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${}_{\mathrm{cg}}$ 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 $H$.

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 a geometric morphism 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.

If ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ is a locally contractible (∞,1)-topos, then we also have a left adjoint of $\mathrm{LConst}$ which we shall see is the operation of forming the bare fundamental ∞-groupoid of an ∞-stack.

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

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

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

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 $\mathrm{\infty }\mathrm{Grpd}$:

$$\mathrm{\infty }\mathrm{CovBund}(X)\simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),\mathrm{\infty }\mathrm{Grpd}).$$\infty CovBund(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 path ∞-groupoid of geometric paths in it.

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

The structured path $\mathrm{\infty }$-groupoid of $X\in H$ is

$$\Pi (X):=\mathrm{LConst}\Pi (X)$$\mathbf{\Pi}(X) := LConst \Pi(X)

forming the adjunction $(\Pi ⊣♭)$.

The unit of the adjunction $(\Pi ⊣\mathrm{LConst})$ provides us with the constant path inclusion

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

All of differential nonabelian cohomology is the theory of obstructions 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 implcit . 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

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

of geometric $k$-simplices built from $R$ and thereby a geometric path $\mathrm{\infty }$-groupoid functor ${\Pi }_R:H\to H$ by

$${\Pi }_R(X):=\underrightarrow{\lim }[{\Delta }_R^{•},X].$$\mathbf{\Pi}_R(X) := \lim_\to [\Delta_R^\bullet, X] \,.

We show below that if all representables $U\in C$ are contractible with respect to the interval object $R$ in that the canonical morphism

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

is an equivalence, then the functor ${\Pi }_R$ obtained this way is equivalent to the canonical structured path $\mathrm{\infty }$-groupoid, in that we have an equivalence

$$({\Pi }_R,{♭}_R)\simeq (\Pi ⊣♭).$$(\mathbf{\Pi}_R, \flat_R) \simeq (\mathbf{\Pi} \dashv \mathbf{\flat}) \,.

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

If $H$ is even a smooth (∞,1)-topos, then the interval object $R$ is accompanied by the infinitesimal interval object $D\subset R$ and the geometric path $\mathrm{\infty }$-groupoid $\Pi (-)$ by the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(-)$.

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

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${}_{\mathrm{cg}}$.

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

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

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

$$\mathrm{LConst}:{\mathrm{sSet}}_{\mathrm{Quillen}}^{\mathrm{fib}}\to \mathrm{sPSh}(C)$$LConst : sSet_{Quillen}^{fib} \to sPSh(C)

consists of objects that are fibrant in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Examples/Proposition

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

• CartSp;

• the site $\mathrm{InfThCartSp}\subset$ 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 $(Id⊣\mathrm{Id})$.

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

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

be the unstructured group object underlying it.

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

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

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

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

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.

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 bifunctors (as discussed there).

So for $X\in \mathrm{sPsh}(C{)}_{\mathrm{proj}}$ cofibrant we have that

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

is a cofibration and also that for $X\to Y$ a (trivial) cofibration in $\mathrm{sPSh}(C{)}_{\mathrm{proj}}$ the induced morphism

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

is a (trivial) cofibration.

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.

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

Restrict attention now to the site $C$ of smooth loci of the form ${ℝ}^n\times D_k$: cartesian spaces times possibly an infinitesimal thickening.

Definition

For $U\in C$ we may then form the infinitesimal singular simplicial complex $U^{({\Delta }_{\inf }^{•})}$, regarded as a simplicial presheaf on $C$. We can form a functorial cofibrant replacement of this that is compatible with the one we have chosen for $U^{{\Delta }_R^{•}}$ by forming the pullback

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

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

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

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

and obtain an adjunction

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

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

$${\Pi }_{\inf }(X)\to {\Pi }_R(X).$$\mathbf{\Pi}_{inf}(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 $\Gamma R=ℝ\in \mathrm{\infty }\mathrm{Grpd}\simeq {\mathrm{Top}}_{\mathrm{cg}}$ etc., with $ℝ$ on the right understood with the discrete topology.

Proposition

This morphism ${\Pi }_{\inf }(X)\to {\Pi }_R(X)$ 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)

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

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

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

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 construtions and references for the moment see path n-groupoid.