d\omega_{i}\omega_{i+1}\cdots\omega_{r}+\sum_{i=1}^{r-1}(-1)^{i}T\omega_{1}\cdots(T\omega_{i}\circ\omega_{i+1})\cdots\omega_{r}$
(28.3)
$\displaystyle+p_{1}^{*}\omega_{1}\circ\int\omega_{2}\cdots\omega_{r}-T(\int\omega_{1}\cdots\omega_{r-1})\circ
p_{0}^{*}\omega_{r}$

###### Proof.

Similar to Lemma 27.15. See also [BS14, Proposition 3.3]. ∎

####  28.2. $\infty$-holonomy of ${\mathbb{Z}}$-connection over
${\mathcal{F}}$

Let $V$ be a ${\mathbb{Z}}$-graded vector bundle with a
${\mathbb{Z}}$-connection $\nabla$ over
${A}^{\bullet}=\mathop{\bigwedge\nolimits^{\\!\bullet}}{\mathcal{F}}^{\vee}$.
Locally, $\nabla=d-\sum_{i=0}^{m}A_{i}$, where
$A_{i}\in\operatorname{End}^{1-i}(V)\otimes_{{A}_{0}}{A}^{i}$. Let
$\omega=\sum_{i=0}^{m}A_{i}$. We define $p$-th holonomy of $\nabla$ to be the
iterative integral

(28.4) $\Psi_{p}=\int\omega^{\otimes
p}\in\Gamma\big{(}(\mathop{\bigwedge\nolimits^{\\!k}}T^{\vee}_{{\mathcal{F}}}P{\mathcal{F}})\otimes\operatorname{End}^{-k}(V)\big{)}$

and $\Psi_{0}=\operatorname{Id}$ for $p=0$.

######  Definition 28.4.

Define the $\infty$-holonomy associated to $\nabla$ to be
$\Psi=\sum_{p=0}^{\infty}\Psi_{p}$.

Since $\omega$ has total degree 1,

$\displaystyle\Psi_{p}=$
$\displaystyle\sum_{i=1,j=p-i}^{p}(-1)^{i+1}\int(\omega^{\otimes
i})d\omega(\omega^{\otimes
j})+\sum_{i=1,j=p-i-1}^{p-1}(-1)^{i}\int(\omega^{\otimes
i})(\omega_{i}\circ\omega_{i+1})(\omega^{\otimes j})$
$\displaystyle+p_{1}^{*}\omega\circ\int\omega^{\otimes(r-1)}-\big{(}\int\omega^{\otimes
r-1}\big{)}\circ p_{0}^{*}\omega$

Summing in $p$, we get

$\displaystyle
d\Psi=\bigg{(}\int\kappa+\big{(}\int\kappa\omega+-\int\omega\kappa\big{)}+\cdots+\sum_{i+j=p-1}(-1)^{i}\int\omega^{i}\kappa\omega^{j}+\cdots\bigg{)}+p_{1}^{*}\omega\circ\Psi-\Psi\circ
p_{0}^{*}\omega$

If $\nabla$ is flat, then locally
$\nabla^{2}=(d-\omega)^{2}=-d\omega-T\omega\circ\omega=-d\omega+\omega\circ\omega$.

Let $\sigma:\Delta^{k}\to{\mathcal{F}}$ be a foliated simplex. We can regard
it as a $k-1$-family of paths into ${\mathcal{F}}$. We can break this into two
parts. First we have a map $\theta_{(k-1)}:I_{k-1}\to
P\Delta^{k}_{(v_{k},v_{0})}$, then there is a canonical map
$P\sigma:P\Delta^{k}_{(v_{k},v_{0})}\to P{\mathcal{F}}_{(x_{k},x_{0})}$. We
define a series of map $\psi_{k}\in\operatorname{End}^{1-k}(V)$ by

$\psi_{k}(\sigma)=\begin{cases}\int_{I^{k-1}}(-1)^{(k-1)(K\Psi)}\theta^{*}_{(k-1)}(P\sigma)^{*}\Psi&\quad
k\geq 1\\\ (V_{x},\nabla^{0}_{x})&\quad k=0\end{cases}$

which is essentially the integral of $I^{k-1}$ of the pullback holonomy of the
${\mathbb{Z}}$-connection $\nabla$.

Now we define the Riemann-Hilbert functor
$\operatorname{RH}:\mathcal{P}_{{A}}\to\mathtt{Rep}(\operatorname{Mon}_{\infty}{\mathcal{F}})$.
On objects we define
$\operatorname{RH}_{0}:\operatorname{Obj}(\mathcal{P}_{{A}})\to\operatorname{Obj}(\mathtt{Rep}(\operatorname{Mon}_{\infty}{\mathcal{F}}))$
by
$\operatorname{RH}_{0}\big{(}(E^{\bullet},\nabla)\big{)}(\sigma_{k})=\psi_{k}(\sigma_{k})$.
We claim that the image of this functor are $\infty$-local systems. Note that,
by our construction

$\displaystyle\operatorname{RH}_{0}\big{(}(E^{\bullet},\nabla)\big{)}_{x}=$
$\displaystyle E_{x}$
$\displaystyle\operatorname{RH}_{0}\big{(}(E^{\bullet},\nabla)\big{)}(x)=$
$\displaystyle\mathbb{E}^{0}_{x}$
$\displaystyle\operatorname{RH}_{0}\big{(}(E^{\bullet},\nabla)\big{)}(\sigma_{k>0})=$
$\displaystyle\int_{I^{k-1}}(-1)^{(k-1)(K\Psi)}\theta^{*}_{(k-1)}(P\sigma)^{*}\Psi$

Write $F$ the image of
$\operatorname{RH}_{0}\big{(}(E^{\bullet})),\nabla)\big{)}$ for simplicity,
i.e. $F(\sigma_{k})=\operatorname{RH}_{0}\big{(}(E^{\bullet}))(\sigma_{k})$.
Since $\mathbb{E}$ is flat, we have

$d\Psi=-p_{0}^{*}A^{0}\circ\Psi+\Psi\circ p_{1}^{*}A^{0}$

Integrate the left side and apply the Stoke’s formula we get

$-\hat{\delta}F-\sum_{i=1}^{k-1}(-1)^{i}F(\sigma_{0\cdots i})F(\sigma_{i\cdots
k})$

Plug in the integration of right side, we get

$\mathbb{E}^{0}\circ
F(\sigma_{k})-(-1)^{k}F(\sigma_{k})-\sum_{i=1}^{k-1}(-1)^{i}F(\sigma_{(0\cdots\hat{i}\cdots
k)})+\sum_{i=1}^{k-1}(-1)^{i}F(\sigma_{0\cdots i})F(\sigma_{i\cdots k})=0$

which is the $k$-th level of the Maurer-Cartan equations for $\infty$-local
system condition. Therefore, $\operatorname{RH}_{0}$ is a well-defined map on
objects.

######  Theorem 28.5 ([Igu09]).

The image of an object under the functor $\operatorname{RH}$ is an
$\infty$-representations of $\operatorname{Mon}_{\infty}({\mathcal{F}})$ if
and only if $\nabla$ is flat.

###### Proof.

By Theorem 4.10 in [Igu09], we have

$\psi_{0}(x_{0})\phi_{k}(\sigma)+(-1)^{k}\psi_{k}(\sigma)\psi(x_{k})=\sum_{i=1}^{k-1}(-1)^{i}\big{(}\psi_{k-1}(\sigma_{(0\cdots\hat{i}\cdots
k)})-\psi_{i}(\sigma_{(0\dots i)})\psi_{k-i}(\sigma_{(i\cdots k)})\big{)}$

which is equivalent to

$\sum_{i=1}^{k-1}(-1)^{i}\psi_{k-1}(\sigma_{(0\cdots\hat{i}\cdots
k)})-\sum_{i=0}^{k}(-1)^{i}\psi_{i}(\sigma_{(0\dots
i)})\psi_{k-i}(\sigma_{(i\cdots k)})$

i.e. $\hat{\delta}\psi+\psi\cup\psi=0$.

For the other direction, we just go back from the definition of $\psi$, and
found that $A_{0}\Psi_{k}-\Psi_{k}A_{0}=d\Psi_{k-1}$ must be equal for all
$k$, which is equivalent to the flatness of $\nabla$. ∎

Now we proceed to $\operatorname{RH}$ on higher simplices

$\operatorname{RH}_{n}:\mathcal{P}_{{A}}(E_{n-1}^{\bullet},E_{n}^{\bullet})\otimes\cdots\otimes\mathcal{P}_{{A}}(E_{0}^{\bullet},E_{1}^{\bullet})\to\mathtt{Rep}(\operatorname{Mon}_{\infty}{\mathcal{F}})(\operatorname{RH}_{0}(E_{0}^{\bullet}),\operatorname{RH}_{0}(E_{n}^{\bullet}))[1-n]$

by

(28.5)
$\operatorname{RH}_{n}(\phi_{n}\otimes\cdots\otimes\phi_{1})(\sigma_{k})=\operatorname{RH}_{0}\big{(}C_{(\phi_{n}\otimes\cdots\otimes\phi_{1})}(\sigma_{k})\big{)}_{n+1,1}$

Next, we will need $\infty$-holonomy with respect to the pre-triangulated
structure of $\mathcal{P}_{{\mathcal{A}}}$. We follow the calculation in
[BS14, Section 3.5] of the following

  * •

$\infty$-holonomy with respect to the shift.

  * •

$\infty$-holonomy with the cone

######  Proposition 28.6.

$\operatorname{RH}$ is an $A_{\infty}$-functor.

###### Proof.

We follow [BS14, Theorem 4.2]. Let
$\phi=\phi_{n}\otimes\cdots\otimes\phi_{1}\in\mathcal{P}_{{A}}(E_{n-1},E_{n})\otimes\cdots\otimes\mathcal{P}_{{A}}(E_{0},E_{1})$
be a tuple of morphisms, denote the holonomy of the associated to the
generalized homological cone $C_{\phi}$ by
$\Psi^{\phi_{n}\otimes\cdots\otimes\phi_{1}}$. Locally, we can write
$D^{\phi}=d-\omega$. By …, we have that on $P{\mathcal{F}}(x_{0},x_{1})$,
where $x_{0},x_{1}$ lie in some leaf. By the $\infty$-holonomy for cones, we
have

$\displaystyle-$ $\displaystyle
d\Psi^{\phi_{n}\otimes\cdots\otimes\phi_{1}}_{n+1,1}-p_{0}^{*}\omega^{0}_{n+1,1}\circ\Psi^{\phi_{n}\otimes\cdots\otimes\phi_{1}}_{n+1,1}+\Psi^{\phi_{n}\otimes\cdots\otimes\phi_{1}}_{n+1,1}\circ
p_{1}^{*}\omega^{0}_{n+1,1}=$
$\displaystyle\sum_{k=1}^{n-1}(-1)^{n-k-1-|\phi_{n}\otimes\cdots\otimes\phi_{k+2}|}\Psi^{\phi_{n}\otimes\cdots\otimes\phi_{k+1}\circ\phi_{k}\otimes\cdots\otimes\phi_{1}}_{n+1,1}+$
$\displaystyle\sum_{k=1}^{n}(-1)^{n-k-|\phi_{n}\otimes\cdots\otimes\phi_{k+1}|}\Psi^{\phi_{n}\otimes\cdots\otimes
d\phi_{k}\otimes\cdots\otimes\phi_{1}}_{n+1,1}$

Now applying $\int(-1)^{K(\Psi J(-)}\theta^{*}(P[-])^{*}(\Psi)$ to both sides
of the equation. For simplicity, we denote
$\phi_{k}\otimes\cdots\otimes\phi_{l}$ by $\phi_{k,l}$. We have

$\displaystyle[$ $\displaystyle RH_{0}(C_{\phi})\cup
RH_{0}(C_{\phi})+\hat{\delta}\operatorname{RH}_{0}(C_{\phi})]_{n+1,1}=$
$\displaystyle\sum_{k=1}^{n-1}(-1)^{n-k-1-|\phi_{n,k+2}|}\operatorname{RH}_{n-1}(\phi_{n}\otimes\cdots\otimes\phi_{k+1}\circ\phi_{k}\otimes\cdots\otimes\phi_{1})$
$\displaystyle+\sum_{k=1}^{n-1}(-1)^{n-k-|\phi_{n,k+1}|}\operatorname{RH}_{n}(\phi_{n}\otimes\cdots\otimes
d\phi_{k}\otimes\cdots\otimes\phi_{1})$

By the matrix decomposition formulas in calculating the holonomy of cones in
[AS14, Section 3.5.2], we get

$\displaystyle[RH_{0}(C_{\phi})\cup RH_{0}(C_{\phi})]_{n+1,1}=$
$\displaystyle\sum_{i+j=n}\operatorname{RH}_{0}(C_{\phi_{n,i+1}})_{j+1,1}\cup\operatorname{RH}_{0}(C_{\phi_{i,1}})_{i+1,1}[(j-\sum_{k=i+1}^{n}p_{k})]$
$\displaystyle+\operatorname{RH}_{0}(E_{n})\cup\operatorname{RH}_{0}(C_{\phi})_{n+1,1}+\operatorname{RH}_{0}(C_{\phi})_{n+1,1}\cup\operatorname{RH}_{0}(E_{0}[n-|\phi|])$
$\displaystyle=$
$\displaystyle\sum_{i+j=n}(-1)^{(j-\sum_{k=i+1}^{n}p_{k})}\operatorname{RH}_{j}(C_{\phi_{n,i+1}})\cup\operatorname{RH}_{i}(C_{\phi_{i,1}})$
$\displaystyle+\operatorname{RH}_{0}(E_{n})\cup\operatorname{RH}_{0}(C_{\phi})_{n+1,1}+(-1)^{n-|\phi|}\operatorname{RH}_{0}(C_{\phi})_{n+1,1}\cup\operatorname{RH}_{0}(E_{0})$

By our construction,

$\displaystyle
D_{\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}(K_{\bullet})}\big{(}\operatorname{RH}_{n}(\phi)\big{)}=$
$\displaystyle\hat{\delta}\big{(}\operatorname{RH}_{n}(\phi)\big{)}+\operatorname{RH}_{0}(E_{n})\cup\operatorname{RH}_{n}(\phi)$
$\displaystyle+(-1)^{n-|\phi|}\operatorname{RH}_{n}(\phi)\cup\operatorname{RH}_{0}(E_{0})$

Put the last two equations into the one above, we get

$\displaystyle\Big{(}\sum_{i+j=n}(-1)^{(j-\sum_{k=i+1}^{n}p_{k})}\operatorname{RH}_{j}(C_{\phi_{n,i+1}})\cup\operatorname{RH}_{i}(C_{\phi_{i,1}})\Big{)}+D\big{(}\operatorname{RH}_{n}(\phi_{n,1})\big{)}$
$\displaystyle=\sum_{k=1}^{n-1}(-1)^{n-k-1-|\phi_{n,k+2}|}\operatorname{RH}_{n-1}(\phi_{n}\otimes\cdots\otimes\phi_{k+1}\circ\phi_{k}\otimes\cdots\otimes\phi_{1})$
$\displaystyle\quad+\sum_{k=1}^{n-1}(-1)^{n-k-|\phi_{n,k+1}|}\operatorname{RH}_{n}(\phi_{n}\otimes\cdots\otimes
d\phi_{k}\otimes\cdots\otimes\phi_{1})$

which is the $A_{\infty}$-relation for an $A_{\infty}$-functor between two dg-
categories. Therefore, $\operatorname{RH}$ is an $A_{\infty}$-functor. ∎

####  28.3. Riemann-Hilbert correspondence

######  Theorem 28.7.

The functor $\operatorname{RH}$ is an $A_{\infty}$-quasi-equivalence.

First, we want to show $\operatorname{RH}$ is $A_{\infty}$-quasi-fully-
faithful. Consider two objects $(E_{1}{\bullet},\mathbb{E}_{1})$,
$(E_{1}{\bullet},\mathbb{E}_{1})\in\mathcal{P}_{{A}}$. The chain map

$\operatorname{RH}_{1}:\mathcal{P}_{{A}}(E_{1},E_{2})\to\mathsf{Loc}^{\infty}_{\mathsf{Ch}_{k}}({\mathcal{F}})(\operatorname{RH}_{0}(E_{1}),\operatorname{RH}_{0}(E_{2}))$

induces a map on the spectral sequence. In $E_{1}$-page, on
$\mathcal{P}_{{A}}$ side,
$H^{\bullet}\big{(}(E_{i},\mathbb{E}_{i}^{0})\big{)}$ are vector bundles with
flat connections, while on the other side
$H^{\bullet}\big{(}(\operatorname{RH}(E_{i}),\mathbb{E}_{i}^{0})\big{)}$ are
${\mathcal{F}}$-local systems. In $E_{2}$-page, the map is

$\displaystyle
H^{\bullet}\bigg{(}M,\operatorname{Hom}\Big{(}H^{\bullet}\big{(}(E_{1},\mathbb{E}_{1}^{0})\big{)},H^{\bullet}\big{(}(E_{2},\mathbb{E}_{2}^{0})\big{)}\Big{)}\bigg{)}\to$
$\displaystyle
H^{\bullet}\bigg{(}M,\operatorname{Hom}\Big{(}H^{\bullet}\big{(}(\operatorname{RH}(E_{1}),\mathbb{E}_{1}^{0})\big{)},H^{\bullet}\big{(}(\operatorname{RH}(E_{2}),\mathbb{E}_{2}^{0})\big{)}\Big{)}\bigg{)}$

which is an isomorphism by the de Rham theorem for foliated local systems.
Next, we shall prove that $\operatorname{RH}$ is $A_{\infty}$-essentially
surjective.

Let $F\in\mathsf{Loc}^{{\operatorname{dg}}}_{{\mathsf{C}}}({\mathcal{F}})$, we
want to construct an object $(E^{\bullet},\nabla)\in\mathcal{P}_{{A}}$ whose
image under $\operatorname{RH}_{0}$ is quasi-isomorphic to $F$. First notice
that $\underline{{\mathbb{R}}}_{{\mathcal{F}}}$ defines a representation of
$\operatorname{Mon}_{\infty}({\mathcal{F}})$ by previous section, which can be
viewed as $\infty$-local system over ${\mathcal{F}}$. Regard
$(M,\underline{{\mathbb{R}}}_{{\mathcal{F}}})$ as a ringed space. Construct a
complex of sheaves $(\underline{C}_{F}^{\bullet},D)$ by

$\big{(}\underline{C}_{F}(U)^{\bullet},D(U)\big{)}=\mathsf{Loc}^{\infty}_{\mathsf{Ch}_{k}}({\mathcal{F}})\big{(}\underline{{\mathbb{R}}}_{{\mathcal{F}}}|_{U},F|_{U}\big{)}^{\bullet}$

We claim that $\underline{C}_{F}^{\bullet}$ is soft. First notice that for
$i>0$, $\underline{C}_{F}^{i}$ is a $\underline{C}_{F}^{0}$-module by cup
products on open sets. By definition,
$\underline{C}_{F}^{0}=\\{\phi:(\mathsf{Loc}^{\infty}_{\mathsf{Ch}_{k}}({\mathcal{F}}))_{0}\to\mathsf{Ch}_{{\mathbb{R}}}^{0}|\phi(x)\in\mathsf{Ch}_{{\mathbb{R}}}^{0}\big{(}\underline{{\mathbb{R}}}_{F}(x),F(x)\big{)}\\}$
which is a sheaf of discontinuous sections, hence soft. Therefore, all
$\underline{C}_{F}^{i}$’s are soft. Recall that for two
$\infty$-representation of a Lie $\infty$-groupoid, the $E_{1}$ term of the
spectral sequence is an ordinary representation. Hence,
$\underline{C}_{F}^{\bullet}$ is a perfect complex of sheaves. Let
$\underline{{A}}^{\bullet}$ be the sheaf of ${\mathcal{C}}^{\infty}$ sections
of
${A}^{\bullet}=\mathop{\bigwedge\nolimits^{\\!\bullet}}{\mathcal{F}}^{\vee}$.
$\underline{{A}}^{0}={{\mathcal{C}}^{\infty}(M)}$ is flat over
$\underline{{\mathbb{R}}}_{{\mathcal{F}}}$ as ${\mathcal{C}}^{\infty}$-rings
since locally the module of smooth functions on $M$ are
${\mathcal{C}}^{\infty}({\mathbb{R}}^{n})$ and the foliated functions are
${\mathcal{C}}^{\infty}({\mathbb{R}}^{n-q})$ where
$q=\operatorname{codim}{\mathcal{F}}$, and
${\mathcal{C}}^{\infty}({\mathbb{R}}^{n-q})\otimes_{\infty}{\mathcal{C}}^{\infty}({\mathbb{R}}^{q})\simeq{\mathcal{C}}^{\infty}({\mathbb{R}}^{n})$
where $\otimes_{\infty}$ is the tensor product for
${\mathcal{C}}^{\infty}$-rings. Therefore,
$\underline{C}_{F}^{\infty}=\underline{C}_{F}^{\bullet}\otimes_{\underline{{\mathbb{R}}}_{{\mathcal{F}}}}\underline{{A}}^{0}$
is a sheaf of perfect ${A}^{0}$-modules.

Again by the flatness of $\underline{{A}}^{0}$ is flat over
$\underline{{\mathbb{R}}}_{{\mathcal{F}}}$. We have a quasi-isomorphism
$(\underline{C}_{F}^{\bullet},D)\simeq(\underline{C}_{F}^{\infty}\otimes_{\underline{{A}}^{0}}\underline{{A}}^{\bullet},D\otimes
1+1\otimes d)$. We need the following proposition from Proposition 2.3.2,
Exposè II, SGA6, [Ber+06].

######  Remark 28.8.

One of the core tool in previous proof is the flatness of
$\underline{{A}}^{0}$ over $\underline{{\mathbb{R}}}_{{\mathcal{F}}}$, or, in
other words, the flatness of $\underline{{A}}^{0}$ over
$H^{0}(\underline{{A}})$. We then expect a natural extension of our results to
arbitrary $L_{\infty}$-algebroids $\mathfrak{g}$ with associated foliation dga
$A$, and $A^{0}$ is flat over $H^{0}(A)$. A natural question will be, given
any dga $A$ (which presents some geometric object), when is $A^{0}$ flat over
$H^{0}$? Or we can consider an even more generalization, given a map of
sheaves of algebras

$d:A^{0}\to A^{1}$

, when $A^{0}$ is flat over $H^{0}(A)=\ker(d)$? We believe that this question
is related to a more general phenomenon in noncommutative geometry.

######  Proposition 28.9.

Let $(X,\underline{\mathcal{S}}_{X})$ be a ringed space, where $X$ is compact
and $\underline{\mathcal{S}}_{X}$ is a soft sheaf of rings. Then

  1. (1)

The global section functor

$\Gamma:\mathsf{Mod}_{\underline{\mathcal{S}}_{X}}\to\mathsf{Mod}_{\underline{\mathcal{S}}_{X}(X)}$

is exact and establishes an equivalence of categories between the category of
sheaves of right $\underline{\mathcal{S}}_{X}$-modules and the category of
right modules over the global sections $\underline{\mathcal{S}}_{X}(X)$ of
$\underline{\mathcal{S}}_{X}$.

  2. (2)

If $\underline{F}\in\mathsf{Mod}_{\underline{\mathcal{S}}_{X}}$ locally has
finite resolutions by finitely generated free
$\underline{\mathcal{S}}_{X}$-modules, then $\Gamma(X,\underline{F})$ has a
finite resolution by finitely generated projective modules.

  3. (3)

The derived category of perfect complexes of sheaves
$\mathsf{D_{Perf}}(\mathsf{Mod}_{\underline{\mathcal{S}}_{X}})$ is equivalent
to the derived category of perfect complexes of modules
$\mathsf{D_{Perf}}(\mathsf{Mod}_{\underline{\mathcal{S}}_{X}(X)})$.

By this theorem, there is a (strict) perfect complex of ${A}^{0}$-modules
$(E,\mathbb{E}^{0})$ and a quasi-isomorphism
$e^{0}:(E^{\bullet},\mathbb{E}^{0})\to(F^{\bullet},\mathbb{F}^{0})=(\Gamma(M,\underline{C}_{F}^{\infty}),D)$.
We shall follow the argument of Theorem 3.2.7 of [Blo05] to construct the
higher components $\mathbb{E}^{i}$ of ${\mathbb{Z}}$-connection along with the
higher components of a morphism $e^{i}$.

On $F^{\bullet}$, we have a ${\mathbb{Z}}$-connection

$\mathbb{F}=D\otimes 1+1\otimes d:F^{\bullet}\to
F^{\bullet}\otimes_{{A}^{0}}{A}^{\bullet}$

. The idea is to transfer this ${\mathbb{Z}}$-connection to $E^{\bullet}$
which is compatible with the quasi-isomorphism on $H^{0}$’s. Note that we have
an induced connection

$\mathbb{H}^{k}:H^{k}(F^{\bullet},\mathbb{F}^{0})\to
H^{k}(F^{\bullet},\mathbb{F}^{0})\otimes_{{A}^{0}}{A}^{1}$

for each $k$. First we will transfer this connection to a connection on
$H^{k}(E^{\bullet},\mathbb{E}^{0})$, and we have the following commutative
diagram

${H^{k}(E^{\bullet},\mathbb{E}^{0})}$${H^{k}(E^{\bullet},\mathbb{E}^{0})\otimes_{{A}^{0}}{A}^{1}}$${H^{k}(F^{\bullet},\mathbb{F}^{0})}$${H^{k}(F^{\bullet},\mathbb{F}^{0})\otimes_{{A}^{0}}{A}^{1}}$$\scriptstyle{\mathbb{H}^{k}}$$\scriptstyle{e^{0}}$$\scriptstyle{e^{0}\otimes
1}$$\scriptstyle{\mathbb{H}^{k}}$

Note that $e^{0}\otimes 1$ is a quasi-isomorphism since ${A}^{\bullet}$ is
flat over ${A}^{0}$. We need the following lemma.

######  Lemma 28.10.

Given a bounded complex of finitely generated projective ${A}^{0}$-modules
$(E^{\bullet},\mathbb{E}^{0})$ with connections
$\mathbb{H}^{k}:H^{k}(F^{\bullet},\mathbb{F}^{0})\to
H^{k}(F^{\bullet},\mathbb{F}^{0})\otimes_{{A}^{0}}{A}^{1}$ for each $k$, there
exists connections

$\tilde{\mathbb{H}}^{k}:E^{k}\to E^{k}\otimes_{{A}^{0}}{A}^{1}$

lifting $\mathbb{H}^{k}$, i.e.

$\tilde{\mathbb{H}}^{k}\mathbb{E}^{0}=(\mathbb{E}^{0}\otimes
1)\tilde{\mathbb{H}}^{k}$

for each $k$ and the connection induced on the cohomology is $\mathbb{H}^{k}$.

###### Proof.

This is Lemma 3.2.8 in [Blo05] and Lemma 4.6 in [BS14]. Since $E^{\bullet}$ is
bounded, let $[N,M]$ be its magnitude. Pick some arbitrary connection $\nabla$
on $E^{M}$. Consider the following diagram whose rows are exact

${E^{M}}$${H^{M}(E^{\bullet},\mathbb{E}^{0})}$${0}$${E^{M}\otimes_{{A}^{0}}{A}^{1}}$${H^{M}(E^{\bullet},\mathbb{E}^{0})\otimes_{{A}^{0}}{A}^{1}}$${0}$$\scriptstyle{j}$$\scriptstyle{\theta}$$\scriptstyle{\nabla}$$\scriptstyle{\mathbb{H}^{M}}$$\scriptstyle{j\otimes
1}$

Here $\theta=\mathbb{H}^{M}\circ j-(j\otimes 1)\circ\nabla$ is
${A}^{0}$-linear and $j\otimes 1$ is surjective. Now, by the projectivity of
$E^{M}$, we can lift $\theta$ to a $\tilde{\theta}:E^{M}\to
E^{M}\otimes_{{A}^{0}}{A}^{1}$ such that $(j\otimes 1)\tilde{\theta}=\theta$.
Let $\tilde{\mathbb{H}}^{k}=\nabla+\tilde{\theta}$. Now replace $\nabla$ by
$\tilde{\mathbb{H}}^{k}$ and the above diagram still commutes.

Now pick some arbitrary connection $\nabla_{M-1}$ on $E^{M-1}$. Note that
$\mathbb{E}^{0}\nabla_{M-1}={\mathbb{H}}^{M-1}\mathbb{E}^{0}=0$ does not
necessarily hold. Set
$\mu={\mathbb{H}}^{M-1}\mathbb{E}^{0}-\mathbb{E}^{0}\nabla_{M-1}$, then $\mu$
is ${A}^{0}$-linear.
$\operatorname{Im}\mu\subset\operatorname{Im}\mathbb{E}^{0}\otimes 1$ since
$\operatorname{Im}{\mathbb{H}}^{M-1}\mathbb{E}^{0}\subset\operatorname{Im}\mathbb{E}^{0}\otimes
1$ as ${\mathbb{H}}$ is a lift of $\mathbb{H}$. Now by the projectivity we can
lift this map to a $\tilde{\mu}:E^{M-1}\to E^{M-1}\otimes_{{A}^{0}}{A}^{1}$
such that $(\mathbb{E}^{0}\otimes 1)\circ\tilde{\mu}=\mu$. Now set
$\tilde{\mathbb{H}}^{M-1}=\nabla_{M-1}+\tilde{\mu}$, then
$(\mathbb{E}^{0}\otimes
1)\tilde{\mathbb{H}}^{M-1}=\tilde{\mathbb{H}}^{M-1}\mathbb{E}^{0}$. We have
the following diagram

${E^{N}}$${E^{N+1}}$${\cdots}$${E^{M-1}}$${E^{M}}$${E^{N}\otimes_{{A}^{0}}{A}^{1}}$${E^{N+1}\otimes_{{A}^{0}}{A}^{1}}$${\cdots}$${E^{M-1}\otimes_{{A}^{0}}{A}^{1}}$${E^{M}\otimes_{{A}^{0}}{A}^{1}}$$\scriptstyle{\mathbb{E}^{0}}$$\scriptstyle{\mathbb{E}^{0}}$$\scriptstyle{\mathbb{E}^{0}}$$\scriptstyle{\mathbb{E}^{0}}$$\scriptstyle{\nabla_{M-1}}$$\scriptstyle{\mu}$$\scriptstyle{\tilde{\mathbb{H}}_{M}}$$\scriptstyle{\mathbb{E}^{0}\otimes
1}$$\scriptstyle{\mathbb{E}^{0}\otimes 1}$$\scriptstyle{\mathbb{E}^{0}\otimes
1}$$\scriptstyle{\mathbb{E}^{0}\otimes 1}$

Now we continue in the same fashion and construct all $\tilde{\mathbb{H}}^{k}$
with $(\mathbb{E}^{0}\otimes
1)\tilde{\mathbb{H}}^{k}=\tilde{\mathbb{H}}^{k}\mathbb{E}^{0}$ for all $k$. ∎

Now let’s continue the proof of the main theorem. Set
$\tilde{\mathbb{E}}^{1}=(-1)^{k}\tilde{\mathbb{H}}_{k}$ on $E^{k}$ for each
$k$. By our construction

$\mathbb{E}^{0}\tilde{\mathbb{E}}^{1}+\tilde{\mathbb{E}}^{1}\mathbb{E}^{0}=0$

but $e^{0}\tilde{\mathbb{E}}^{1}=\mathbb{F}^{1}e^{0}$ might not hold. We will
correct this by modifying $\tilde{\mathbb{E}}^{1}$. Consider the map
$\psi=e^{0}\tilde{\mathbb{E}}^{1}-\mathbb{F}^{1}e^{0}:E^{\bullet}\to
F^{\bullet}\otimes_{{A}^{0}}{A}^{1}$. It is easy to verify that $\psi$ is
${A}^{0}$-linear and a map of chain complexes. Now we have the following
diagram

${(E^{\bullet}\otimes_{{A}^{0}}{A}^{1},\mathbb{E}^{0}\otimes
1)}$${E^{\bullet}}$${(F^{\bullet}\otimes_{{A}^{0}}{A}^{1},\mathbb{F}^{0}\otimes
1)}$$\scriptstyle{e^{0}\otimes
1}$$\scriptstyle{\psi}$$\scriptstyle{\tilde{\psi}}$

here $e^{0}\otimes 1$ is a quasi-isomorphism since $e^{0}$ is a homotopy
equivalence. $\tilde{\psi}$ is a lift of $\psi$ and there exists a homotopy
$e^{1}:E^{\bullet}\to F^{\bullet-1}\otimes_{{A}^{0}}{A}^{1}$

$\phi-(e^{0}\otimes 1)\tilde{\psi}=(e^{1}\mathbb{E}^{0}+\mathbb{F}^{0}e^{1})$

Now let $\mathbb{E}^{1}=\tilde{\mathbb{E}}$. We have

$\mathbb{E}^{0}\mathbb{E}^{1}+\mathbb{E}^{1}\mathbb{E}^{0}=0$

and

$e^{1}\mathbb{E}^{0}+\mathbb{F}^{0}e^{1}=e^{0}\mathbb{E}^{1}+\mathbb{F}^{1}e^{0}$

Now we have constructed the first two components $\mathbb{E}^{0}$ and
$\mathbb{E}^{1}$ of the ${\mathbb{Z}}$-connection, and the first two
components $e^{0}$ and $e^{1}$ of the quasi-isomorphism
$E^{\bullet}\otimes_{{A}^{0}}{A}^{\bullet}\to
F^{\bullet}\otimes_{{A}^{0}}{A}^{\bullet}$.

Now let’s proceed to construct the rest components. Consider the mapping cone
$C_{e^{0}}^{\bullet}$ of $e^{0}$, i.e.
$C_{e^{0}}^{\bullet}=E[1]^{\bullet}\oplus F^{\bullet}$. Now let
$\mathbb{L}^{0}$ be defined as the matrix

$\begin{pmatrix}\mathbb{E}^{0}[1]&0\\\ e^{0}[1]&\mathbb{F}^{0}\end{pmatrix}$

Define $\mathbb{L}^{1}$ as the matrix $\begin{pmatrix}\mathbb{E}^{1}[1]&0\\\
e^{1}[1]&\mathbb{F}^{1}\end{pmatrix}$ Now $\mathbb{L}^{0}\mathbb{L}^{0}=0$ and
$[\mathbb{L}^{0},\mathbb{L}^{1}]=0$ by construction. Let

$D=\mathbb{L}^{1}\mathbb{L}^{1}+\begin{pmatrix}0&0\\\
\mathbb{F}^{2}e^{0}&[\mathbb{F}^{0},\mathbb{F}^{2}]\end{pmatrix}$

It is easy to check that $D$ is ${A}$-linear,

  1. (1)

$[\mathbb{L}^{0},D]=0$,

  2. (2)

$D|_{0\oplus F^{\bullet}}$.

Note that $(C_{e^{0}}^{\bullet},\mathbb{L}^{0})$ is acyclic since it is a
mapping cone of a quasi-isomorphism. By flatness of ${A}^{\bullet}$ over
${A}^{0}$, $(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{2},\mathbb{L}^{0}\otimes
1)$ is also acyclic. In addition,

$\operatorname{Hom}^{\bullet}_{{A}^{0}}\big{(}(E^{\bullet},\mathbb{E}^{0}),(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{2},\mathbb{L}^{0}\otimes
1)\big{)}$

is a subcomplex of

$\operatorname{Hom}^{\bullet}_{{A}^{0}}\big{(}C_{e^{0}}^{\bullet},(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{2},[\mathbb{L}^{0},-])\big{)}$

How
$D\in\operatorname{Hom}^{\bullet}_{{A}^{0}}\big{(}(E^{\bullet},\mathbb{E}^{0}),(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{2},\mathbb{L}^{0}\otimes
1)\big{)}$ is a cycle, so there exists some
$\tilde{\mathbb{L}}^{2}\in\operatorname{Hom}^{\bullet}_{{A}^{0}}\big{(}(E^{\bullet},\mathbb{E}^{0}),(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{2},\mathbb{L}^{0}\otimes
1)\big{)}$ such that $-D=[\mathbb{L}^{0},\tilde{\mathbb{L}}^{2}]$. Define
$\mathbb{L}^{2}$ by

$\mathbb{L}^{2}=\tilde{\mathbb{L}}^{2}+\begin{pmatrix}0&0\\\
0&\mathbb{F}^{2}\end{pmatrix}$

We have

$\displaystyle[\mathbb{L}^{0},\mathbb{L}^{2}]=$
$\displaystyle\Big{[}\mathbb{L}^{0},\tilde{\mathbb{L}}^{2}+\begin{pmatrix}0&0\\\
0&\mathbb{F}^{2}\end{pmatrix}\Big{]}$ $\displaystyle=$
$\displaystyle-D+\Big{[}\mathbb{L}^{0},\begin{pmatrix}0&0\\\
0&\mathbb{F}^{2}\end{pmatrix}\Big{]}$ $\displaystyle=$
$\displaystyle-\mathbb{L}^{1}\mathbb{L}^{1}$

Therefore we get

$\displaystyle\mathbb{L}^{0}\mathbb{L}^{2}+\mathbb{L}^{1}\mathbb{L}^{1}+\mathbb{L}^{2}\mathbb{L}^{0}=0$

Following this pattern, we continue by setting

$D=\mathbb{L}^{1}\mathbb{L}^{2}+\mathbb{L}^{2}\mathbb{L}^{1}+\begin{pmatrix}0&0\\\
\mathbb{F}^{3}e^{0}&[\mathbb{F}^{0},\mathbb{F}^{3}]\end{pmatrix}$

Again it is easy to verify that $D$ is ${A}^{0}$-linear, and

  1. (1)

$[\mathbb{L}^{0},D]=0$,

  2. (2)

$D|_{0\oplus F^{\bullet}}$.

By the same reasoning as before, so there exists some
$\tilde{\mathbb{L}}^{3}\in\operatorname{Hom}^{\bullet}_{{A}^{0}}\big{(}(E^{\bullet},\mathbb{E}^{0}),(C_{e^{0}}^{\bullet}\otimes_{{A}^{0}}{A}^{3},\mathbb{L}^{0}\otimes
1)\big{)}$ such that $-D=[\mathbb{L}^{0},\tilde{\mathbb{L}}^{3}]$. Define
$\mathbb{L}^{3}$ by

$\mathbb{L}^{3}=\tilde{\mathbb{L}}^{3}+\begin{pmatrix}0&0\\\
0&\mathbb{F}^{3}\end{pmatrix}$

By easy verification we get $\sum_{i=0}^{3}\mathbb{L}^{i}\mathbb{L}^{3-i}=0$.

Now suppose we have constructed $\mathbb{L}^{0},\cdots,\mathbb{L}^{n}$ which
satisfy

$\sum_{i=0}^{k}\mathbb{L}^{i}\mathbb{L}^{k-i}=0$

for $k=0,\cdots,n$. Then we define

$D=\sum_{i=1}^{n}\mathbb{L}^{i}\mathbb{L}^{n+1-i}+\begin{pmatrix}0&0\\\
\mathbb{F}^{n+1}e^{0}&[\mathbb{F}^{0},\mathbb{F}^{n+1}]\end{pmatrix}$

Again we have $D$ is ${A}^{0}$-linear, and

  1. (1)

$[\mathbb{L}^{0},D]=0$,

  2. (2)

$D|_{0\oplus F^{\bullet}}$.

We can continue the inductive construction of $\mathbb{L}$ to get a
${\mathbb{Z}}$-connection satisfying $\mathbb{L}\mathbb{L}=0$. Then we have
constructed both components of the ${\mathbb{Z}}$-connection and the morphism
from $(E^{\bullet},\mathbb{E})$ to $(F^{\bullet},\mathbb{F})$.

Now we have shown that $\operatorname{RH}$ is $A_{\infty}$-essentially
surjective. Therefore, $\operatorname{RH}$ is an $A_{\infty}$-quasi-
equivalence.

######  Corollary 28.11.

The $\infty$-category $\mathtt{Loc}^{\infty}_{\mathsf{Ch}_{k}}{\mathcal{F}}$
is equivalent to the $\infty$-category
$\mathtt{Mod}_{{A}}^{{\operatorname{coh}}}$, for
${A}=\operatorname{CE}({\mathcal{F}})$.

####  28.4. Integrate $\infty$-representations of $L_{\infty}$-algebroids

$\operatorname{RH}$ is a functor from cohesive modules over the foliation dga
$A$, which can also be regarded as cohesive modules over the foliation Lie
algebroid $T{\mathcal{F}}$. It is not hard to generalize the
$\operatorname{RH}$ as a functor from cohesive modules over any
$L_{\infty}$-algebroids, where we only need to refine the iterated integrals
to the corresponding vector bundles over the foliations, i.e. we only
integrate along leaves of the (singular) foliations generated by
$L_{\infty}$-algebroids. On the other hand, the monodromy $\infty$-groupoid of
a perfect singular foliation ${\mathcal{F}}$ is the truncation of the
integration of the $L_{\infty}$-algebroid $\mathfrak{g}$ associated
${\mathcal{F}}$. Therefore, given a perfect singular foliation ${\mathcal{F}}$
with its associated $L_{\infty}$-algebroid $\mathfrak{g}$, we get the
following commutative diagram

${{\mathfrak{g}}}$${{\operatorname{Mon}(\mathfrak{g})}}$${{\operatorname{Mon}({\mathcal{F}})}}$${{\mathsf{Mod}^{{\operatorname{coh}}}_{\mathfrak{g}}}}$${{\mathsf{Loc}^{\infty}(\mathfrak{g})}}$${{\mathsf{Loc}^{\infty}({\mathcal{F}})}}$$\scriptstyle{\int}$$\scriptstyle{\mathsf{Rep}}$$\scriptstyle{\mathsf{Rep}}$$\scriptstyle{\tau}$$\scriptstyle{\mathsf{Rep}}$$\scriptstyle{\operatorname{RH}}$$\scriptstyle{\tau}$

where $\tau$ denotes the truncation functor. A natural question to ask is when
$\operatorname{RH}$ will be an $A_{\infty}$-quasi-equivalence, or induce an
$\infty$-equivalence at the $\infty$-category level. This will be studied in a
future paper.

### Index

  * $(\infty,1)$-category Definition 7.7
  * $(\infty,n)$-category Definition 7.5
  * $\acute{e}$tale morphism item 2
  * $\infty$-action Definition 17.2
  * $\infty$-groupoid Example 5.3
    * in simplicial sets Definition 11.2
    * weak Definition 7.3
  * $\infty$-representation
    * of a $L_{\infty}$-algebroid Definition 16.2
    * of a Lie $\infty$-groupoid Definition 17.5
    * of a simplicial set Definition 16.6
  * 2-category Example 7.1
  * action groupoid §17.1
  * acyclic cofibration
    * in a model category item 4
  * acyclic fibration
    * in a model category item 4
    * stalkwise §11.1
  * acyclic fibrations
    * in a category of fibrant objects item 3
  * algebraic $K$ theory sheaves §19.6
  * algebraic $K$-theory
    * of a singular foliation Definition 19.42
  * almost Lie algebroid Proposition 21.5
  * Atiyah class §18.4
  * atlas item 3
    * full holonomy §19.3
    * leaf-preserving §19.3
    * path holonomy Definition 19.23
  * Banach manifold §6.3.2
  * Berezinian map §18.5
  * bicomplete §5.1
  * bisubmersion Definition 19.18
  * boundary extension Definition 11.36
  * category of fibrant object Definition 12.1
  * Čech $\infty$-groupoid Definition 20.2
  * Chevalley-Eilenberg algebra Definition 16.1
  * cofibrant §5.1
  * cofibration
    * in a model category Definition 5.1
    * in a semi-model category Definition 12.32
  * coherent
    * $\overline{\partial}$-analytic §21.1
    * $\overline{\partial}_{V}$-coherent analytic sheaf §22.2
  * cohesive module Definition 15.12
    * quasi- Definition 15.12
  * collapsible extension Definition 11.35
  * connection
    * ${\mathbb{Z}}$- Definition 15.11
  * cotangent sheaf item 2
  * curved stacky dga Definition 15.8
  * de Rham theorem Part VII
    * $A_{\infty}$ for foliation Theorem 27.24
    * for foliations Theorem 27.1
  * derived Banach manifold Definition 6.12
  * derived cobordism ring item 2
  * derived intersection item 1
  * derived Lie $\infty$-category Definition 11.8
  * derived Lie $\infty$-groupoid Definition 11.6
  * derived manifold §8.2
    * affine §8.2
    * simplicial §11
  * descent category §11
    * homotopy Definition 11.10
  * dg manifold Definition 9.4
  * dg-$A$-module Definition 6.2
  * dg-${\mathcal{C}}^{\infty}$-ring Definition 6.9
  * dg-Lie algebroid Definition 13.1
  * differentiable stack §1
  * differential graded algebra Definition 6.1
  * differential graded category Definition 6.3
  * differential graded manifold Example 8.10
  * differential ideal item 4
  * differentialble cohomology Definition 17.7
  * Dwyer-Kan equivalence Definition 7.28
  * ${\mathcal{D}}_{X}$-module §26.1
  * Ehresmann connection
    * for $L_{\infty}$-algebroid fibration Definition 24.18
    * for singular foliations Definition 24.29
  * elliptic involutive structure Definition 22.2
  * elliptic regular foliation §21.4, §22.1
  * enough points 2nd item
  * equivalence
    * of bisumsersion morphisms Definition 20.5
  * fiber
    * of a singular foliation Definition 19.9
  * fibrant §5.1
  * fibration
    * BLX derived manifolds 2nd item
    * in a category of fibrant objects Definition 12.1
    * in a model category Definition 5.1
    * in a semi-model category Definition 12.32
    * Reedy item 1
  * foliated manifold Definition 19.1
  * foliated submersion Definition 19.36
  * foliated transverse Definition 19.34
  * foliation Definition 19.1
    * higher Definition 25.10
  * foliation atlas item 1
  * free category §7.6
  * functor
    * $(\infty,1)$- Definition 11.3
  * global equivalence item 2
  * global fibration item 1
  * global model structure
    * for simplicial presheaves §10.1.1
  * graded manifold Example 8.10, Definition 9.2
  * Grothendieck pretopology Definition 10.1
  * Grothendieck topology §10.1.1
    * ($\infty$,1)- §10.1.2
  * Haefliger cocycle item 2
  * hammock localization Definition 7.27
  * Hausdorff Morita equivalent Definition 19.30
  * higher differential geometry §1
  * holonomy
    * $\infty$-holonomy Definition 28.4
    * $n$-th holonomy of singular foliations Definition 24.35
    * transformation Definition 19.16
  * holonomy embedding Definition 19.28
  * holonomy group §19.3
  * holonomy groupoid §19.3, Definition 19.13
    * of singular foliations Definition 19.24
  * holonomy homomorphism §19.3
  * homotopy category §15.2
  * homotopy coherent nerve Remark 8.2
  * homotopy equivalence
    * $C^{*}$-algebras Example 12.5
  * homotopy $K$-matching object Definition 11.4
  * homotopy mapping space Definition 7.32
  * homotopy pullback
    * category of fibrant objects Theorem 12.9
  * homotopy sheaves Definition 10.4
  * hypercomplete $\infty$-sheaves §10.1.1
  * hypercovering Definition 10.5
  * hypersheaves §10.1.1
  * immersion item 1
  * infinitesimal ideal system Definition 18.22
  * isomorphism of bisumsersion morphism Definition 20.5
  * isotropy Lie algebra §19.2
  * iterated integrals Definition 27.12
    * for vector bundles §28.1
  * jointly conservative 2nd item
  * K-theory machine item 5
  * Kan complex Example 5.3, Definition 7.12
  * Kan fibration §11
  * $L_{\infty}$-algebroid Definition 13.2
    * differential geometric Definition 9.1
    * sub Definition 13.6
  * $L_{\infty}$-algebroids
    * morphism Definition 13.7
  * $L_{\infty}$-morphism Definition 13.9
  * $L_{\infty}$-pair Definition 18.7
  * leaf item 1
  * Lie groupoid §1
  * Lie integration functor §3
  * linear holonomy group Definition 19.12
  * linear holonomy homomorphism Definition 19.12
  * local cofibration item 2
  * local equivalence item 1
    * of bisumsersion morphisms Definition 20.5
  * local isomorphism
    * of bisumsersion morphisms Definition 20.5
  * local model structure §10.1.1
  * local morphism
    * bisubmersion Definition 19.19
  * local stalkwise cover Definition 11.23
  * local surjection Definition 11.14
  * Locally ${\mathcal{C}}^{\infty}$-ringed spaces Definition 8.7
  * locally cocartesian fibration Definition 8.4
  * locally cocartesian lift Definition 8.4
  * locally presentable Definition 8.1
  * locally stalkwise pretoplogy Definition 11.24
  * manifold of short geodesic paths Proposition 19.38
  * model category §5.1
  * model structure Definition 5.1
    * for chain complexes Example 5.5
    * Quillen Example 5.2
    * Reedy §5.2
  * monodromy §19.3
  * monodromy groupoid Definition 19.13, §23.1
  * monodromy morphism §24.2
  * morphism
    * bisubmersion Definition 19.19
  * $n$-category Definition 7.2
  * normalization Definition 14.1
  * path object item 6
  * perfect complex Definition 13.16
  * perfect singular foliation
    * weakly §21.4
  * plaques item 1
  * Poincare lemma
    * for foliations Lemma 27.2
  * point 1st item
  * Pontryagin algebra Definition 18.6
  * Pontryagin character Definition 18.6
  * projective model structure §6.1
  * Q mainfold §9.2
  * quasi-category Definition 7.16
  * relative curvature §15.2
  * retract item 3
  * Riemann-Hilbert correspondence Part VII
  * scalar Atiyah class §18.5
  * Schochet fibration Example 12.5
  * semi-locally simply connected Definition 23.1
  * semi-model category Definition 12.32
  * sieve Definition 10.11
  * simplicial category §5.2, Definition 7.19
  * simplicial localization Definition 7.25
  * simplicial model category Definition 7.22
  * simplicial structure §5.2
  * singular foliaiton Definition 19.8
  * singular foliation
    * elliptic Definition 21.11
    * perfect Definition 21.9
  * singular sub-foliaiton §19.2
  * site §10.1.1
    * $(\infty,1)$- Definition 10.8
  * slice Definition 19.14
  * smooth morphism item 3
  * smooth stack §1
  * stable $\infty$-category Example 1.4
  * stack
    * $\infty$- §10.1.1
  * stacky cdga Definition 15.2
  * stacky dga Definition 15.2, §6.1
  * stalk §11.1
  * stalkwise surjection Definition 11.15
  * stalkwise weak equivalence Definition 11.17
  * standard simplicial resolution Definition 7.24
  * Stoke’s theorem Lemma 28.3
  * structured spaces §8.1
  * submersion item 1, §6.3.2
  * sufficiently close
    * $s$ Definition 20.11
    * $t$ Definition 20.11
  * synthetic differential geometry §8.2
  * tangent space
    * of the leaf Definition 19.9
  * Todd class §18.5
  * topological $\infty$-groupoid Definition 23.7
  * transeversal §19.1
  * transeversal section Definition 19.11
  * transversal Definition 19.11
  * transversal basis Definition 20.1
  * twisted perfect complex §21.4
  * twisting cochain Definition 13.8
  * universal $L_{\infty}$-algebroid Theorem 21.2
  * $V$-analytic sheaf Definition 22.7
  * weak equaivalence
    * in a semi-model category Definition 12.32
  * weak equivalence
    * BLX derived manifolds 1st item
    * in a category of fibrant objects Definition 12.1
    * in a model category Definition 5.1
    * Reedy item 2
  * Yoneda embedding
    * $\infty,1)$- Definition 10.10

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# The alignment of the C3 Accelerator Structures with the Rasnik alignment
system

Harry van der Graaf11footnotetext: retired; corresponding author , Niels van
Bakel  , Bram Bouwens  , Martin Breidenbach  , Andrew Haase  , Joris van
Heijningen  , Anoop Nagesh Koushik  , Emilio Nanni  , Tristan du Pree  , Nick
van Remortel  , Caterina Vernieri

###### Abstract

The Rasnik 3-point alignment system, widely applied in particle physics
experiments and in the instrumentation of gravitational wave experiments, can
be used as N-point alignment system by ’leap frog’ N individual 3-point
systems. The conceptual implementation of Rasnik chains in C3 is presented.
Then, the proper operation of a laser diode and a CMOS image sensor in liquid
nitrogen has been verified. Finally, next plans for testing a small but
complete system, immersed in liquid nitrogen, are presented.

##  1 The Rasnik 3-point alignment system

The first Red Alignment System Nikhef (Rasnik) was developed in 1983 for the
alignment of the muon chambers in the Muon Spectrometer of the L3 experiment
at CERN [1]. As sensor, 4-quadrant photodiodes were used. In 1993, CMOS image
sensors became available, and Rasnik evolved into a system in which a back-
illuminated coded mask is projected, by means of a positive singlet lens, onto
the image sensor. Since 2005, some 8000 of these Rasnik system are flawlessly
operational in the ATLAS experiment at CERN. With these Rasnik systems,
composed from low-cost, commercially available components, the 2D alignment of
three points can be measured with a spatial resolution of 1 nm. The best
resolution power of 7 pm/$\surd$Hz was obtained with an image frame rate of
250 Hz [2].

Figure 1: Principle of the RasDif system. The monochromatic waves, arriving at
the zone plate, cause a diffraction pattern on the image sensor.

For the alignment of active beam elements of future CLIC and ILC linear
colliders, systems with a span between the image sensor and the mask of a
Rasnik system larger than 20 m are required. The diameter and focal length of
the required lens become impractically large, and its replacement by a zone
plate was considered. In the so-called RasDif system, depicted in figure 1,
the back-illuminated coded mask is replaced by a monochromatic point-like
light source generating spherical waves onto the zone plate. This results in a
typical diffraction pattern onto the image sensor: the position of this
pattern on the sensor is a measure for the 3-point alignment of light source,
zone plate and image sensor, respectively.

##  2 Rasnik in Liquid Nitrogen

So far, Rasnik systems have been operated in ambient air and in vacuum. In C3,
Rasnik must operate in ambient air, in vacuum and in boiling liquid nitrogen
(LN2), at 79 K [3] [4][5]. Given the index of refraction of LN2 of 1.20,
lenses can not be applied because a sharp image in LN2 will be out of focus in
vacuum. This is the main reason why the RasDif system is applied.

Laser diodes were tested to operate in LN2: they work, albeit that their
supply voltage of 2 V at 293 K usually needs to be raised to 6 V when immersed
in LN2. With a current of 10 mA, the heat dissipation causes local boiling and
therefore the formation of bubbles. The glass cover seal provides local
thermal shielding, so bubble formation in the critical light exit zone is
locally reduced. The following three laser diodes were found to be LN2 proof:
Laser Components ADL65074TR, Laser Components ADL65055TL, and ROHM RDL65MZT7.
The long term stability of the sealed package in vacuum, in ambient air and in
LN2 remains to be verified.

Modern CMOS image sensors are known not to operate below 90 K. In 2014, two
groups tested CMOS image sensors in populair low-cost webcams by simply
immersing them in LN2, while reading them out via their USB connection [6, 7].
Eventually, two webcams were found to operate unaffected in LN2: the Floureon
"car reversing camera", and the Microsoft HD-3000 Model 1456 (or earlier). The
latter can still be acquired, and proper operation in LN2 has been recently
confirmed, including a cold start, avoiding the self heating of the circuitry
due to dissipation. For the coming R&D projects, some 8 of these image sensors
will be applied in experimental set-ups.

It is well known, in CMOS design, which circuitry could be applied for a given
constraint in temperature [8, 9, 10, 11]. There is commercial interest in the
development of a state-of-the-art CMOS image sensor for monitoring the inside
of LH2 tanks.

##  3 Rasnik in C3

Figure 2: Principle of the leapfrog multipoint alignment system. Each chain
plate is equipped with a laser diode, a zone plate and an image sensor.

In figure 2, the assembly of chain plates is shown: the position, in X and Y,
of any plate with respect to any other plate is known. The alignment of
Accelerator Structures and Quads of C3 can be realised by linking them with
chain plates.

Figure 3: The Stick. Each Stick includes the three basic Rasnik components. At
one side, a Stick is fixed onto the object-to-align. It will be a challenge to
reduce the associated ’contact error’ to below 0.1 µm.

In C3, the chain plate takes the form of a stick, shown in figure 3. In a
chain, all Sticks are practically identical: individual differences are known
after a calibration procedure. On a Stick a CMOS image sensor chip is fixed, a
pattern forming a zone lens is milled out, and laser diode is mounted. A Stick
is mounted, using its mechanical interface, onto an Accelerator Structure or a
Quad, the latter being an assembly of a static quadrupole magnet and a Beam
Position Monitor.

Figure 4: Top view of the possible placement of Sticks onto the Accelerator
Structures. Note that a rotation of a Structure around its Y axis
(perpendicular to the plane of view) is not recorded by Rasnik.

A first approach towards the application of Rasnik in the alignment of the 1 m
long copper Accelerator Structures is shown in figure 4. Although there is
some redundancy, the Rasnik data does not cover all 6 degrees of freedom of
the Structures. Since two adjacent Structures are confined by means of a raft
structure, this may be acceptable. In figure 5 the number of Sticks is
doubled, providing the redundant measurement of all 6 degrees of freedom of
all Structures.

Figure 5: Top view of the possible placement of Sticks onto the Accelerator
Structures, providing complete and redundant data. The alignment of the Quads
will be integrated in this chain.

##  4 The effective precision of a chain of Rasnik systems

The error of a 3-point Rasnik system is defined as the variation in the
sagitta, defined as the distance, in X and Y, of the optical centre of the
zone plate to the optical axis, being the line through the optical centres of
the laser source and the image sensor. Assuming that the medium is
homogeneous, light propagates in a straight line. For simple systems using low
cost laser diodes and image sensors extracted from popular webcams, the
(Gaussian) noise in the sagitta measurement due to quantum fluctuations in
light falling onto pixels, can be as good as 100 nm. With modern low-cost
image sensors, the frame rate of 100 Hz results in a resolution power of 10
nm/$\sqrt{Hz}$ which is the equivalent of 100 nm resolution of one measurement
per second. With a custom designed image sensor optimised for application in
C3, a resolution power of 5 nm/$\surd$Hz should be within reach [1, 2, 12].

The 2D position of a Quad or Accelerator Structure is transferred to Rasnik by
means of Sticks. The mechanical interface between the Stick and the object
onto which it is mounted is associated with a set up error of order 1 µm if
state-of-the-art suspension mechanics are applied. In addition, the positions
of the optical centres of laser source, zone plate and image sensor with
respect to the mechanical interface, which are measured in a calibration
procedure (see Calibration of the Sticks) are subject to a similar offset
error of 1 µm.

Combining the two errors together, each Rasnik data point from a particular
system is subject to an offset error of order 1 µm, constant in time, and a
random Gaussian error much smaller than 0.1 µm.

Figure 6: The error propagation in a chain of N = 16 Rasnik systems. The
maximum error reaches D times Rasnik’s intrinsic error.

Figure 6 shows the errors associated with a chain of $N$ = 16 Rasnik systems.
In this calculation the positions of chainplate 0, 1 and 15 are fixed, and the
positions of the in-between chainplates are calculated from the MonteCarlo
simulated Rasnik data, assuming a spatial resolution of 1.0 (RMS, unitless)
for each system. The largest error occurs, as expected, at the central plate
since it is far away from the fixed chainplates. The largest spatial error
equals $\sqrt{N}$ times the sagitta error of a single Rasnik system, where D =
$\sqrt{N}$.

The Quarter Cryogenic Module (QCM) will be equipped with 4-fold Rasnik chains
placed at both sides. Given the offset error of an individual Rasnik system of
1 µm, and a degradation factor $\sqrt{N}$ = 2, the relative positions of the
two Accelerator Structures and the Quad will be known within 1.5 µm after the
eight calibrated Sticks are placed.

In C3, a SuperSector includes 728 Accelerator Structures. If these are
equipped with two continuous Rasnik chains, the largest uncertainty in the
middle of each chain is $\sqrt{728}$ / $\sqrt{2}$ = 19 µm. By adding Rasnik
systems on Sticks 1, 364 and 728, this largest error can be reduced by a
factor $\sqrt{2}$. As long as the light path area, now confined by the raft’s
main beams, is respected, there is room for improvement.

##  5 Conclusions and future plans

Figure 7: The InPlaneDemo setup: it includes two mutually coupled parallel
Rasnik systems, and two ’diagonal’ systems. With this 2 to 4 fold redundant
system, the quality of Rasnik systems can be verified in ambient air, vacuum,
and fully immersed in LN2 [1]. This setup fits in a foam cryostat with inner
fiducial dimension of 60 x 10 x 10 cm3 Figure 8: The raft unit. Figure 9:
Layout of the 8 Sticks in QCM. With calibrated Sticks, the relative positions
of the two Accelerator Structures and the Quad are known. Figure 10: Cross
section of the QCM. Since all Sticks are identical, interfaces are required
between a Stick and either an Accelerator Structure (left) or the Quad
(right). These interfaces may be an integral part of the Accelerator Structure
or the Quad.

One CMOS image sensor has been found and identified to operate flawlessly
immersed in LN2 at 77 K. With some 10 of these sensors, demonstrations of
complete Rasnik systems, in air and immersed in LN2, could be performed,
within a year. Most likely, new CMOS image sensors will become available,
capable to operate in a cryogenic environment, thanks to commercial interest.
The proper operation of laser diodes in LN2 has been certified.

By applying Rasnik instead of a system in which the local position of a
stretched wire is determined by multiple Wire Position Sensors (WPS)
[13],[14], severe problems are avoided: no risk of a hard-accessible broken
wire, no error in the vertical coordinate due to uncertainties in wire sag,
and no risk of variations in the wire position due to flowing LN2 medium.

Other benefits of using Rasnik alignment systems are:

  * •

electronics: benefits of using only off-the-shelf USB Muxs, CPUs or GPUs;

  * •

precision is limited by (mechanical) offset; intrinsic 2D spatial resolution
of 10 nm;

  * •

mechanical noise, due to bubble formation and flow of LN2 can be well studied
given the excellent spatial resolution power;

  * •

no scale calibration required;

  * •

low cost: € 30 per system, excluding readout;

  * •

no drift: 1/f noise is not measurable;

  * •

radiation-proof components can be selected.

Figure 7 shows the InPlaneDemo setup. It includes two parallel and two
diagonal Rasnik systems. With this, performance-limiting effects of Rasnik in
LN2 can be studied in detail:

  * •

the formation of N2 gas bubbles on the surface of the heat-dissipating laser
diodes and CMOS image sensors;

  * •

local variations in the index of refraction of LN2 due to the laminar or
turbulent flow, and due to convection near the warmer laser diodes and CMOS
image sensors;

  * •

mechanical noise due to bubble formation and bubble transport in LN2. Here the
10 nm spatial resolution and a high frame rate of Rasnik will be useful;

  * •

the long-term quality degradation of laser diodes and CMOS image sensors.

Rasnik for the Quarter Cryogenic Module (QCM). Figure 8 shows the basic raft
unit: an assembly of two sequential Accelerator Structures, one Quad module,
waveguides and a frame keeping these items in their proper position. Placed in
a cryostat filled with LN2, the functionality of QCM can be fully tested [15].
For this, the relative positions of the Quad and the Accelerator Structures
can be monitored with Rasnik, as is shown in figures 9 and 10. The optical
paths are confined in the raft’s construction bars. Since only the laser
diodes and image sensors dissipate their power there is little bubble
formation inside these bars, and the LN2 flow in the bar is expected to be
directed in Z, and laminar. Variations in the index of refraction in the X and
Y directions should be small.

Figure 11: The Calibration Station. Three special Sticks have their mechanical
references at both ends, such that the distance between the two references is
precisely identical for all three Sticks. After rotating all three Sticks
around the Z axis by 180 deg, the image shift on the sensor on the right-hand
Stick equals 4x the sagitta, being the alignment error in X and Y of the
Station.

Calibration of the Sticks. With calibrated Sticks, the 6D relative positions
of the two Accelerator Structures and the Quad in the QCM are known as soon as
the eight Sticks are mounted in position. With the Calibration Station, shown
in figure 11, placed in a normal lab environment, the 3x X and 3x Y
coordinates of the source, zone plate and sensor, respectively, with respect
to the mechanical reference, can be obtained for each individual Stick [16].
After measuring the alignment error of the Station, the calibration of any
Stick is obtained by placing sets of three Sticks on the station, such that
each Stick will pass at least one time the left-, middle- and right position
on the Station.

## Acknowledgments

We thank Nikhef for facilitating tests of crucial components in their dewar,
and we are grateful to Amolf to provide us with LN2. We thank Oscar van Petten
for producing mechanical supports, Berend Munneke for his practical
assistance, and Nico Rem for keeping us working safe.

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