$(p-1)$-dimensional data of $v^{p}$ coincides with that already given by
$v^{0},\ldots,v^{p-1}$; the degeneracy conditions will tell us that it
suffices to consider non-degenerate intersections
$U_{\alpha_{0}\ldots\alpha_{p}}$, since if $\alpha_{i}=\alpha_{i+1}$ for some
$0\leqslant i<p$ then the corresponding edge (containing a $0$-simplex and two
$1$-simplices) will be trivially labelled. As a notational shorthand, given a
simplex $I=\\{i_{0}<\ldots<i_{k}\\}$ of indices, we write
$\alpha_{I}\coloneqq\alpha_{i_{0}}\ldots\alpha_{i_{k}}$. Since we are
evaluating on the Čech nerve, rather than labelling the simplices of
$\Delta[p]_{\mathrm{pair}}$ by subsets of $[p]$, it makes sense to label them
with subsets of $\\{\alpha_{0}<\ldots<\alpha_{p}\\}$ for each fixed
$U_{\alpha_{0}\ldots\alpha_{p}}\in\check{{\mathcal{N}}}{\mathcal{U}}_{p}$.
Following Definition 3.3.4, each $0$-simplex
$\alpha_{i_{0}}\ldots\alpha_{i_{k}}\leqslant\alpha_{0}\ldots\alpha_{p}$ is
labelled with a $k$-simplex of
$\llbracket{\mathcal{N}}^{\mathrm{dg}}\operatorname{\mathsf{Free}}(U_{\alpha_{I}})\rrbracket$,
i.e. by complexes

$C_{i_{0}}(\alpha_{I}),\ldots,C_{i_{k}}(\alpha_{I})$

along with, for all non-empty subsets
$J=\\{{j_{0}}<\ldots<{j_{\ell}}\\}\subseteq\\{{i_{0}}<\ldots<{i_{k}}\\}$,
morphisms

$\varphi_{J}(\alpha_{I})\in\operatorname{Hom}_{\operatorname{\mathsf{Free}}(U_{\alpha_{I}})}^{1-\ell}\big{(}C_{j_{\ell}}(\alpha_{I}),C_{j_{0}}(\alpha_{I})\big{)}$

such that, for all $J$ with $|J|\geqslant 3$,

$\partial\varphi_{J}(\alpha_{I})=\sum_{m=1}^{\ell-1}(-1)^{m-1}\varphi_{J\setminus\\{j_{m}\\}}(\alpha_{I})+\sum_{m=1}^{\ell-1}(-1)^{\ell(m-1)+1}\varphi_{j_{0}<\ldots<j_{m}}(\alpha_{I})\circ\varphi_{j_{m}<\ldots<j_{k}}(\alpha_{I}).$

Next, setting $I=\\{i_{0}<\ldots<i_{k}\\}$ and
$J=\\{j_{0}<\ldots<j_{\ell}\\}\subset I$, each $(k-\ell)$-cell
$\alpha_{J}<\alpha_{I}$ in $\Delta[p]_{\mathrm{pair}}$ is labelled with an
$(\ell+1)$-tuple of objects

$\Big{(}C_{j_{m}}^{\perp\alpha_{I}}(\alpha_{J})\in\operatorname{\mathsf{Free}}(U_{\alpha_{I}})\Big{)}_{0\leqslant
m\leqslant\ell}$

where each $C_{j_{m}}^{\perp\alpha_{I}}(\alpha_{J})$ is elementary, and such
that we obtain a direct-sum decomposition

$C_{J_{m}}(\alpha_{I})\cong C_{J_{m}}(\alpha_{J})\oplus
C_{J_{m}}^{\perp\alpha_{I}}(\alpha_{J})$

for all $0\leqslant m\leqslant\ell$.

Now we introduce the change of notation that will recover the definition of
simplicial twisting cochain from [TT86, §3], which we have already seen in
Section 5.4. Given $U_{\alpha_{0}\ldots\alpha_{p}}$, some specific
$\alpha\coloneqq\alpha_{i}$, and some subset $I\subseteq[p]$ containing $i$,
let $\sigma=\alpha_{I}$; for $J\subseteq I$ also containing $i$, let
$\tau=\alpha_{J}$, so that

$\\{\alpha_{i}\\}=\\{\alpha\\}\subseteq\tau\subseteq\sigma\subseteq\alpha_{[p]}=\alpha_{0}\ldots\alpha_{p}.$

We then define

$\displaystyle E_{\sigma,\alpha}^{\bullet}$ $\displaystyle\coloneqq
C_{i}(\sigma)$ $\displaystyle E_{\sigma,\tau,\alpha}^{\bullet}$
$\displaystyle\coloneqq C_{i}^{\perp\sigma}(\tau)$
$\displaystyle{}^{\sigma}\mathfrak{a}_{\alpha_{J}}^{\ell,1-\ell}$
$\displaystyle\coloneqq\varphi_{J}(\sigma)$
$\displaystyle{}^{\sigma}\mathfrak{a}$
$\displaystyle\coloneqq\sum_{i=0}^{k-1}{}^{\sigma}\mathfrak{a}^{i,1-i}$

where $k=|\sigma|-1$ and $\ell=|\tau|-1$. It remains to show that conditions
(STC 1) to (STC 4) in [TT86, §3] are satisfied. As mentioned in Section 5.4,
conditions (STC 3) and (STC 4) are satisfied by construction, following the
definition of a GTT-labelling, and condition (STC 1) is not really necessary,
if we consider simplicial twisting cochains that resolve _complexes_ of
coherent sheaves. What remains to show is that (STC 2) is satisfied, but this
is exactly the content of 4.1.1 (b), i.e. that the $\varphi_{J}(\sigma)$
(which constitute the ${}^{\sigma}\mathfrak{a}$) satisfy the Maurer–Cartan
equation.

####  B.2 Proof of Theorem 4.2.2 —
$\operatorname{Tot}^{1}{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{\bullet})$
recovers weak equivalences

We start by unravelling the definition of a weak equivalence of twisting
cochains ([Wei16, Definition 2.27]) and spelling out the explicit conditions
in the first three degrees; we then do the same for the definition of a
$1$-simplex in the totalisation, and show that the conditions agree with those
of a weak equivalence. Finally, we give a general combinatorial argument that
applies in arbitrary degree.

Let $(E^{\bullet},\varphi)$ be a twisting cochain: the data of
$E_{\alpha}^{\bullet}\in\operatorname{\mathsf{Free}}(U_{\alpha})$ for all
$U_{\alpha}\in{\mathcal{U}}$, and
$\varphi=\sum_{p\in\mathbb{N}}\varphi^{p,1-p}$ satisfying the Maurer–Cartan
equation, where

$\varphi_{\alpha_{0}\ldots\alpha_{p}}^{p,1-p}\colon
E_{\alpha_{p}}^{\bullet}\to E_{\alpha_{0}}^{\bullet}[p-1]$

and with $\varphi_{\alpha}^{0,1}$ being exactly the differential of
$E_{\alpha}$. If $(F^{\bullet},\psi)$ is another twisting cochain, then a
degree-$0$ morphism $\Lambda\colon(F^{\bullet},\psi)\to(E^{\bullet},\varphi)$
is, following [Wei16, Definition 2.12], the data of maps (that do not
necessarily commute with the differentials)

$\Lambda_{\alpha_{0}\ldots\alpha_{p}}^{p,-p}\colon F_{\alpha_{p}}^{\bullet}\to
E_{\alpha_{0}}^{\bullet}[p].$

This morphism is a _weak equivalence_ ([Wei16, Definition 2.27]) if the
$\Lambda_{\alpha}^{0,0}$ are quasi-isomorphisms, and if the $\Lambda^{p,-p}$
satisfy

$\hat{\delta}\Lambda+\varphi\cdot\Lambda-\Lambda\cdot\psi=0.$

(Morally, this is asking that
$\operatorname{D}\Lambda\coloneqq(\hat{\delta}+\partial)\Lambda=0$, where we
think of $\partial$ as an analogue to the differential in the category of
chain complexes, given by the difference between pre- and post-composition
with the two “differentials”, which are now the twisting cochains $\varphi$
and $\psi$). All the terms in this equation are of total degree $1$, and so we
can consider what happens in each different bidegree.282828We omit explicitly
writing the degrees of the $\varphi$, $\psi$, and $\Lambda$ terms from now on,
since they can be deduced from the degree of the simplex
$\alpha_{0}\ldots\alpha_{p}$ in the subscript, knowing that
$\deg\varphi=\deg\psi=1$ and $\deg\Lambda=0$. As is often the case, ensuring
that the signs are correct is the majority of the work, and there are two
things to be aware of: the composition of a $(p,q)$-term with an $(r,s)$-term
has a sign of $(-1)^{qr}$ ([Wei16, §2.2, Equation 3]); and the differential of
a morphism $f\colon A\to B$ of degree $n$ in a dg-category of chain complexes
is given by $\partial f=f\circ\mathrm{d}_{A}+(-1)^{n+1}\mathrm{d}_{B}\circ f$.

#####  B.2.1 The first three terms

To simplify notation, we will write $E_{\alpha_{0}\ldots\alpha_{p}}$ (resp.
$F_{\alpha_{0}\ldots\alpha_{p}}$) instead of
$\varphi_{\alpha_{0}\ldots\alpha_{p}}$ (resp.
$\psi_{\alpha_{0}\ldots\alpha_{p}}$). For the sake of clarity, we usually
denote the complex by $E_{\alpha_{i}}^{\bullet}$ so as not to confuse it with
its differential $E_{\alpha_{i}}$.

  * •

The $(0,1)$-terms tell us that

$E_{\alpha}\circ\Lambda_{\alpha}-\Lambda_{\alpha}\circ F_{\alpha}=0$

which simply says that $\Lambda_{\alpha}$ is a chain map from
$F_{\alpha}^{\bullet}$ to $E_{\alpha}^{\bullet}$.

  * •

The $(1,0)$-terms tell us that

$\displaystyle 0=$ $\displaystyle
E_{\alpha\beta}\Lambda_{\beta}-\Lambda_{\alpha}F_{\alpha\beta}$
$\displaystyle-$ $\displaystyle
E_{\alpha}\Lambda_{\alpha\beta}-\Lambda_{\alpha\beta}F_{\beta}$

which we can rewrite as

$\partial\Lambda_{\alpha\beta}=E_{\alpha\beta}\Lambda_{\beta}-\Lambda_{\alpha}F_{\alpha\beta}.$

  * •

The $(2,-1)$ terms tell us that

$\displaystyle 0=$ $\displaystyle
E_{\alpha\beta\gamma}\Lambda_{\gamma}-\Lambda_{\alpha}F_{\alpha\beta\gamma}$
$\displaystyle+$ $\displaystyle
E_{\alpha\beta}\Lambda_{\beta\gamma}+\Lambda_{\alpha\beta}F_{\beta\gamma}$
$\displaystyle+$ $\displaystyle
E_{\alpha}\Lambda_{\alpha\beta\gamma}-\Lambda_{\alpha\beta\gamma}F_{\gamma}$
$\displaystyle+$ $\displaystyle\Lambda_{\alpha\gamma}$

which we can rewrite as

$\displaystyle\partial\Lambda_{\alpha\beta\gamma}=$ $\displaystyle
E_{\alpha\beta\gamma}\Lambda_{\gamma}-\Lambda_{\alpha}F_{\alpha\beta\gamma}$
$\displaystyle+$ $\displaystyle
E_{\alpha\beta}\Lambda_{\beta\gamma}+\Lambda_{\alpha\beta}F_{\beta\gamma}$
$\displaystyle+$ $\displaystyle\Lambda_{\alpha\gamma}.$

Now we look at the first three terms of a $1$-simplex in
$\operatorname{Tot}{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{\bullet})$.
To do this, we need to understand the simplicial structure of
$\Delta[p]\times\Delta[1]$ for $p\in\mathbb{N}$. Fortunately there is a simple
combinatorial description of all sub-simplices, given by considering certain
paths in a two-dimensional grid: a non-degenerate $q$-simplex in
$\Delta[p]\times\Delta[1]$ is given by a strictly increasing sequence of
$(q+1)$ pairs $(i,j)$, for $0\leqslant i\leqslant p$ and $0\leqslant
j\leqslant 1$, where “strictly increasing” means that at least one of $i$ and
$j$ increases with each successive element. We write such a sequence as
$\left[\begin{smallmatrix}i_{0}&i_{1}&\ldots&i_{q}\\\
j_{0}&j_{1}&\ldots&j_{q}\end{smallmatrix}\right]$. For example, in
$\Delta[1]\times\Delta[1]$,

  * •

the $0$-simplices are $\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$, $\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$, $\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$, and $\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$;

  * •

the $1$-simplices are $\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]$, $\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]$, $\left[\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right]$, $\left[\begin{smallmatrix}0&1\\\
1&1\end{smallmatrix}\right]$, and $\left[\begin{smallmatrix}0&1\\\
0&1\end{smallmatrix}\right]$;

  * •

the $2$-simplices are $\left[\begin{smallmatrix}0&1&1\\\
0&0&1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}0&0&1\\\
0&1&1\end{smallmatrix}\right]$.

Thinking of $\left[\begin{smallmatrix}i\\\ j\end{smallmatrix}\right]$ as a
coordinate, $\left[\begin{smallmatrix}i_{0}&i_{1}\\\
j_{0}&j_{1}\end{smallmatrix}\right]$ as the path from
$\left[\begin{smallmatrix}i_{0}\\\ j_{0}\end{smallmatrix}\right]$ to
$\left[\begin{smallmatrix}i_{1}\\\ j_{1}\end{smallmatrix}\right]$, and
$\left[\begin{smallmatrix}i_{0}&i_{1}&i_{2}\\\
j_{0}&j_{1}&j_{2}\end{smallmatrix}\right]$ as the triangle given by the
vertices $\left[\begin{smallmatrix}i_{0}\\\ j_{0}\end{smallmatrix}\right]$,
$\left[\begin{smallmatrix}i_{1}\\\ j_{1}\end{smallmatrix}\right]$, and
$\left[\begin{smallmatrix}i_{2}\\\ j_{2}\end{smallmatrix}\right]$, we can draw
$\Delta[1]\times\Delta[1]$ as a square with diagonal, as in Figure B.2.i.

$\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
1&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&0&1\\\
0&1&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1&1\\\
0&0&1\end{smallmatrix}\right]$ Figure B.2.i: The canonical simplicial
structure of $\Delta[1]\times\Delta[1]$.

The story is entirely analogous in higher dimensions: see Figure B.2.ii for
the case of $\Delta[2]\times\Delta[1]$, and [GMTZ22, Appendix B.2] for the
general description of $\Delta[p]\times\Delta[q]$ (though here we are only
concerned with $\Delta[p]\times\Delta[1]$).

$\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}2\\\
0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}2\\\
1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}2&2\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1&2\\\
0&0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&2\\\
0&0\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
1&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1&2\\\
1&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&2\\\
1&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&1\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}1&2\\\
0&1\end{smallmatrix}\right]$$\left[\begin{smallmatrix}0&2\\\
0&1\end{smallmatrix}\right]$

$E_{\alpha}^{\bullet}$$E_{\beta}^{\bullet}$$E_{\gamma}^{\bullet}$$F_{\alpha}^{\bullet}$$F_{\beta}^{\bullet}$$F_{\gamma}^{\bullet}$$\lambda_{\alpha}$$\lambda_{\beta}$$\lambda_{\gamma}$$E_{\alpha\beta}$$E_{\beta\gamma}$$E_{\alpha\gamma}$$F_{\alpha\beta}$$F_{\beta\gamma}$$F_{\alpha\gamma}$$d_{\alpha\beta}$$d_{\beta\gamma}$$d_{\alpha\gamma}$

Figure B.2.ii: _Left:_ The canonical simplicial structure of
$\Delta[2]\times\Delta[1]$. _Right:_ The data of a morphism
$\Delta[2]\times\Delta[1]\to{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{2})$.
Note that the $2$\- and $3$-simplices are not labelled in either figure, for
the sake of legibility, but the three constituent $3$-simplices are drawn in
Figure B.2.iii.

  * •

Since $\Delta[0]\times\Delta[1]\cong\Delta[1]$, the degree-$0$ component of a
$1$-simplex in
$\operatorname{Tot}{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{\bullet})$
is given by

$F_{\alpha}^{\bullet}$$E_{\alpha}^{\bullet}$$\lambda_{\alpha}$

which, by the definition of the dg-nerve, is exactly the data of a quasi-
isomorphism292929Recall that morphisms in the dg-nerve go in the “backwards”
direction, i.e. from $x_{i_{k}}$ to $x_{i_{0}}$ (Definition 2.5.2).
$\lambda_{\alpha}\colon
F_{\alpha}^{\bullet}\xrightarrow{\raisebox{-3.01385pt}[0.0pt][0.0pt]{$\sim$}}E_{\alpha}^{\bullet}$.
This means that $\lambda_{\alpha}$ satisfies

$E_{\alpha}\circ\lambda_{\alpha}-\lambda_{\alpha}\circ F_{\alpha}=0$

since quasi-isomorphisms are, in particular, chain maps.

  * •

As described above (and in Figure B.2.i), the product
$\Delta[1]\times\Delta[1]$ is a square with diagonal. This means that the
degree-$1$ component is of the form

$E_{\alpha}^{\bullet}$$E_{\beta}^{\bullet}$$F_{\alpha}^{\bullet}$$F_{\beta}^{\bullet}$$\lambda_{\alpha}$$\lambda_{\beta}$$F_{\alpha\beta}$$E_{\alpha\beta}$$d_{\alpha\beta}$$h_{\alpha\beta}^{F}$$h_{\alpha\beta}^{E}$

where $\lambda_{\alpha}$ and $\lambda_{\beta}$ are the degree-$0$ components
from above, $E_{\alpha\beta}$ and $F_{\alpha\beta}$ are the degree-$(1,0)$
components of the twisting cochains, $d_{\alpha\beta}$ is some quasi-
isomorphism, and $h_{\alpha\beta}^{E}$ and $h_{\alpha\beta}^{F}$ are
homotopies satisfying

$\displaystyle\partial h_{\alpha\beta}^{E}$
$\displaystyle=d_{\alpha\beta}-E_{\alpha\beta}\lambda_{\beta}$
$\displaystyle\partial h_{\alpha\beta}^{F}$
$\displaystyle=d_{\alpha\beta}-\lambda_{\alpha}F_{\alpha\beta}.$

As mentioned in Remark 4.2.4, this is where we need to “forget” that we are
working simplicially and construct something cubical. To do this, we define

$\lambda_{\alpha\beta}\coloneqq h_{\alpha\beta}^{F}-h_{\alpha\beta}^{E}$

giving us the diagram

$E_{\alpha}^{\bullet}$$E_{\beta}^{\bullet}$$F_{\alpha}^{\bullet}$$F_{\beta}^{\bullet}$$\lambda_{\alpha}$$\lambda_{\beta}$$F_{\alpha\beta}$$E_{\alpha\beta}$$\lambda_{\alpha\beta}$

where $\lambda_{\alpha\beta}$ satisfies

$\partial\lambda_{\alpha\beta}=E_{\alpha\beta}\lambda_{\beta}-\lambda_{\alpha}F_{\alpha\beta}$

by linearity of $\partial$.

  * •

The degree-$2$ component is given by a labelling of the canonical
triangulation of $\Delta[2]\times\Delta[1]$, as shown in Figure B.2.ii. There
are three $3$-simplices (i.e. tetrahedra) that make up this triangular prism,
each one labelled with a $3$-simplex in the dg-nerve: the equations that these
three homotopies satisfy are shown in Figure B.2.iii. Applying the linearity
of $\partial$ to the equations in Figure B.2.iii, we see that

$\displaystyle\partial\left(f_{\left[\begin{smallmatrix}0&1&2&2\\\
0&0&0&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&1&1&2\\\
0&0&1&1\end{smallmatrix}\right]}+f_{\left[\begin{smallmatrix}0&0&1&2\\\
0&1&1&1\end{smallmatrix}\right]}\right)=$ $\displaystyle
f_{\left[\begin{smallmatrix}0&2&2\\\
0&0&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&0&2\\\
0&1&1\end{smallmatrix}\right]}$ $\displaystyle+$ $\displaystyle
f_{\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]}\left(f_{\left[\begin{smallmatrix}1&1&2\\\
0&1&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}1&2&2\\\
0&0&1\end{smallmatrix}\right]}\right)$ $\displaystyle+$
$\displaystyle\left(f_{\left[\begin{smallmatrix}0&0&1\\\
0&1&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&1&1\\\
0&0&1\end{smallmatrix}\right]}\right)f_{\left[\begin{smallmatrix}1&2\\\
1&1\end{smallmatrix}\right]}$ $\displaystyle-$ $\displaystyle
f_{\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}0&1&2\\\
1&1&1\end{smallmatrix}\right]}+f_{\left[\begin{smallmatrix}0&1&2\\\
0&0&0\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}2&2\\\
0&1\end{smallmatrix}\right]}.$

This means that, if we define $\lambda_{\alpha\beta\gamma}$ to be the
alternating sum of these three homotopies

$\lambda_{\alpha\beta\gamma}\coloneqq f_{\left[\begin{smallmatrix}0&1&2&2\\\
0&0&0&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&1&1&2\\\
0&0&1&1\end{smallmatrix}\right]}+f_{\left[\begin{smallmatrix}0&0&1&2\\\
0&1&1&1\end{smallmatrix}\right]}.$

then, using Figure B.2.ii, the above translates to

$\displaystyle\partial\lambda_{\alpha\beta\gamma}=$ $\displaystyle
h_{\alpha\gamma}^{F}-h_{\alpha\gamma}^{E}$ $\displaystyle+$ $\displaystyle
E_{\alpha\beta}\left(h_{\beta\gamma}^{F}-h_{\beta\gamma}^{E}\right)$
$\displaystyle+$
$\displaystyle\left(h_{\alpha\beta}^{F}-h_{\alpha\beta}^{E}\right)F_{\beta\gamma}$
$\displaystyle-$
$\displaystyle\lambda_{\alpha}F_{\alpha\beta\gamma}+E_{\alpha\beta\gamma}\lambda_{\gamma}$

which rearranges to give

$\displaystyle\partial\lambda_{\alpha\beta\gamma}=$ $\displaystyle
E_{\alpha\beta\gamma}\lambda_{\gamma}-\lambda_{\alpha}F_{\alpha\beta\gamma}$
$\displaystyle+$ $\displaystyle
E_{\alpha\beta}\lambda_{\beta\gamma}+\lambda_{\alpha\beta}F_{\beta\gamma}$
$\displaystyle+$ $\displaystyle\lambda_{\alpha\gamma}.$

But then equations $(\star_{i})$ and $(\ast_{i})$ are identical for $i=0,1,2$.
In other words, _up to degree $2$_, a $1$-simplex
$(\lambda_{\alpha},\lambda_{\alpha\beta},\lambda_{\alpha\beta\gamma})$ in the
totalisation defines exactly a weak equivalence
$(\Lambda_{\alpha},\Lambda_{\alpha\beta},\Lambda_{\alpha\beta\gamma})$ of
twisting cochains. It remains only to give a general argument for arbitrary
degree.

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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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0&1\end{smallmatrix}\right]$}} }}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}\pgfsys@moveto{8.26079pt}{8.25447pt}\pgfsys@lineto{120.78432pt}{120.796pt}\pgfsys@stroke\pgfsys@invoke{
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{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\scriptsize$\left[\begin{smallmatrix}0&2\\\
0&1\end{smallmatrix}\right]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\qquad\begin{aligned}
\partial f_{\left[\begin{smallmatrix}0&1&1&2\\\
0&0&1&1\end{smallmatrix}\right]}=\phantom{+}&f_{\left[\begin{smallmatrix}0&1&2\\\
0&1&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&1&2\\\
0&0&1\end{smallmatrix}\right]}\\\ -&f_{\left[\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}1&1&2\\\
0&1&1\end{smallmatrix}\right]}+f_{\left[\begin{smallmatrix}0&1&1\\\
0&0&1\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}1&2\\\
1&1\end{smallmatrix}\right]}\end{aligned}$

$\leavevmode\hbox to147.52pt{\vbox
to181.79pt{\pgfpicture\makeatletter\hbox{\hskip 8.77745pt\lower-9.09412pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\left[\begin{smallmatrix}0\\\ 0\end{smallmatrix}\right]$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\scriptsize$\left[\begin{smallmatrix}0&2\\\
1&1\end{smallmatrix}\right]$}} }}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
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}\hbox{{\scriptsize$\left[\begin{smallmatrix}0&1\\\
0&1\end{smallmatrix}\right]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}\pgfsys@moveto{8.26079pt}{8.25447pt}\pgfsys@lineto{120.78432pt}{120.796pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.70705}{0.70715}{-0.70715}{0.70705}{120.78432pt}{120.796pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{56.66422pt}{56.7254pt}{17.5549pt}{17.43823pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{59.99722pt}{61.80841pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$\left[\begin{smallmatrix}0&2\\\
0&1\end{smallmatrix}\right]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\qquad\begin{aligned}
\partial f_{\left[\begin{smallmatrix}0&0&1&2\\\
0&1&1&1\end{smallmatrix}\right]}=\phantom{+}&f_{\left[\begin{smallmatrix}0&1&2\\\
0&1&1\end{smallmatrix}\right]}-f_{\left[\begin{smallmatrix}0&0&2\\\
0&1&1\end{smallmatrix}\right]}\\\ -&f_{\left[\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}0&1&2\\\
1&1&1\end{smallmatrix}\right]}+f_{\left[\begin{smallmatrix}0&0&1\\\
0&1&1\end{smallmatrix}\right]}f_{\left[\begin{smallmatrix}1&2\\\
1&1\end{smallmatrix}\right]}\end{aligned}$

Figure B.2.iii: _Left:_ The three $3$-simplices of $\Delta[2]\times\Delta[1]$.
_Right:_ The equations satisfied by the corresponding homotopy in the dg-
nerve, where $f_{[-]}$ is the morphism labelling the simplex $[-]$.

#####  B.2.2 Full proof

Now we give the argument for arbitrary degree. For $0\leqslant m\leqslant p$,
we define (cf. Figure B.2.iv)

$\Delta_{m}^{p+1}\coloneqq\left[\begin{smallmatrix}0&1&2&\ldots&m&m&m+1&\ldots&p-1&p\\\
0&0&0&\ldots&0&1&1&\ldots&1&1\end{smallmatrix}\right]$

i.e. the non-degenerate $(p+1)$-simplex of $\Delta[p]\times\Delta[1]$ given by
travelling along the bottom $p$-simplex (corresponding to
$\Delta[p]\times\\{0\\}$) for $m$ steps (between the first $m+1$ vertices),
then travelling straight up to the top $p$-simplex (corresponding to
$\Delta[p]\times\\{1\\}$), before continuing on along the remaining vertices.
Note that these are _all_ of the non-degenerate $(p+1)$-simplices of
$\Delta[p]\times\Delta[1]$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta_{0}^{3}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta_{1}^{3}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta_{2}^{3}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta_{3}^{3}$
Figure B.2.iv: The four non-degenerate $4$-simplices of
$\Delta[3]\times\Delta[1]$, represented as paths between vertices: in each
diagram, the bottom row consists of the vertices
$\left[\begin{smallmatrix}0\\\ 0\end{smallmatrix}\right]$ to
$\left[\begin{smallmatrix}3\\\ 0\end{smallmatrix}\right]$ and the top row of
the vertices $\left[\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right]$ to
$\left[\begin{smallmatrix}3\\\ 1\end{smallmatrix}\right]$.

We can think of a morphism
$\Delta[p]\times\Delta[1]\to{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{p})$
as a labelling of the “prism” $\Delta[p]\times\Delta[1]$, where the vertices
are labelled with objects
$x_{i_{j}}\in{\mathscr{Twist}}(\check{{\mathcal{N}}}{\mathcal{U}}_{p})^{0}$,
and each simplex $I=\\{{i_{0}}<\ldots<{i_{k}}\\}$ with $k\geqslant 2$ is
labelled by $f_{I}\in\operatorname{Hom}^{k-1}(x_{i_{k}},x_{i_{0}})$ satisfying
the equation • ‣ Definition 2.5.2 defining the dg-nerve, which here is exactly

$\partial
f_{I}+\sum_{j=1}^{k-1}(-1)^{j}f_{I\setminus\\{i_{j}\\}}+\sum_{j=1}^{k-1}(-1)^{k(j-1)}f_{\\{i_{0}<\ldots<i_{j}\\}}f_{\\{i_{j}<\ldots<i_{k}\\}}=0.$

Generalising the notation of Figure B.2.ii, we write
$E_{\alpha_{i}}^{\bullet}$ for the object labelling the vertex
$\left[\begin{smallmatrix}i\\\ 0\end{smallmatrix}\right]$, and
$F_{\alpha_{i}}^{\bullet}$ for the object labelling the vertex
$\left[\begin{smallmatrix}i\\\ 1\end{smallmatrix}\right]$; given
$I=\\{i_{0}<\ldots<i_{k}\\}\subseteq[p]$ with $k\geqslant 1$, we write
$E_{\alpha_{i_{0}}\ldots\alpha_{i_{k}}}$ for the morphism
$f_{\left[\begin{smallmatrix}I\\\ 0\end{smallmatrix}\right]}$, and
$F_{\alpha_{i_{0}}\ldots\alpha_{i_{k}}}$ for the morphism
$f_{\left[\begin{smallmatrix}I\\\ 1\end{smallmatrix}\right]}$, where
$\left[\begin{smallmatrix}I\\\
j\end{smallmatrix}\right]=\left[\begin{smallmatrix}i_{0}&i_{1}&\ldots&i_{k}\\\
j&j&\ldots&j\end{smallmatrix}\right]$.

Given any simplex $(\alpha_{0}\ldots\alpha_{p})$, we define

$\lambda_{\alpha_{0}\ldots\alpha_{p}}\coloneqq\sum_{m=0}^{p}(-1)^{m}f_{\Delta_{m}^{p+1}}.$

By $(\star)$, we know that

$\displaystyle\sum_{m=0}^{p}(-1)^{m}\partial f_{\Delta_{m}^{p+1}}$
($\circledast_{\partial}$) $\displaystyle+$
$\displaystyle\sum_{m=0}^{p}\sum_{j=1}^{p}(-1)^{m}(-1)^{j}f_{\Delta_{m}^{p+1}\setminus\operatorname{ver}_{j}}$
($\circledast_{\hat{\delta}}$) $\displaystyle+$
$\displaystyle\sum_{m=0}^{p}\sum_{j=1}^{p}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m}^{p+1}(0,j)}f_{\Delta_{m}^{p+1}(j,p+1)}$
($\circledast_{\circ}$) $\displaystyle=$ $\displaystyle\,0$

where we introduce two notational shorthands: given a simplex $\sigma$ we
write $\sigma\setminus\operatorname{ver}_{j}$ to mean
$\sigma\setminus\operatorname{ver}_{j}\sigma$, and $\sigma(i,j)$ to mean
$\\{\operatorname{ver}_{i}\sigma<\ldots<\operatorname{ver}_{j}\sigma\\}$. We
will now examine each of $(\circledast_{\partial})$,
$(\circledast_{\hat{\delta}})$, and $(\circledast_{\circ})$ in turn.

Firstly, $(\circledast_{\partial})$. By the definition of $\partial$, along
with its linearity, and the definition of
$\lambda_{\alpha_{0}\ldots\alpha_{p}}$, we see that this is exactly

$\displaystyle\sum_{m=0}^{p}(-1)^{m}\partial f_{\Delta_{m}^{p+1}}$
$\displaystyle=\sum_{m=0}^{p}(-1)^{m}\left(f_{\Delta_{m}^{p+1}}\circ
F_{\alpha_{p}}+(-1)^{p+1}E_{\alpha_{0}}\circ f_{\Delta_{m}^{p+1}}\right)$
$\displaystyle=\left(\sum_{m=0}^{p}(-1)^{m}f_{\Delta_{m}^{p+1}}\right)\circ
F_{\alpha_{p}}+(-1)^{p+1}E_{\alpha_{0}}\circ\left(\sum_{m=0}^{p}(-1)^{m}f_{\Delta_{m}^{p+1}}\right)$
$\displaystyle=\lambda_{\alpha_{0}\ldots\alpha_{p}}F_{\alpha_{p}}+(-1)^{p+1}E_{\alpha_{0}}\lambda_{\alpha_{0}\ldots\alpha_{p}}.$
($\boxtimes_{\partial}$)

Next, $(\circledast_{\hat{\delta}})$. A table showing all terms in the case
$p=3$ is given in Figure B.2.v, and might prove useful in visualising some of
the combinatorics. First of all, note that
$\Delta_{m}^{p+1}\setminus\operatorname{ver}_{m}=\Delta_{m-1}^{p+1}\setminus\operatorname{ver}_{m}$
for all $m=1,\ldots,p$, and the corresponding morphisms in
$(\circledast_{\hat{\delta}})$ have opposite signs, since $j=m+1$ is fixed for
both, and so they cancel. Now the remaining morphisms can all303030Indeed, the
only morphisms labelling simplices that pass through one of
$\left[\begin{smallmatrix}j\\\ 0\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}j\\\ 1\end{smallmatrix}\right]$ but not the other
are exactly those with diagonal paths, but $\Delta_{m}^{p+1}$ consists
entirely of horizontal and vertical paths (cf. Figure B.2.iv) and so the only
terms that contain diagonal paths are
$\Delta_{m}^{p+1}\setminus\operatorname{ver}_{m}$ and
$\Delta_{m}^{p+1}\setminus\operatorname{ver}_{m+1}$, which are exactly the
previously mentioned terms that cancel each other out. be grouped together:
those labelling simplices that skip over both $\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$; those labelling simplices that skip over both
$\left[\begin{smallmatrix}2\\\ 0\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}2\\\ 1\end{smallmatrix}\right]$; and so on, up to
those that skip over both $\left[\begin{smallmatrix}p-1\\\
0\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}p-1\\\
1\end{smallmatrix}\right]$. But the set of all simplices that skip over both
$\left[\begin{smallmatrix}j\\\ 0\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}j\\\ 1\end{smallmatrix}\right]$ is exactly the set
of all $\Delta_{m}^{p}$ for $m=0,\ldots,p-1$ where we label the top and bottom
$p-1$-simplices in $\Delta[p-1]\times\Delta[1]$ by
$\\{0<1<\ldots<\hat{j}<\ldots<p\\}$. For example,
$\Delta_{0}^{4}\setminus\operatorname{ver}_{2}$,
$\Delta_{2}^{4}\setminus\operatorname{ver}_{1}$, and
$\Delta_{3}^{4}\setminus\operatorname{ver}_{1}$ are exactly $\Delta_{0}^{3}$,
$\Delta_{1}^{3}$, and $\Delta_{2}^{3}$ (respectively) on the copy of
$\Delta[2]\times\Delta[1]$ given by $\\{0<2<3\\}\times\\{0<1\\}$ (see Figure
B.2.v). We denote these copies of $\Delta_{m}^{p}$ on
$\\{0<\ldots<\hat{j}<\ldots<p\\}\times\\{0<1\\}$ by $\Delta_{m}^{p+1\setminus
j}$.

In the bottom half of Figure B.2.v we represent how the 12 terms in the case
$p=3$ either cancel out or group together, and this pattern generalises: by
the above, $(\circledast_{\hat{\delta}})$ is exactly313131To check that all
terms are indeed recovered, note that the right-hand side partitions the set
of indices $\\{(m,j)\mid 0\leqslant m\leqslant p,1\leqslant j\leqslant p\\}$
into three sets: the “thick diagonal” $\\{(m,j)\mid m=j\\}\cup\\{(m,j)\mid
m=j-1\\}$, the lower triangle $\\{(m,j)\mid m<j-1\\}$, and the upper triangle
$\\{(m,j)\mid m>j\\}$ (with this last partition being slightly obscured, since
there is a shift $j\mapsto j-1$ in the indexing of the removed vertex).

$\displaystyle\sum_{m=0}^{p}\sum_{j=1}^{p}(-1)^{m}(-1)^{j}f_{\Delta_{m}^{p+1}\setminus\operatorname{ver}_{j}}$
$\displaystyle=\sum_{m=1}^{p}(-1)^{m}\left(f_{\Delta_{m}^{p+1}\setminus\operatorname{ver}_{m}}-f_{\Delta_{m-1}^{p+1}\setminus\operatorname{ver}_{m}}\right)$
$\displaystyle+\sum_{j=2}^{p}(-1)^{j}\left(\sum_{m=0}^{j-2}(-1)^{m}f_{\Delta_{m}^{p+1}\setminus\operatorname{ver}_{j}}+\sum_{m=j}^{p}(-1)^{m+1}f_{\Delta_{m}^{p+1}\setminus\operatorname{ver}_{j-1}}\right)$

but the first sum on the right vanishes (since
$\Delta_{m}^{p+1}\setminus\operatorname{ver}_{m}=\Delta_{m-1}^{p+1}\setminus\operatorname{ver}_{m}$)
and the second simplifies (following the pattern of Figure B.2.v) to

$\sum_{j=1}^{p-1}\sum_{m=0}^{p-1}(-1)^{j-1}(-1)^{m}f_{\Delta_{m}^{p+1\setminus
j}}=\sum_{j=1}^{p-1}(-1)^{j-1}\left(\sum_{m=0}^{p-1}f_{\Delta_{m}^{p+1\setminus
j}}\right)$

which is

$-\sum_{j=1}^{p-1}(-1)^{j}\lambda_{\alpha_{0}\ldots\widehat{\alpha_{j}}\ldots\alpha_{p}}$

precisely by our definition of $\lambda$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta_{m}^{4}$$\Delta_{m}^{4}\setminus\operatorname{ver}_{1}$$\Delta_{m}^{4}\setminus\operatorname{ver}_{2}$$\Delta_{m}^{4}\setminus\operatorname{ver}_{3}$$m=0$$m=1$$m=2$$m=3$

$0$$1$$2$$3$$1$$2$$3$${}_{j}\setminus^{m}$$\times$$\Delta_{0}^{4\setminus
1}$$\Delta_{0}^{4\setminus 2}$$\times$$\times$$\Delta_{1}^{4\setminus
2}$$\Delta_{1}^{4\setminus 1}$$\times$$\times$$\Delta_{2}^{4\setminus
1}$$\Delta_{2}^{4\setminus 2}$$\times$

Figure B.2.v: _Top:_ The 12 terms of $(\circledast_{\hat{\delta}})$ in the
case $p=3$. _Bottom:_ The $(j,m)$ entry corresponds to
$\Delta_{m}^{4}\setminus\operatorname{ver}_{j}$. The entries marked with a
$\times$ cancel out with the other $\times$ to which it is connected by a
line; the remaining entries are labelled with $\Delta_{m}^{4\setminus j}$,
denoting the copy of $\Delta_{m}^{3}$ on
$\\{0<\ldots<\hat{j}<\ldots<3\\}\times\\{0<1\\}$, and the ones joined with
lines are the ones that add to give the
$\lambda_{\alpha_{0}\ldots\widehat{\alpha_{j}}\ldots\alpha_{3}}$ term. Note
that entries that are horizontally or vertically adjacent to one another in
this table have opposite signs in $(\circledast_{\hat{\delta}})$.

Finally, $(\circledast_{\circ})$. A table showing all terms in the case $p=3$
is given in Figure B.2.vi. To start, we split up the sum in
$(\circledast_{\circ})$ into an “upper triangular” part and a “lower
triangular” part:

$\begin{gathered}\sum_{m=0}^{p}\sum_{j=1}^{p}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m}^{p+1}(0,j)}f_{\Delta_{m}^{p+1}(j,p+1)}\\\
=\sum_{j=1}^{p}\sum_{m=0}^{j-1}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m}^{p+1}(0,j)}f_{\Delta_{m}^{p+1}(j,p+1)}\\\
+\sum_{j=1}^{p}\sum_{m=j}^{p}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m}^{p+1}(0,j)}f_{\Delta_{m}^{p+1}(j,p+1)}.\end{gathered}$

Now, in the first sum on the right-hand side, the
$f_{\Delta_{m}^{p+1}(j,p+1)}$ term will be constant over all values of $m$,
since if $m<j$ then $\Delta_{m}^{p+1}(j,p+1)$ is simply
$\left[\begin{smallmatrix}j-1&j&\ldots&p\\\
1&1&\ldots&1\end{smallmatrix}\right]$, and similarly for the
$f_{\Delta_{m}^{p+1}(0,j)}$ term in the second sum. In other words, we can
write $(\circledast_{\circ})$ as

$\displaystyle\sum_{j=1}^{p}\left(\sum_{m=0}^{j-1}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m}^{j}(0,j)}\right)f_{\left[\begin{smallmatrix}j-1&j&\ldots&p\\\
1&1&\ldots&1\end{smallmatrix}\right]}$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{p}f_{\left[\begin{smallmatrix}0&1&\ldots&j\\\
0&0&\ldots&0\end{smallmatrix}\right]}\left(\sum_{m=j}^{p}(-1)^{m}(-1)^{(p+1)(j-1)}f_{\Delta_{m-j}^{p+1-j}+\left[\begin{smallmatrix}j\\\
0\end{smallmatrix}\right]}\right)$

where we write $\Delta_{m-j}^{p+1-j}+\left[\begin{smallmatrix}j\\\
0\end{smallmatrix}\right]$ to mean repeated vertex-wise addition, i.e.

$\displaystyle\Delta_{n}^{q}+\left[\begin{smallmatrix}i\\\
0\end{smallmatrix}\right]$
$\displaystyle=\left[\begin{smallmatrix}0&1&2&\ldots&n&n&n+1&\ldots&q-1&q\\\
0&0&0&\ldots&0&1&1&\ldots&1&1\end{smallmatrix}\right]+\left[\begin{smallmatrix}i\\\
0\end{smallmatrix}\right]$
$\displaystyle\coloneqq\left[\begin{smallmatrix}i&1+i&2+i&\ldots&n+i&n+i&n+1+i&\ldots&q-1+i&q+i\\\
0&0&0&\ldots&0&1&1&\ldots&1&1\end{smallmatrix}\right].$

But this is exactly

$\sum_{j=1}^{p}(-1)^{(p+1)(j-1)}\lambda_{\alpha_{0}\ldots\alpha_{j-1}}F_{\alpha_{j-1}\ldots\alpha_{p}}+\sum_{j=1}^{p}(-1)^{(p+1)(j-1)}(-1)^{j}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}$

and shifting the first sum from $\sum_{j=1}^{p}$ to $\sum_{j=0}^{p-1}$ and the
second sum from $\sum_{m=j}^{p}$ to $\sum_{m=0}^{p-j}$ gives

$\sum_{j=0}^{p-1}(-1)^{(p+1)j}\lambda_{\alpha_{0}\ldots\alpha_{j}}F_{\alpha_{j}\ldots\alpha_{p}}+\sum_{j=1}^{p}(-1)^{(p+1)(j-1)}(-1)^{j}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}.$

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$0$$0$$1$$2$$3$$0$$0$$1$$1$$2$$3$$0$$0$$1$$2$$2$$3$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$1$$1$$1$$2$$3$$0$$1$$1$$1$$2$$3$$0$$1$$1$$2$$2$$3$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$1$$1$$2$$2$$3$$0$$1$$2$$2$$2$$3$$0$$1$$2$$2$$2$$3$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$1$$1$$2$$3$$3$$0$$1$$2$$2$$3$$3$$0$$1$$2$$3$$3$$3$$\Delta_{m}^{4}$$\Delta_{m}^{4}(0,1)(1,4)$$\Delta_{m}^{4}(0,2)(2,4)$$\Delta_{m}^{4}(0,3)(3,4)$$m=0$$m=1$$m=2$$m=3$

$0$$1$$2$$3$$1$$2$$3$${}_{j}\setminus^{m}$$\lambda_{0}\circ
F_{0123}$$\Delta_{0}^{2}\circ F_{123}$$\Delta_{0}^{3}\circ
F_{23}$$E_{01}\circ(\Delta_{0}^{3}+\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right])$$\Delta_{1}^{2}\circ F_{123}$$\Delta_{1}^{3}\circ
F_{23}$$E_{01}\circ(\Delta_{1}^{3}+\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right])$$E_{012}\circ(\Delta_{0}^{2}+\left[\begin{smallmatrix}2\\\
0\end{smallmatrix}\right])$$\Delta_{2}^{3}\circ
F_{23}$$E_{01}\circ(\Delta_{2}^{3}+\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right])$$E_{012}\circ(\Delta_{1}^{2}+\left[\begin{smallmatrix}2\\\
0\end{smallmatrix}\right])$$E_{0123}\circ\lambda_{3}$

Figure B.2.vi: _Top:_ The 12 terms of $(\circledast_{\circ})$ in the case
$p=3$, where we write $\Delta_{m}^{4}(0,j)(j,4)$ as shorthand for
$\Delta_{m}^{4}(0,j)$ followed by $\Delta_{m}^{4}(j,4)$. _Bottom:_ The entries
are labelled with $f_{\Delta_{m}^{4}(0,j)}f_{\Delta_{m}^{4}(j,4)}$; those
joined with a line are those that add to give a
$\lambda_{\alpha_{0}\ldots\alpha_{j}}$ (or
$\lambda_{\alpha_{j}\ldots\alpha_{p}}$) term. Note that the entries above (or
on) the diagonal and on the same line all share the same first component, and
those below the diagonal on the same line all share the same second component.

Putting this all together, we know that
$(\boxtimes_{\partial})+(\boxtimes_{\hat{\delta}})+(\boxtimes_{\circ})=0$, and
so

$\displaystyle 0$
$\displaystyle=-(\boxtimes_{\hat{\delta}})-(\boxtimes_{\partial})-(\boxtimes_{\circ})$
$\displaystyle=\sum_{j=1}^{p-1}(-1)^{j}\lambda_{\alpha_{0}\ldots\widehat{\alpha_{j}}\ldots\alpha_{p}}$
$\displaystyle-\lambda_{\alpha_{0}\ldots\alpha_{p}}F_{\alpha_{p}}-(-1)^{p+1}E_{\alpha_{0}}\lambda_{\alpha_{0}\ldots\alpha_{p}}$
$\displaystyle-\sum_{j=0}^{p-1}(-1)^{(p+1)j}\lambda_{\alpha_{0}\ldots\alpha_{j}}F_{\alpha_{j}\ldots\alpha_{p}}-\sum_{j=1}^{p}(-1)^{(p+1)(j-1)}(-1)^{j}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}$

but the first sum is exactly
$(\hat{\delta}\lambda)_{\alpha_{0}\ldots\alpha_{p}}$, so it remains only to
show that $-(\boxtimes_{\partial})-(\boxtimes_{\circ})=E\lambda-\lambda F$,
since then $\lambda$ satisfies

$\hat{\delta}\lambda+E\lambda-\lambda F=0$

which is exactly the condition necessary in order for $\lambda$ to be a weak
equivalence (since the $\lambda_{\alpha}$ terms live in the maximal Kan
complex of the dg-nerve, and are thus quasi-isomorphisms by Lemma 2.5.6). We
can merge the two terms of $(\boxtimes_{\partial})$ into the two sums of
$(\boxtimes_{\circ})$ to obtain

$\displaystyle-(\boxtimes_{\partial})-(\boxtimes_{\circ})$
$\displaystyle=-\sum_{j=0}^{p}(-1)^{(p+1)(j-1)}(-1)^{j}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}-\sum_{j=0}^{p}(-1)^{(p+1)j}\lambda_{\alpha_{0}\ldots\alpha_{j}}F_{\alpha_{j}\ldots\alpha_{p}}$
$\displaystyle=\sum_{j=0}^{p}\left[(-1)^{(p+1)(j-1)+j+1}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}-(-1)^{(p+1)j}\lambda_{\alpha_{0}\ldots\alpha_{j}}F_{\alpha_{j}\ldots\alpha_{p}}\right].$

But $(p+1)(j-1)+j+1\equiv(1-j)(p-j)\mod 2$ and $(p+1)j\equiv-j(p-j)\mod 2$, so
we can write the above as

$\sum_{j=0}^{p}\left[(-1)^{(1-j)(p-j)}E_{\alpha_{0}\ldots\alpha_{j}}\lambda_{\alpha_{j}\ldots\alpha_{p}}-(-1)^{-j(p-j)}\lambda_{\alpha_{0}\ldots\alpha_{j}}F_{\alpha_{j}\ldots\alpha_{p}}\right]$

and this is exactly $E\lambda-\lambda F$.

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