\\ & & \downarrow & & \downarrow \\ & & U^{({\Delta }_D^{•})}& \to & U^{{\Delta }_R^{•}}\end{array}$$\array{ &&&& U \\ &&&\swarrow& \downarrow \\ \mathbf{\Pi}_{D}(U)&\to&Q &\to& \mathbf{\Pi}_R(U) \\ &&\downarrow && \downarrow \\ &&U^{(\Delta^\bullet_{D})} &\to& U^{\Delta^\bullet_R} }

and forming one more functorial cofibant replacement $\varnothing \hookrightarrow {\Pi }_D(U)\stackrel{\simeq }{\to }Q$.

Using this we define as before for $X\in \mathrm{sPSh}(C)$

$${\Pi }_D(X):={\int }^{U\in C}{\Pi }_D(U)\cdot X(U)$$\mathbf{\Pi}_{D}(X) := \int^{U \in C} \mathbf{\Pi}_{D}(U) \cdot X(U)

and obtain an adjunction

$${\Pi }_D:\mathrm{sPSh}(C)\overleftarrow{\to }\mathrm{sPSh}(C):{♭}_D.$$\mathbf{\Pi}_{D} : sPSh(C) \stackrel{\leftarrow}{\to} sPSh(C) : \flat_{D} \,.

The componentwise inclusions ${\Pi }_D(U)\to {\Pi }_R(U)$ induce a natural morphism

$${\Pi }_D(X)\to {\Pi }_R(X).$$\mathbf{\Pi}_{D}(X) \to \mathbf{\Pi}_R(X) \,.

Properties

As above, write $R\in \mathrm{sPSh}(C)$ for the object represented by ${ℝ}^1\in C$. Recall from the discussion above the notation $\Gamma R=ℝ\in \mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$ etc., with $ℝ$ on the right understood with the discrete topology.

Proposition

We have a natural equivalence

$${\Pi }_D\simeq {\Pi }_{\inf }.$$\mathbf{\Pi}_{D} \simeq \mathbf{\Pi}_{inf} \,.

Proof

The crucial observation is that ${\Pi }_D$ makes infinitesimal neighbour points isomorphic in that for $\hat{U}\in {\mathrm{CartSp}}_{\mathrm{th}}$ an infinitesimal thickening $\mathrm{Red}\hat{U}=U$ of $U\in \mathrm{CartSp}$, the canonical morphism

$${\mathrm{LConst}}_{\inf }(U)\to {\Pi }_D(\hat{U})$$LConst_{inf}(U) \to \mathbf{\Pi}_D(\hat U)

is a weak equivalence in $\mathrm{sPSh}({\mathrm{CartSp}}_{\mathrm{th}}{)}_{\mathrm{proj}}$ . This is the analog of the statement we had before, that the morphism $*\to {\Pi }_R(U)$ is a weak equivalence.

Accordingly, the proof of the proposition here follows by the same logic as the proof above that ${\Pi }_R\simeq \Pi$: we notice that every $X\in \mathrm{sPSh}({\mathrm{CartSp}}_{\mathrm{th}}{)}_{\mathrm{proj}}$ has a cofibrant incarnation that is degreewise a coproduct of representables

$$X\simeq {\int }^n{\Delta }^n\left(\coprod _{i_n}{\hat{U}}_{i_n}\right).$$X \simeq \int^{n} \Delta^n \left( \coprod_{i_n} \hat U_{i_n} \right) \,.

The functor ${\Pi }_{\inf }$ takes of this componentwise the reduced component and extends it by ${\mathrm{LConst}}_{\inf }$ back to ${\mathrm{CartSp}}_{\mathrm{th}}$.

$${\Pi }_X\simeq {\int }^n{\Delta }^n(\coprod _{i_n}{\mathrm{LConst}}_{\inf }(U_{i_n})).$$\mathbf{\Pi}_{X} \simeq \int^{n} \Delta^n \left( \coprod_{i_n} LConst_{inf}(U_{i_n}) \right) \,.

Applying componentwise the weak equivalence ${\mathrm{LConst}}_{\inf }{U}_i\to {\Pi }_D(U_i)$ noticed above this produces the natural equivalence ${\Pi }_{\inf }\to {\Pi }_D$.

de Rham theorem

Proposition

The natural morphism ${\Pi }_D(X)\to {\Pi }_R(X)$ induces an isomorphism on $R$-cohomology

$${\pi }_{0}H({\Pi }_R(X),B^{n}R)\stackrel{\simeq }{\to }{\pi }_{0}H({\Pi }_D(X),B^{n}R).$$\pi_0 \mathbf{H}(\mathbf{\Pi}_R(X), \mathbf{B}^n R) \stackrel{\simeq}{\to} \pi_0 \mathbf{H}(\mathbf{\Pi}_{D}(X), \mathbf{B}^n R) \,.

sketch of a Proof

We use that $B^{n}R$ satisfies descent on our geometrically contractible representables $U$ and hence is fibrant also in the local model structure. So for $U$ representable we have

$${\pi }_{0}H({\Pi }_D(U),B^{n}R)\simeq {\pi }_0{\mathrm{sPSh}}_C({\Pi }_D(U),B^{n}R)$$\pi_0 \mathbf{H}(\mathbf{\Pi}_{D}(U), \mathbf{B}^n R) \simeq \pi_0 sPSh_C( \mathbf{\Pi}_{D}(U), \mathbf{B}^n R)

and we know that this is the de Rham cohomology of $U$, as discussed at infinitesimal path ∞-groupoid. So given the nature of $U$ here this is $ℝ$ if $n=0$ and 0 otherwise.

So again using the cofibrant replacement $X={\int }^n\left({\coprod }_{i_n}{U}_{i_n}\right)$ as above we find that

$$\begin{array}{rl}H({\Pi }_D(X),B^{n}R)& \simeq \mathrm{sPSh}({\Pi }_D(X),B^{n}R)\\ & \simeq {\int }_n\prod _{i_n}\mathrm{sPSh}({\Pi }_D(U_{i_n}),B^{n}R)\\ & \simeq {\int }_n\prod _{i_n}{ℬ}^nℝ\end{array}$$\begin{aligned} \mathbf{H}(\mathbf{\Pi}_{D}(X), \mathbf{B}^n R) & \simeq sPSh(\mathbf{\Pi}_{D}(X), \mathbf{B}^n R) \\ & \simeq \int_n \prod_{i_n} sPSh( \mathbf{\Pi}_{D}(U_{i_n}), \mathbf{B}^n R) \\ & \simeq \int_n \prod_{i_n} \mathcal{B}^n \mathbb{R} \end{aligned}

as for $\Pi (X)$.

Here we use the discussion at Chevalley-Eilenberg algebra/Deligne cohomology/infinitesimal path ∞-groupoid which for a manifold $X$ identifies $\mathrm{sPSh}({\Pi }_D(X),B^{n}R)$ under Dold-Kan correspondence with the complex

$$(C^{\mathrm{\infty }}(X)\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^1(X)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^{n-1}(X)\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^n(X))$$\left( C^\infty(X) \stackrel{d_{dR}}{\to} \Omega^1(X) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to}\Omega^{n-1}(X) \stackrel{d_{dR}}{\to} \Omega^n(X) \right)

canonically quasi-isomorphic to

$$[{ℬ}^nℝ]=(ℝ\stackrel{0}{\to }0\cdots \stackrel{0}{\to }0\stackrel{0}{\to }0).$$[\mathcal{B}^n \mathbb{R}] = \left( \mathbb{R} \stackrel{0}{\to} 0 \cdots \stackrel{0}{\to} 0 \stackrel{0}{\to} 0 \right) \,.

Corollary

(de Rham theorem)

In the $(\mathrm{\infty }\mathrm{,1})$-topos $H$ we have for naturally for $X\in H$ an isomorphism

$$H_{\mathrm{dR}}^n(X)\simeq H^n(\mid X\mid ,ℝ).$$H^n_{dR}(X) \simeq H^n(|X|,\mathbb{R}) \,.

Proof

With the above results we have

$$\begin{array}{rl}{H}_{\mathrm{dR}}^n(X)& :={\pi }_{0}H({\Pi }_{\inf }(X),B^{n}R)\\ & \simeq {\pi }_{0}H({\Pi }_D(X),B^{n}R)\\ & \simeq {\pi }_{0}H(\Pi (X),B^{n}R)\\ & \simeq {\pi }_0\mathrm{Top}(\mid X\mid ,{ℬ}^nℝ)\\ & \simeq H^n(\mid X\mid ,ℝ)\end{array}.$$\begin{aligned} H^n_{dR}(X) & := \pi_0 \mathbf{H}(\mathbf{\Pi}_{inf}(X),\mathbf{B}^n R) \\ & \simeq \pi_0 \mathbf{H}(\mathbf{\Pi}_D(X),\mathbf{B}^n R) \\ & \simeq \pi_0 \mathbf{H}(\mathbf{\Pi}(X),\mathbf{B}^n R) \\ & \simeq \pi_0 Top(|X|,\mathcal{B}^n\mathbb{R}) \\ & \simeq H^n(|X|,\mathbb{R}) \end{aligned} \,.

Truncations of the path $\mathrm{\infty }$-groupoid

For $X\in X$ a 0-truncated object, denote by

$$\Pi (X)\to \cdots \to {\Pi }_2(X)\to {\Pi }_1(X)\to {\Pi }_0(X)$$\mathbf{\Pi}(X) \to \cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)

the internal Postnikov tower of $\Pi (X)$.

raw material , to be expanded and polished

In applications it is often convenient to consider truncations of the path $\mathrm{\infty }$-groupoid: if $A$ is an n-groupoid, i.e. an n-truncated object of our (∞,1)-topos then morphisms $\Pi (X)\to A$ will be in bijection with morphisms ${\Pi }_n(X)\to A$, where ${\Pi }_n(X)$ is a suitable truncation of $\Pi (X)$.

One variant of such a truncation is a coskeleton truncation, obtaining objects $P_n(X)$ that have only trivial (degenerate) cells iin degree $k>n$. Maps out of the $P_n(X)$ don’t impose a flatness constraint in degree $n$.

For $Y\to X$ a cover by a Cech nerve, the object

• $P_1(X)$ is given in terms of generators and relations in ScWaI

• $P_2(X)$ is given in terms of generators and relations in ScWaIII

  Another construction of this (in the related case of the full fundamental bigroupoid) is in

  • David Roberts A fundamental bigroupoid for topological groupoids , (pdf, Lab)

  There, any groupoid internal to $\mathrm{Top}$ can be passed as an argument, not just that associated to the cover.

For more constructions and references for the moment see path n-groupoid.