Schreiber
path ∞-groupoid

Idea

The fundamental ∞-groupoid $\Pi (X)$ of a topological space $X$ is the ∞-groupoid whose k-morphisms are continuous $k$-dimensional paths in $X$.

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

Unstructured

Let $C$ be some site and let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ be the (∞,1)-topos of ∞-stacks 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;

• and its right adjoint $p^{*}=\mathrm{LConst}$, the constant ∞-stack functor.

But we also have

• its left adjoint, the inverse image $p_{!}$

which we shall see is the operation $p_{!}=\Pi$ of forming the bare fundamental ∞-groupoid of an ∞-stack.

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

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

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

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.

Model given by a line object

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 $\Pi :H\to H$ by

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

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

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

$$\begin{array}{ccccc}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)& & \stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Pi }{\to }}& & \mathrm{\infty }\mathrm{Grpd}\\ & {}_{\Pi }\searrow {\nwarrow }^{(-{)}_{\mathrm{flat}}}& & {}^{\mathrm{LConst}}\swarrow {\nearrow }_{\Gamma }\\ & & {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\end{array}.$$\array{ Sh_{(\infty,1)}(C) &&\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi}{\to}}&& \infty Grpd \\ & {}_{\mathllap{\mathbf{\Pi}}}\searrow \nwarrow^{\mathrlap{(-)_{flat}}} && {}^{\mathllap{LConst}}\swarrow \nearrow_{\mathrlap{\Gamma}} \\ && Sh_{(\infty,1)}(C) } \,.

Infinitesimal

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 ${\Pi }^{\inf }(-)$.

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 local model structures.

Remark

Of particular interest is this setup for the case that $(\mathrm{SPSh}(C{)}^{\mathrm{loc}}{)}^{\circ }$ is a smooth (∞,1)-topos. See there for more details.

General definition

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

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

induced from the interval object.

The ∞-Lie groupoid presented by this object in $H=(\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{)}^{\circ }$ we shall call the path $\mathrm{\infty }$-groupoid of $U$.

To extend this construction from representables $U$ to general ∞-Lie groupoids $X\in \mathrm{SPSh}(C)$ we make use of an equivalent but better behaved model. There is a canonical morphism $U\to U^{{\Delta }_R^{•}}$ and we are guaranteed a factorization

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

functorally $\Pi (-):C\to \mathrm{SPSh}(C)$ for all $U\in C$ into a cofibration $U\hookrightarrow \Pi (U)$ and a weak equivalence $\Pi (U)\stackrel{\simeq }{\to }{U}^{{\Delta }_R^{•}}$ 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 $\Pi (U)$ is cofibrant in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ and hence also in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Therefore for any such choice we may form the (∞,1)-Yoneda extension of $\Pi (-)$ to a Quilen adjunction

$$\Pi (-):\mathrm{SPSh}(C{)}_{\mathrm{proj}}\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{proj}}:(-{)}_{\mathrm{flat}}$$\Pi(-) : SPSh(C)_{proj} \stackrel{\leftarrow}{\to} SPSh(C)_{proj} : (-)_{flat}

with respect to the global model structure on simplicial presheaves.

For every $X\in \mathrm{SPSh}(C)$ the object presented in $H$ by ${\Pi }^{\inf }(X)$ we call the infinitesimal $\mathrm{\infty }$-groupoid of $X$.

Convenient model

We now describe a particular such model by making use of the standard Bousfield-Kan resolution

$$\Delta :\Delta \to \mathrm{SSet}$$\mathbf{\Delta} : \Delta \to SSet

$${\Delta }^n:=N([n]/{\Delta }^{\mathrm{op}}{)}^{\mathrm{op}}$$\mathbf{\Delta}^n := N([n]/\Delta^{op})^{op}

of the cosimplicial simplicial object $\Delta (-,-)$.

Definition

(infinitesimal path $\mathrm{\infty }$-groupoid of a smooth locus)

For $U\in C$ write

$$\Pi (U):={\int }^{[n]\in \Delta }{\Delta }^n\cdot U^{({\Delta }^n)}\in \mathrm{SPSh}(C),$$\Pi(U) := \int^{[n]\in \Delta} \mathbf{\Delta}^n \cdot U^{(\Delta^n)} \;\;\;\in SPSh(C) \,,

where in the integrand of the coend we have the tensoring of the simplicial model category $\mathrm{SPSh}(C)$ by simplicial sets.

for further discussion see for the moment the analogous discussion at infinitesimal path ∞-groupoid.

Properties

Preservation of weak equivalences between fibrants

Proposition

The functor $\Pi :\mathrm{SPSh}(C)\to \mathrm{SPSh}(C)$ preserves weak equivalences between fibrant objects in the both the projective and the global as well as in the local, in the projective as well as in the injective model structure.

Proof

Since all representables are cofibrant in all the model structures, it follows that $\mathrm{SPSh}(U\times {\Delta }_C^n,-)$ preserves weak equivalences between fibrant objects. For each $V\in C$ The homotopy colimit ${\int }^n{\Delta }^n\cdot (-)(V)$ preserves weak equivalences between the diagrams of (necessarily cofibrant) simplicial sets.

The local Quillen adjunction

We have seen so far that the adjunction $\Pi ⊣(-{)}_{\mathrm{flat}}$ is a Quillen adjunction with respect to the global projective model structure $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$. We now discuss how under certain conditions this is also a Quillen equivalence with respect to a local model structure, so that it does produce morphisms

$$\Pi :H\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}$$\Pi : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat}

of ∞-stack (∞,1)-toposes.

Definition

(Cech localization)

Let $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{cov}}$ be the Cech localization of the global projective model structure on simplicial presheaves on the site $C$ : the left Bousfield localization of $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ at the set of good Cech covers of representables, i.e. at the set of morphisms

$$C(\coprod _i{U}_i)\to U_i$$C(\coprod_i U_i) \to U_i

for $\{U_i\to U\}$ any covering sieve in the topology on $C$ such that all intersections $({\coprod }_i{U}_i{)}^{{\times }_U^n}$ are coproducts of representables, and for $C({\coprod }_i{U}_i)$ the Cech nerve of the corresponding morphism ${\coprod }_i{U}_i\to U$.

Remark

The usual Jardine/Blander local model structure is the hypercompletion of $\mathrm{SPSh}(C{)}^{\mathrm{cov}}$, where descent is satisfied not just for Cech covers, but for all hyypercovers.

Proposition

If

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

then

• the adjunction $\Pi ⊣(-{)}_{\mathrm{flat}}$ for the convenient model of $\Pi$ above is a Quillen adjunction with respect to the Cech localization:

$$\Pi :\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{cov}}\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{cov}}.$$\Pi : SPSh(C)_{proj}^{cov} \stackrel{\leftarrow}{\to} SPSh(C)_{proj}^{cov} \,.

Proof

We first notice two lemmas.

Lemma 1 $\Pi$ sends covers $(C({\coprod }_i{U}_i)\to U)$ to global weak equivalences.

Proof of lemma 1: By assumption $\Pi (U)(V)$ is contractible for all $V$. It therefore suffices to show that $\Pi ({\coprod }_i{U}_i)(V)$ is, too. For that it suffices to check that the geometric realization $\mid C({\coprod }_i{U}_i)\mid$ is a contractible space. We have

$$\begin{array}{rl}\mid C(\coprod _i{U}_i)\mid & =\mid {\int }^n\Pi ((\coprod _i{U}_i{)}^{{\times }_U^{(n+1)}})(V)\times {\Delta }^n\mid \\ & \simeq {\int }^n\mid \Pi ((\coprod _i{U}_i{)}^{{\times }_U^{(n+1)}})(V)\mid \times \mid {\Delta }^n\mid \\ & \simeq {\int }^n\{n\mathrm{fold}\mathrm{intersections}\}\times \mid {\Delta }^n\mid \\ & \simeq \underset{{\to }^n}{\lim }\{n\mathrm{fold}\mathrm{intersections}\}\\ & \simeq \mathrm{pt}\end{array},$$\begin{aligned} |C(\coprod_i U_i)| & = |\int^n \Pi((\coprod_i U_i)^{\times_U^{(n+1)}})(V) \times \mathbf{\Delta^n}| \\ & \simeq \int^n |\Pi((\coprod_i U_i)^{\times_U^{(n+1)}})(V)|\times |\mathbf{\Delta^n}| \\ & \simeq \int^n \{n\;fold\;intersections\}\times |\mathbf{\Delta^n}| \\ & \simeq \lim_{\to^n} \{n\;fold\;intersections\} \\ & \simeq pt \end{aligned} \,,

where in the second but last step we used that by assumption $\{U_i\to U\}$ is a good cover, hence all intersections are representable, and therefore contractible, by assumption.

Lemma 2: The right adjoint $(-{)}_{\mathrm{flat}}$ preserves the fibrant objects of $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{cov}}$.

Proof of lemma 2. By the general mechanism of left Bousfield localization, the fibrant objects of $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{cov}}$ 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 $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$, $(-{)}_{\mathrm{flat}}$ preserves the global fibrancy of objects. To show moreover that $A_{\mathrm{flat}}$ is a Cech cover local object we need to show that for all $C({\coprod }_i{U}_i)\to U$ we have that $\mathrm{SPSh}(U,A_{\mathrm{flat}})\to \mathrm{SPSh}(C({\coprod }_i{U}_i),A_{\mathrm{flat}})$ is a weak equivalence.

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

Proof of the proposition

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

So it is sufficient to show that $\Pi$ 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 $\mathrm{SPSh}(\Pi (Y),A)\to \mathrm{SPSh}(\Pi (X),A)$ is an acyclic Kan fibration for all fibrant and local $A$. By the adjunction $\Pi ⊣(-{)}_{\mathrm{flat}}$ this is equivalent to $\mathrm{SPSh}(Y,A_{\mathrm{flat}})\to \mathrm{SPSh}(\hat{X},A_{\mathrm{flat}})$ being a weak equivalence. But by lemma 2 $A_{\mathrm{flat}}$ is still locally fibrant. Therefore this is indeed a local weak equivalence, by their above mentioned characterization.

The constant path inclusion

Observation

For every $X$ in $\mathrm{SPSh}(C)$ there is naturally a morphism

$$X\to \Pi (X).$$X \to \Pi(X) \,.

When $X$ is cofibrant then this is a cofibration. This is the inclusion of constant paths into all paths on $X$.

The morphism is the morphism of coends

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

which is induced componentwise from the cofibrations $U\hookrightarrow \Pi (U)$. This is also the image under the left Quillen bifunctor

$$\int (-)\cdot (-)[C,\mathrm{SPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}\times [C^{\mathrm{op}},\mathrm{SSet}{]}_{\mathrm{proj}}$$\int (-)\cdot (-) [C,SPSh(C)_{proj}]_{inj} \times [C^{op},SSet]_{proj}

of $(Y(-)\to \Pi (-),X)$, where $Y:C\to \mathrm{SPSh}(C)$ is the Yoneda embedding. When $X$ is cofibrant, this respects cofibrations in the first argument. But $Y\to \Pi$ is componentwise $Uhookrigharrow\Pi (U)$ a cofibation in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$, hence a cofibration in $[C,\mathrm{SPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}$. Therefore $X\to \Pi (X)$ is a cofibration when $X$ is cofibrat.

Remark (relation to differential cohomology)

The inclusion

$$X\hookrightarrow \Pi (X)$$X \hookrightarrow \Pi(X)

induces a natural morphism

$$A_{\mathrm{flat}}\to A$$A_{flat} \to A

which is a fibration if $A$ is fibrant.

At differential cohomology it is discussed how

• cohomology with coefficients in $A_{\mathrm{flat}}$ describes flat differential cohomology with coefficients in $A$.

• the obstruction problem to lifts through $A_{\mathrm{flat}}\to A$ describes general differential cohomology with coefficients in $A$.

The infinitesimal path inclusion

We discuss the inclusion of the infinitesimal path ∞-groupoid into the path $\mathrm{\infty }$-groupoid and the relation to integration of ∞-Lie algebroid valued differential forms.

When the ambient (∞,1)-topos is a smooth (∞,1)-topos the path $\mathrm{\infty }$-groupoid functor $\Pi$ is accompanied by the infinitesimal path ∞-groupoid functor ${\Pi }^{\inf }$.

In this case there is a natural inclusion

$$\iota :{\Pi }^{\inf }(-)\hookrightarrow \Pi (-)$$\iota : \Pi^{inf}(-) \hookrightarrow \Pi(-)

as described here at infinitesimal singular simplicial complex.

Compatibility with the bare path $\mathrm{\infty }$-groupoid

Claim

The structured path $\mathrm{\infty }$-groupoid functor $\Pi :H\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}$ constructed above is compatible with the canonical bare path $\mathrm{\infty }$-groupoid functor $\Pi :H\overleftarrow{\to }\mathrm{\infty }\mathrm{Grpd}:\mathrm{LConst}$ in that it fits into a commuting diagram

$$\begin{array}{ccccc}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)& & \stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Pi }{\to }}& & \mathrm{\infty }\mathrm{Grpd}\\ & {}_{\Pi }\searrow {\nwarrow }^{(-{)}_{\mathrm{flat}}}& & {}^{\mathrm{LConst}}\swarrow {\nearrow }_{\Gamma }\\ & & {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\end{array}$$\array{ Sh_{(\infty,1)}(C) &&\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi}{\to}}&& \infty Grpd \\ & {}_{\mathllap{\mathbf{\Pi}}}\searrow \nwarrow^{\mathrlap{(-)_{flat}}} && {}^{\mathllap{LConst}}\swarrow \nearrow_{\mathrlap{\Gamma}} \\ && Sh_{(\infty,1)}(C) }

if the underlying site $C$ for $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ has the property that all objects $U$ of $C$ are contractible with respect to the chosen line object in that $\Pi (U)\to *$ is weak equivalence.

This is the case in particlar for the site $\mathrm{InfFatCartSpace}$ of “infinitesimally fatened cartesian spaces”, the category of smooth loci of the form ${ℝ}^n\times D_{(k)}^r$.

Idea of 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 $\Pi (\tilde{X})\simeq {\int }^n{\Delta }^n\Pi ({\tilde{X}}_n)$. Hence with the above

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

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

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

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

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

and have therefore

$$H(\Pi (X),\mathrm{LConst}𝒜)\simeq H(\mathrm{LConst}\Pi (X),\mathrm{LConst}𝒜)\simeq H(\Pi (X),\Gamma \mathrm{LConst}𝒜)\simeq H(\Pi (X),𝒜).$$\mathbf{H}(\mathbf{\Pi}(X),LConst \mathcal{A}) \simeq \mathbf{H}(LConst \Pi(X), LConst \mathcal{A}) \simeq \mathbf{H}(\Pi(X), \Gamma LConst \mathcal{A}) \simeq \mathbf{H}(\Pi(X), \mathcal{A}) \,.

Similarly one sees that

$$(\mathrm{LConst}𝒜{)}_{\mathrm{flat}}\simeq \mathrm{LConst}𝒜.$$(LConst \mathcal{A})_{flat} \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

• Dan 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.

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

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.

Thanks. I thank Richard Williamson for alerting me of the fact that from every $(\mathrm{\infty }\mathrm{,1})$-topos there should canonically be an unstructured path $\mathrm{\infty }$-groupoid functor, and for discussion about this point.