Schreiber
infinitesimal path ∞-groupoid

The infinitesimal path $\mathrm{\infty }$-groupoid ${\Pi }^{\inf }(X)$ of a space $X$ is the ∞-Lie groupoid that has as k-morphisms the infinitesimal $k$-dimensonal paths in $X$.

It is in fact the ∞-Lie algebroid associated by ∞-Lie differentiation to the finite path ∞-groupoid $\Pi (X)$ of $X$.

For $X$ an ordinary manifold ${\Pi }^{\inf }(X)$ is the tangent Lie algebroid $TX$ of $X$. Regarded as a simplicial object, ${\Pi }^{\inf }(X)$ is a simplicial resolution of the deRham space $\mathrm{dR}(X)$ of $X$.

Idea

In a smooth (∞,1)-topos $H$ the path ∞-groupoid operation $\Pi (-):H\to H$ is accompanied by its infinitesimal version, the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(-):H\to H$.

The interplay between these two is the study of ∞-Lie differentiation and integration.

When $X$ is a manifold or a smooth locus then ${\Pi }^{\inf }(X)$ is, as a simplicial smooth space, the infinitesimal singular simplicial complex of $X$.

Where a morphism $\Pi (X)\to A$ out of the path ∞-groupoid describes flat A-valued parallel transport along paths in $X$ – a local system on $X$ with coefficients in $A$ – a morphism ${\Pi }^{\inf }(X)\to A$ describes flat $A$-valued parallel transport along infinitesimal paths in $X$.

Morphisms out of ${\Pi }^{\inf }(X)$ describe the infinitesimal part of morphisms out of $\Pi (X)$:

$$\begin{array}{ccccc}{\Pi }^{\inf }(X)& \stackrel{}{\to }& 𝔞& & \mathrm{flat}\mathrm{transport}\mathrm{along}\mathrm{infinitesimal}\mathrm{paths}\\ \downarrow & & \downarrow \\ \Pi (X)& \stackrel{}{\to }& A& & \mathrm{flat}\mathrm{transport}\mathrm{along}\mathrm{finite}\mathrm{paths}\end{array}$$\array{ \Pi^{inf}(X) &\stackrel{}{\to}& \mathfrak{a} && flat\;transport\;along\;infinitesimal\;paths \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{}{\to}& A && flat\;transport\;along\;finite\;paths }

Such infinitesimal parallel transport encodes ∞-Lie algebroid valued differential forms.

In low degrees a morphism out of ${\Pi }^{\inf }(X)$ is essentially what is known as a Grothendieck connection.

Definition

Let $H=((\mathrm{SSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{)}^{\circ },R)$ be a smooth (∞,1)-topos. Recall that by assumption all representables are smooth loci.

General definition

For $U\in C\subset \mathrm{Sh}(C)$ a representable write

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

for the infinitesimal singular simplicial complex of $U$.

The ∞-Lie groupoid presented by this object in $H=(\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{)}^{\circ }$ we shall call the infinitesimal 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 $U\to U^{({\Delta }_{\inf }^{•})}$ and we are guaranteed a factorization

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

functorally ${\Pi }^{\inf }(-):C\to \mathrm{SPSh}(C)$ for all $U\in C$ into a cofibration $U\hookrightarrow {\Pi }^{\inf }(U)$ and a weak equivalence ${\Pi }^{\inf }(U)\stackrel{\simeq }{\to }{U}^{({\Delta }_{\inf }^{•})}$ 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 ${\Pi }^{\inf }(U)$ 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 ${\Pi }^{\inf }(-)$ to a Quilen adjunction

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

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

The constant path inclusion

Observation

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

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

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

The morphism is the morphism of coends

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

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

Remark (relation to differential cohomology)

The inclusion

$$X\hookrightarrow {\Pi }^{\inf }(X)$$X \hookrightarrow \Pi^{inf}(X)

induces a natural morphism

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

which is a fibration if $A$ is fibrant.

At differential cohomology it is discussed how

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

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

Convenient model

It is useful in applications to choose the cofibrant replacement ${\Pi }^{\inf }(U)$ of $U^{({\Delta }_{\inf }^{•})}$ such that it Yoneda extension ${\Pi }^{\inf }(-):\mathrm{SPSh}(C)\to \mathrm{SPSh}(C)$ preserves as many weak equivalences in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ as possible.

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

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

Proposition

The object ${\Pi }^{\inf }(U)$ is a cofibrant replacement of $U^{({\Delta }_{\inf }^{•})}$ in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$.

$$\varnothing \hookrightarrow {\Pi }^{\inf }(U)\stackrel{\simeq }{\to }{U}^{({\Delta }^{•}{)}_{\inf }}$$\emptyset \hookrightarrow \Pi^{inf}(U) \stackrel{\simeq}{\to} U^{(\Delta^\bullet)_{inf}}

Proof

1. – the weak equivalence ${\Pi }^{\inf }(U)\stackrel{\simeq }{\to }{U}^{({\Delta }_{\inf }^{•})}$ –

   It will be sufficient to show that for each $V\in C$ the simplicial simplicial set $U^{{\Delta }_{\inf }^{•}}(V):{\Delta }^{\mathrm{op}}\to \mathrm{Set}\hookrightarrow \mathrm{SSet}$ is Reedy cofibrant. In that case the standard statement about the Bousfield-Kan map gives us the weak equivalence

   $${\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}(V)\stackrel{\simeq }{\to }{\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}(V)$$\int^{n} \mathbf{\Delta^n} \cdot U^{(\Delta^n_{inf})}(V) \;\; \stackrel{\simeq}{\to} \;\; \int^{n} \Delta^n \cdot U^{(\Delta^n_{inf})}(V)

   of simplicial sets. Since colimits of presheaves are computed objectwise this says that the morphism of simplicial presheaves

   $${\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}\stackrel{\simeq }{\to }{\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}$$\int^{n} \mathbf{\Delta^n} \cdot U^{(\Delta^n_{inf})} \;\; \stackrel{\simeq}{\to} \;\; \int^{n} \Delta^n \cdot U^{(\Delta^n_{inf})}

   is objectwise a weak equivalence of simplicial sets and hence a weak equivalence in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$.

   To see that $U^{({\Delta }_{\inf }^{•})}(V)$ is indeed Reedy cofibrant notice that the latching object for $[n]\in {\Delta }^{\mathrm{op}}$ is

   $$L_{[n]}{U}^{({\Delta }_{\inf }^{•})}(V)={\mathrm{colim}}_{[n]\stackrel{\sigma }{\to }[k]}{U}^{({\Delta }_{\inf }^k)}(V)$$L_{{[n]}}U^{(\Delta^\bullet_{inf})}(V) = colim_{[n] \stackrel{\sigma}{\to} [k]} U^{(\Delta^k_{inf})}(V)

   where the colimit is over all surjections $[n]\to [k]$, i.e. all composites of co-degeneracy maps out of $[n]$. Accordingly the canonical morphism $L_{[n]}{U}^{({\Delta }_{\inf }^{•})}(V),X)\hookrightarrow U^{({\Delta }_{\inf }^n)}(V)$ is the inclusion of simplicial sets of degenerate infinitesimal $n$-simplicies in $U$ into all infinitesimal $n$-simplices and hence in particular a monomorphism of simplicial sets, hence a cofibration of simplicial sets.

   This says that $U^{({\Delta }_{\inf }^{•})}(V)$ is Reedy cofibrant.

2. the cofibrancy $\varnothing \hookrightarrow {\Pi }^{\inf }(U)$

   The coend over the tensor

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

   is a Quillen bifunctor for the global model structures on functors as indicated.

   It is a standard fact that the object ${\Delta }^{•}$ is cofibrant in $\in [\Delta ,\mathrm{SSet}{]}_{\mathrm{proj}}$.

   It follows from the properties of a Quillen bifunctor that the functor

   $${\int }^n{\Delta }^n\cdot (-{)}_n:[\Delta ,\mathrm{SPSh}(C{)}_{\mathrm{proj}}]\to \mathrm{SPSh}(C{)}_{\mathrm{proj}}$$\int^{n} \mathbf{\Delta}^n \cdot (-)_n : [\Delta,SPSh(C)_{proj}] \to SPSh(C)_{proj}

   respects cofibrations (and acyclic cofibrations).

   Moreover, we have that all the objects $U^{({\Delta }_{\inf }^k)}$ of infinitesimal simplices in a representable are again representable (see MSIA) and all representables are cofibrant in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$. Therefore $U^{({\Delta }_{\inf }^{•})}$ is degreewise cofibrant in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ and hence cofibrant in $[\Delta ,\mathrm{SPSh}(C{)}_{\mathrm{proj}}{]}_{\mathrm{inj}}$.

   Therefore also ${\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^{•})}$ is cofibrant in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ and hence in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Definition

(infinitsimal path $\mathrm{\infty }$-groupoid functor)

We write

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

for the (∞,1)-Yoneda extension of ${\Pi }^{\inf }:C\to \mathrm{SPSh}(C{)}_{\mathrm{proj}}$.

So for $X\in \mathrm{SPSh}(C)$ we have

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

where in the integrand we have again the tensoring of $\mathrm{SPSh}(C)$ by $\mathrm{SSet}$.

We call the object presented by ${\Pi }^{\inf }(X)$ in $X$ the infinitesimal path $\mathrm{\infty }$-groupoid of $X$.

Remark

The object ${\Pi }^{\inf }(X)$ is built in degree $k$ from $k$-fold 1-jets of $X$. But the objectwise Kan fibrant replacement that it presents under the model structure on simplicial presheaves has jets of arbitrary order in each degree.

Remark

For $X$ an ordinary manifold, ${\Pi }^{\inf }(X)$ is the ∞-Lie algebroid known as the tangent Lie algebroid of $X$.

For general $X$, there is a canonical inclusion

$$\mathrm{Lie}(\Pi (X))\hookrightarrow {\Pi }^{\inf }(X)$$Lie(\Pi(X)) \hookrightarrow \Pi^{inf}(X)

but ${\Pi }^{\inf }(X)$ is not in general an ∞-Lie algebroid and in particular is not the ∞-Lie algebroid $\mathrm{Lie}(\Pi (X))$ of the path ∞-groupoid $\Pi (X)$ of $X$ if there are non-infinitesimal morphisms in $X$.

Properties of the convenient model

Respect for furtherweak equivalences

Being a left Quillen functor by the above statement, the functor ${\Pi }^{\inf }:\mathrm{SPSh}(C{)}_{\mathrm{proj}}\to \mathrm{SPSh}(C{)}_{\mathrm{proj}}$ preserves at least those objectwise weak equivalences that are also cofibrations in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ and hence in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

But ${\Pi }^{\inf }$ preserves more weak equivalences.

Proposition

(respect for weak equivalences between degreewise representables)

The functor ${\Pi }^{\inf }:\mathrm{SPSh}(C)\to \mathrm{SPSh}(C)$ preserves global weak equivalences between objects that are degreewise representable. If $\mathrm{Sh}(C)$ has enough topos points then it preserves also local weak equivalences between degreewise representable objects.

Lemma

(totalization expression on degreewise representables)

If $X\in \mathrm{SPSh}(C)$ is degreewise representable, then there is a natural weak equivalence

$${\Pi }^{\inf }(X)\stackrel{\simeq }{\to }{\int }^{[n]\in \Delta }{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)},$$\Pi^{inf}(X) \stackrel{\simeq}{\to} \int^{[n] \in \Delta} \Delta^n \cdot X^{(\Delta^n_{inf})} \,,

where on the right in the integrand we form degreewise the representable object of infinitesimal simplices.

Proof

We may rewite the defining coend as

$$\begin{array}{rl}{\Pi }^{\inf }(X)& ={\int }^{U\in C}\Pi (U)\cdot X(U)\\ & ={\int }^{U\in C}{\int }^{[n]\int \Delta }{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}\cdot X(U)\\ & ={\int }^{[n]\in \Delta }{\int }^{U\in C}{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}\cdot X(U)\\ & ={\int }^{[n]\in \Delta }{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)}\end{array},$$\begin{aligned} \Pi^{inf}(X) &= \int^{U\in C} \Pi(U) \cdot X(U) \\ &= \int^{U \in C} \int^{[n] \int \Delta} \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} \cdot X(U) \\ &= \int^{[n] \in \Delta} \int^{U \in C} \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} \cdot X(U) \\ &= \int^{[n] \in \Delta} \mathbf{\Delta}^n \cdot X^{(\Delta^n_{inf})} \end{aligned} \,,

where all equality signs denote isomorphisms (to be distinguished from weak equivalences). We have used the co-Yoneda lemma and in the last line $X^{({\Delta }_{\inf }^n)}$ denotes the object obtained from the simplicial representable $X$ by taking degreewise the object of infnitesimal $n$-simplices.

By arguments as already used above, we see that we have a global weak equivalence

$${\int }^{[n]\in \Delta }{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)}\stackrel{\simeq }{\to }{\int }^{[n]\in \Delta }{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)}$$\int^{[n] \in \Delta} \mathbf{\Delta}^n \cdot X^{(\Delta^n_{inf})} \;\; \stackrel{\simeq}{\to} \;\; \int^{[n] \in \Delta} \Delta^n \cdot X^{(\Delta^n_{inf})}

to the ordinary totalization of the bisimplicial object $X_{•}^{({\Delta }_{\inf }^{•})}$.

Proof

(respect for weak equivalences between degreewise representables)

Let now $X\to Y$ be a global weak equivalence of simplicial representable objects in that for all representables $U$ the morphism $X(U)\to Y(U)$ is a weak equivalence of simplicial sets. Then also $X^{{\Delta }_{\inf }^n}(U)\to Y^{{\Delta }_{\inf }^n}(U)$ is a weak equivalence of simplicial sets: by the definition of infinitesimal simplices it is just a restriction of the morphism $X^{(n+1)}(U)\to Y^{(n+1)}(U)$

$$\begin{array}{ccccc}{X}^{{\Delta }_{\inf }^n}(U)& \to & X^{(n+1)}(U)& \to & X^{{\Delta }_{\inf }^n}(U)\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }& & \downarrow \\ Y^{{\Delta }_{\inf }^n}(U)& \to & Y^{(n+1)}(U)& \to & Y^{{\Delta }_{\inf }^n}(U)\end{array},$$\array{ X^{\Delta^n_{inf}}(U) &\to& X^{(n+1)}(U) &\to& X^{\Delta^n_{inf}}(U) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow \\ Y^{\Delta^n_{inf}}(U) &\to& Y^{(n+1)}(U) &\to& Y^{\Delta^n_{inf}}(U) } \,,

where the right horizontal morphism are taken to be the identity on pairwise first order infinitesimal neighbouring $(n+1)$-tuples of $U$-points in each degree and send all other $(n+1)$-tuples to, say, the totally degenerate $(n+1)$-tuple on their first element. Then the horizontal composites are identity morphisms and the left vertical morphism is a retract of a weak equivalence and hence itself a weak equivalence.

The image under the totalization of a morphism of bisimplicial sets which is degreewise in one degree a weak equivalence is a weak equivalence, hence for $X\to Y$ a global weak equivalence also ${\int }^n{\Delta }^n\cdot X^{{\Delta }_{\inf }^n}\to {\int }^n{\Delta }^n\cdot Y^{{\Delta }_{\inf }^n}$ is one and hence by 2-out-of-3 also the left vertical morphism in

$$\begin{array}{ccc}{\int }^n{\Delta }^n\cdot X^{{\Delta }_{\inf }^n}& \stackrel{\simeq }{\to }& {\int }^n{\Delta }^n\cdot X^{{\Delta }_{\inf }^n}\\ \downarrow & & {\downarrow }^{\simeq }\\ {\int }^n{\Delta }^n\cdot Y^{{\Delta }_{\inf }^n}& \stackrel{\simeq }{\to }& {\int }^n{\Delta }^n\cdot Y^{{\Delta }_{\inf }^n}\end{array}$$\array{ \int^n \mathbf{\Delta}^n \cdot X^{\Delta^n_{inf}} &\stackrel{\simeq}{\to}& \int^n \Delta^n \cdot X^{\Delta^n_{inf}} \\ \downarrow && \downarrow^{\mathrlap{\simeq}} \\ \int^n \mathbf{\Delta}^n \cdot Y^{\Delta^n_{inf}} &\stackrel{\simeq}{\to}& \int^n \Delta^n \cdot Y^{\Delta^n_{inf}} }

is a weak equivalence.

This shows the claim for global weak equivalences. If the topos $\mathrm{Sh}(C)$ has enough points, then local weak equivalences are tested on stalks given by some colimits

$${\mathrm{colim}}_{U_i}X(U_i)\stackrel{\simeq }{\to }{\mathrm{colim}}_{U_i}Y(U_i).$$colim_{U_i} X(U_i) \stackrel{\simeq}{\to} colim_{U_i} Y(U_i) \,.

We need to check that a morphism $X\to Y$ is a weak equivalence on all stalks this way, it still induces a local weak equivalence of totalizations. But since these are built from colimits themselves, we can take the stalk colimit inside the coend

$${\mathrm{colim}}_{U_i}{\int }^n{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)}(U_i)\stackrel{=}{\to }{\int }^n{\Delta }^n\cdot {\mathrm{colim}}_{U_i}{X}^{({\Delta }_{\inf }^n)}(U_i)\stackrel{\simeq }{\to }{\int }^n{\Delta }^n\cdot {\mathrm{colim}}_{U_i}{Y}^{({\Delta }_{\inf }^n)}(U_i)\stackrel{=}{\to }{\mathrm{colim}}_{U_i}{\int }^n{\Delta }^n\cdot Y^{({\Delta }_{\inf }^n)}(U_i),$$colim_{U_i} \int^n \Delta^n \cdot X^{(\Delta^n_{inf})}(U_i) \stackrel{=}{\to} \int^n \Delta^n \cdot colim_{U_i} X^{(\Delta^n_{inf})}(U_i) \stackrel{\simeq}{\to} \int^n \Delta^n \cdot colim_{U_i} Y^{(\Delta^n_{inf})}(U_i) \stackrel{=}{\to} colim_{U_i} \int^n \Delta^n \cdot Y^{(\Delta^n_{inf})}(U_i) \,,

where again equality signs denote isomorphisms. This then shows the claim also for local weak equivalences.

Cofibrancy of the constant path inclusion

Proposition

(cofibrancy of the constant path inclusion)

For $U\in C\subset \mathrm{SPSh}(C)$ any representable, the canonical morphism $U\to U^{({\Delta }_{\inf }^{•})}$ factors through ${\Pi }^{\inf }(U)$ by a cofibration in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$:

$$U\hookrightarrow {\Pi }^{\inf }(U)\stackrel{\simeq }{\to }{U}^{({\Delta }_{\inf }^{•})}$$U \hookrightarrow \Pi^{inf}(U) \stackrel{\simeq}{\to} U^{(\Delta^\bullet_{inf})}

Proof

The morphism $U\to {\Pi }^{\inf }(U)$ is the composite

$$U\stackrel{\mathrm{pt}\times U}{\to }({\int }^n{\Delta }^n)\cdot U\to {\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}={\Pi }^{\inf }(U).$$U \stackrel{pt \times U}{\to} (\int^n \mathbf{\Delta^n}) \cdot U \to \int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} = \Pi^{inf}(U) \,.

Here the first morphism, being of the form $U\to S\cdot U$ for $U$ representable and $S$ any simplicial set is a cofibration, since for all acyclic fibrations $A\to B$ in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ we have by Yoneda

$$\begin{array}{ccc}U& \to & A\\ \downarrow & & {\downarrow }^{\in W\cap F\subset \mathrm{SPSh}(C{)}_{\mathrm{proj}}}\\ U\cdot S& \to & B\end{array}\iff \begin{array}{ccc}{\Delta }^0& \to & A(U)\\ \downarrow & & {\downarrow }^{\in W\cap F\subset \mathrm{SSet}}\\ S& \to & B(U)\end{array}$$\array{ U &\to& A \\ \downarrow && \downarrow^{\mathrlap{\in W \cap F \subset SPSh(C)_{proj}}} \\ U \cdot S &\to& B } \;\;\;\; \;\;\;\; \;\;\;\; \;\;\;\; \Leftrightarrow \;\;\;\; \array{ \Delta^0 &\to& A(U) \\ \downarrow && \downarrow^{\mathrlap{\in W \cap F \subset SSet}} \\ S &\to& B(U) }

and a lift on the right exists because all monomorphism of simplicial sets are cofibrations in SSet.

It remains to show that

$$({\int }^n{\Delta }^n\cdot U)\to ({\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)})$$(\int^n \mathbf{\Delta}^n \cdot U) \to (\int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})})

is a cofibratiokn.

To do so, we want to demonstrate the existence of lifts in all diagrams

$$\begin{array}{ccc}{\int }^n{\Delta }^n\cdot U& \stackrel{a}{\to }& A\\ \downarrow & & {\downarrow }^{\in W\cap F\subset \mathrm{SPSh}(C{)}_{\mathrm{proj}}}\\ {\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}& \stackrel{b}{\to }& B\end{array}.$$\array{ \int^n \mathbf{\Delta}^n \cdot U &\stackrel{a}{\to}& A \\ \downarrow && \downarrow^{\mathrlap{\in W \cap F \subset SPSh(C)_{proj}}} \\ \int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} &\stackrel{b}{\to}& B } \,.

By the cofibrany of ${\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}$ itself we are guaranteed a lift

$$\begin{array}{ccc}& & A\\ & \hat{b}\nearrow & {\downarrow }^{\in W\cap F\subset \mathrm{SPSh}(C{)}_{\mathrm{proj}}}\\ {\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}& \stackrel{b}{\to }& B\end{array}.$$\array{ && A \\ &{\hat b}\nearrow& \downarrow^{\in \mathrlap{W \cap F \subset SPSh(C)_{proj}}} \\ \int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} &\stackrel{b}{\to}& B } \,.

that makes the lower triangle commute. The idea is to start with this lift and then inductively adjust it using that $A\to B$ is an acyclic fibration, such that it finally makes also the top triangle commute.

By

$${\mathrm{SPSh}}_C({\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)},B)={\int }_n\mathrm{SSet}({\mathrm{Delta}}^n,B(U^{({\Delta }_{\inf }^n)}))$$SPSh_C(\int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})}, B) = \int_n SSet( \mathbf{Delta}^n , B(U^{(\Delta^n_{inf})}) )

we have that the bottom morphism $b$ is a collection of horizontal morphisms making the following diagrams commute

$$\begin{array}{ccc}⋮& & ⋮\\ {\Delta }^2& \stackrel{b_0}{\to }& B(U^{({\Delta }_{\inf }^2)})\\ \uparrow \downarrow \uparrow \downarrow \uparrow & & \uparrow \downarrow \uparrow \downarrow \uparrow \\ \\ {\Delta }^1& \stackrel{b_1}{\to }& B(U^{({\Delta }_{\inf }^1)})\\ \uparrow \downarrow \uparrow & & \uparrow \downarrow \uparrow \\ {\Delta }^0& \stackrel{b_0}{\to }& B(U)\end{array}$$\array{ \vdots && \vdots \\ \mathbf{\Delta}^2 &\stackrel{b_0}{\to}& B(U^{(\Delta^2_{inf})}) \\ \uparrow \downarrow \uparrow \downarrow \uparrow && \uparrow \downarrow \uparrow \downarrow \uparrow \\ \\ \mathbf{\Delta}^1 &\stackrel{b_1}{\to}& B(U^{(\Delta^1_{inf})}) \\ \uparrow \downarrow \uparrow && \uparrow \downarrow \uparrow \\ \mathbf{\Delta}^0 &\stackrel{b_0}{\to}& B(U) }

and similarly for $a$ and $\hat{b}$. We adjust ${\hat{b}}_k$ to $a_k$ by induction on $k$.

We start by setting

$${\hat{b}}_0\prime :=a_0.$$\hat b_0' := a_0 \,.

Then we need to find a “morphism” that connects the old choice ${\hat{b}}_0$ to the adjusted one ${\hat{b}}_0^{\prime }=a_0$. This is obtained by lifting in the diagram

$$\begin{array}{ccccc}{\Delta }^0\coprod {\Delta }^0& & \stackrel{(a_0,{\hat{b}}_0)}{\to }& & A(U)\\ \downarrow & & {}^{{\gamma }_0}\nearrow & & \downarrow \\ {\Delta }^1& \to & {\Delta }^0& \stackrel{b_0}{\to }& B(U)\end{array}.$$\array{ \mathbf{\Delta}^0 \coprod \mathbf{\Delta}^0 &&\stackrel{(a_0, \hat b_0)}{\to}&& A(U) \\ \downarrow &&{}^{\gamma_0}\nearrow&& \downarrow \\ \mathbf{\Delta}^1 &\to& \mathbf{\Delta}^0 &\stackrel{b_0}{\to}& B(U) } \,.

Analogously, we find a weak inverse ${\gamma }^{-1}$ to $\gamma$ using that $\gamma$ maps to a degenerate cell through $A\to B$. Analogously we find a composite

$${\hat{b}}_1\prime :={\gamma }^{-1}\circ {\hat{b}}_1\circ \gamma .$$\hat b_1' := \gamma^{-1} \circ \hat b_1 \circ \gamma \,.

This way we continue, to finally obtain a lift

$$\begin{array}{ccc}{\int }^n{\Delta }^n\cdot U& \stackrel{a}{\to }& A\\ \downarrow & {}^{\hat{b}\prime }\nearrow & {\downarrow }^{\in W\cap F\subset \mathrm{SPSh}(C{)}_{\mathrm{proj}}}\\ {\int }^n{\Delta }^n\cdot U^{({\Delta }_{\inf }^n)}& \stackrel{b}{\to }& B\end{array}.$$\array{ \int^n \mathbf{\Delta}^n \cdot U &\stackrel{a}{\to}& A \\ \downarrow &{}^{\hat b'}\nearrow& \downarrow^{\mathrlap{\in W \cap F \subset SPSh(C)_{proj}}} \\ \int^n \mathbf{\Delta}^n \cdot U^{(\Delta^n_{inf})} &\stackrel{b}{\to}& B } \,.

The Chevalley-Eilenberg algebra of ${\Pi }^{\inf }(X)$: simplicial deRham complex

Corollary

(CE-algebra of ${\Pi }^{\inf }(X)$ is simplicial deRham)

For $X$ a simplicial manifold, the Chevalley-Eilenberg algebra of ${\Pi }^{\inf }(X)$ is quasi-isomorphic to the simplicial deRham complex of $X$.

Proof

By the discussion at simplicial deRham complex we have that the total complex of the double Moore cochain complex $C(\mathrm{Hom}(X_{•}^{({\Delta }_{\inf }^{•})},R))$ is quasi-isomorphic to the simplicial deRham complex of $X$. By the Eilenberg-Zilber theorem this total complex is quasi-isomorphic to the the Moore complex of the totalization $C(\mathrm{Hom}({\int }^n{\Delta }^n{X}^{({\Delta }_{\inf }^n)},R))$.

The statement then follows using the lemma ( totalization expression on degreewise representables ) used above that (since a simplicial manifold is in particular degreewise representable) ${\Pi }^{\inf }(X)\simeq {\int }^n{\Delta }^n\cdot X^{({\Delta }_{\inf }^n)}$.

Remark

This justifies that for general $X$ we write

$${\Omega }^{•}(X):=\mathrm{CE}({\Pi }^{\inf }(X))$$\Omega^\bullet(X) := CE(\Pi^{inf}(X))

and speak of the deRham complex of the $\mathrm{\infty }$-Lie groupoid $X$.

History and references

The idea of considering the infinitesimal singular simplicial complex as a simplicial object in a smooth topos seems to go back at least to Andre Joyal. Its usefulness for synthetic differential geometry, in particular for the definition of differential forms in synthetic differential geometry has been particularly worked out by Anders Kock. Its concrete realization on schemes was already prominently considered by Alexander Grothendieck in the context of Grothendieck connections/ deRham descent and in that of deRham spaces. The full realization of the infinitesimal singular simplicial complex on spaces formally dual to algebras was spelled out in much detail by Larry Breen and William Messing.

Relevant references on this are collected at infinitesimal singular simplicial complex.

The explicit interpretation of the infinitesimal singular simplicial complex of a manifold or scheme as an object presenting an ∞-Lie groupoid by use of the model structure on simplicial sheaves must have been obvious to Andre Joyal, who found this model structure, but I am not aware of explicit statements to that extent in the literature, along the lines followed here. Similarly, the construction considered here of Yoneda-extending to a Quillen functor on all ∞-Lie groupoids that sits inside the finite path ∞-groupoid-functor, seems to have not yet been conceived explicitly in the literature before.