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where the vertical morphism lies in $cell(\mathcal{H})$, i.e. is an FSAE and
the horizontal morphism has the right lifting property with respect to
$\mathcal{H}$, making it a fibration. ∎

Corollary 2.2.29 is particularly useful because it sheds light on the
interaction of FSAEs and the homotopy category of $\textnormal{s{Set}}_{P}$,
$\mathcal{H}\textnormal{s{Set}}_{P}$. This is summarized in the following
proposition which we are using all throughout the remainder of this work.

######  Proposition 2.2.30.

Let $X,Y\in\textnormal{s{Set}}_{P}$.

  1. (i)

Let $X\xrightarrow{\alpha}Y$ be a morphism in
$\mathcal{H}\textnormal{s{Set}}_{P}$. Then there exists a morphism
$f:X\xhookrightarrow{Z}$ in $\textnormal{s{Set}}_{P}$ and an FSAE
$Y\xrightarrow{s}Z$ such that

$\alpha=[s]^{-1}\circ[f].$

If $X$ and $Y$ are finite, then $Z$ can be taken to be finite.

  2. (ii)

Let $X\xrightarrow{f,g}Y$ be two morphisms in $\textnormal{s{Set}}_{P}$. Then
$[f]=[g]$ if and only if there exists an FSAE $Y\xhookrightarrow{s}Z$ such
that $s\circ f$ and $s\circ g$ are strictly simplicially (stratum preserving,
elementary) homotopic. If $X$ and $Y$ are finite, then the same statement
holds with $Z$ finite.

###### Proof.

Denote by $F,1\xrightarrow{i}F$ the fibrant replacement functor from Corollary
2.2.29. We start with the proof of $(i)$. We only do the proof in the finite
case, as the infinite one is a strict simplification of it. Let
$X,Y\in\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ and
$X\xrightarrow{\alpha}Y$ be a morphism in
$\mathcal{H}\textnormal{s{Set}}_{P}$. Then, as $F(Y)$ is fibrant and $X$
cofibrant, $\alpha$ fits into a commutative diagram

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where $f$ is a morphism $X\xrightarrow{f}F(Y)$ in $\textnormal{s{Set}}_{P}$
(see Proposition 1.1.27). Now, apply Lemma 2.2.25 to get a factorization

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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-19.07779pt}{21.21284pt}\pgfsys@lineto{11.2214pt}{0.34888pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.82362}{-0.56714}{0.56714}{0.82362}{11.38611pt}{0.23547pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{[f^{\prime}]}$} }}\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{20.84024pt}{19.63791pt}\pgfsys@lineto{20.84024pt}{3.87773pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{1.0}{0.0}{20.84024pt}{19.63791pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{20.84024pt}{3.67775pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.19301pt}{10.28891pt}\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{$\scriptstyle{[i_{Y}^{\prime}]}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{30.68872pt}{21.96693pt}\pgfsys@curveto{48.5717pt}{14.51242pt}{52.72589pt}{-17.59676pt}{36.50212pt}{-29.98781pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.92302}{0.38477}{0.38477}{0.92302}{30.68872pt}{21.96692pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.79472}{-0.60696}{0.60696}{-0.79472}{36.34319pt}{-30.10918pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{48.53233pt}{1.98245pt}\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{$\scriptstyle{[i_{Y}]}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{20.84024pt}{-14.07042pt}\pgfsys@lineto{20.84024pt}{-29.8306pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{1.0}{0.0}{20.84024pt}{-14.07042pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{20.84024pt}{-30.03058pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\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}},$

where we know of the outer diagram and all of the triangle diagrams but the
most upper one that they are commutative, and where $i^{\prime}_{Y}$ is an
FSAE and $Y^{\prime}$ is finite. Using the fact that the morphism
$Y^{\prime}\to F(Y)$ is an FSAE and hence an isomorphism in the homotopy
category and chasing the diagram shows that the most upper triangle also
commutes. To summarize, we have shown

$[i^{\prime}_{Y}]\circ\alpha=[f^{\prime}]$

and hence

$\alpha=[i^{\prime}_{Y}]^{-1}[f^{\prime}]$

for the FSAE $i_{Y}^{\prime}$ between finite filtered simplicial sets.

The if part is immediate from the fact that every FSAE is a weak equivalence
and again Proposition 1.1.27. For the only if statement, consider the fibrant
replacement $Y\xhookrightarrow{i_{Y}}F(Y)$. Then $[i_{Y}\circ f]=[i_{Y}\circ
g]$. As $X$ is cofibrant and $F(Y)$ is fibrant, we have a simplicial homotopy
$H:X\otimes\Delta^{1}\to F(Y)$ between $i_{Y}\circ f$ and $i_{Y}\circ g$ (see
for ex. [Hir09]). In other words there is a commutative diagram

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48.63605pt\lower-25.40482pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-48.63605pt}{-25.30498pt}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\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{${Y}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.31943pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
20.518pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.518pt\hfil&\hfil\hskip 40.11803pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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16.11806pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{i_{0}\sqcup i_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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g}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{32.51799pt}{-13.74529pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{i_{Y}}$}
}}\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@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.99995pt}{-22.80498pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\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<EMAIL_ADDRESS>
If we apply Lemma 2.2.25 to this we obtain:

$\leavevmode\hbox to139.45pt{\vbox
to84.42pt{\pgfpicture\makeatletter\hbox{\hskip
69.72546pt\lower-42.25899pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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18.93048pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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0.0pt\hfil&\hfil\hskip 32.3194pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.31943pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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0.0pt\hfil&\hfil\hskip 33.0894pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $i_{Y}^{\prime}$ is an FSAE of finite filtered simplicial sets. A quick
diagram chase together with the fact that $Y^{\prime}\to F(Y)$ is a
monomorphism shows that the upper left part of the diagram commutes. In
particular, $i_{Y}^{\prime}\circ f$ and $i_{Y}^{\prime}\circ g$ are homotopic
through a simplicial homotopy. ∎

####  2.2.4 The interaction of FSAEs with mapping cylinders and products

Vital to many of the arguments we need for the construction of a Whitehead
group for filtered simplicial sets are the various interactions of strong
anodyne extensions with mapping cylinders. In the classical cellular setting,
this can be found in [Coh73, Ch. 2, §5] and from a simplicial perspective in
Whiteheads original publication in [Whi39, Sec. 6]. We have already seen that
the class of strict elementary expansions is rather limiting in this sense,
c.f. Example 2.2.8. In this subsection we show that the same is not the case
for general elementary expansions.

Recall that in a simplicial model category the simplicial mapping cylinder,
$M_{f}$ of a morphism $f:X\to Y$ is defined via the pushout

${X\cong
X\otimes\Delta^{0}}$${Y}$${X\otimes\Delta^{1}}$${M_{f}}$$\scriptstyle{f}$$\scriptstyle{i_{0}}$

.

Recall further that if $X$ is cofibrant, then $Y\to M_{f}$ is an acylic
cofibration and $X\xhookrightarrow{i_{1}}X\otimes\Delta^{1}\to M_{f}$ is a
cofibration. This allows to factor any such $f$ into an acyclic cofibration
followed by the induced weak equivalence $p_{Y}:M_{f}\to Y$. For the category
of simplicial sets with the Kan-Quillen model structure, the statement that
$Y\hookrightarrow M_{f}$ is a acyclic cofibration (i.e. an anodyne extension)
can be slightly strengthened. This is a consequence of the following result.

######  Proposition 2.2.31.

[Mos19, Prop. 16] Let $A\xhookrightarrow{s}B$ be a strong anodyne extensions
and $X\xhookrightarrow{i}Y$ be any cofibration (both in sSet). Then

$B\times X\cup_{A\times X}A\times Y\hookrightarrow B\times Y$

is also a strong anodyne extension.

In particular, one obtains that the inclusions $X\hookrightarrow
X\times\Delta^{1}$ at both endpoints is a strong anodyne extensions. Hence,
for $X\xrightarrow{f}Y$ a morphism in $\textnormal{s{Set}}_{P}$ the inclusion
of $Y\hookrightarrow M_{f}$, is a strong anodyne extension too, by stability
under pushouts. This is of course the simplicial analogue to the respective
result for mapping cylinders of cellular maps and cellular expansions (see for
example [Coh73, Cor. 5.1A]). It turns out this result generalizes to the
filtered setting. To obtain this in a fairly general form, we need the
following outer product construction.

######  Definition 2.2.32.

First, note that for two partially ordered sets $P$ and $P^{\prime}$ there is
a natural isomorphism $N(P)\times N(P^{\prime})\cong N(P\times P^{\prime})$.
Under this identification, we obtain an outer product functor

$-\otimes-:\textnormal{s{Set}}_{P}\times\textnormal{s{Set}}_{P}\longrightarrow\textnormal{s{Set}}_{P\times
P^{\prime}},$

by simply taking the product of arrows into $N(P)$ and $N(P^{\prime})$
respectively. In case where $P^{\prime}$ is a point, this corresponds to the
simplicial copower on $\textnormal{s{Set}}_{P}$ under
$\textnormal{s{Set}}_{P^{\prime}}\cong\textnormal{s{Set}}$, justifying the
overload of notation. From the presheaf perspective of 1.1.25 this
construction is given by:

$X\otimes Y:\Delta(P\times
P^{\prime})^{op}\hookrightarrow\Delta(P)^{op}\times\Delta(P^{\prime})^{op}\xrightarrow{X\times
Y}\textnormal{{Set}}\times\textbf{{{Set}}}\xrightarrow{-\times-}\textbf{Set}.$

######  Proposition 2.2.33.

Let $A\xhookrightarrow{s}B$ be a strong anodyne extension in
$\textnormal{s{Set}}_{P}$ and let $X\hookrightarrow Y$ be a cofibration in
$\textnormal{sSet}_{P^{\prime}}$. Then

$B\otimes X\cup_{A\otimes X}A\otimes Y\hookrightarrow B\otimes Y$

is a strong anodyne extension in $\textnormal{sSet}_{P\times P^{\prime}}$.

###### Proof.

This is really just a modification of the proof of [Mos19, Prop. 16]. The
proof goes consists of two steps. First we show that the class of
$X\xhookrightarrow{s}Y$ that have this property for some fixed $i$ is closed
under:

  1. (i)

arbitrary coproducts,

  2. (ii)

pushouts,

  3. (iii)

transfinite composition,

and that the analogous statement holds with rolls of $s$ and $i$ swapped. For
the second step, note that the class of cofibrations in
$\textnormal{sSet}_{P^{\prime}}$ is generated under these operations by the
boundary inclusions $\partial\Delta^{\mathcal{J}^{\prime}}$
$\hookrightarrow\Delta^{\mathcal{J}^{\prime}}$, for d-flags
$\mathcal{J}^{\prime}$ in $P^{\prime}$ and the class of FSAEs is generated by
the admissible horn inclusions
$\Lambda^{\mathcal{J}}_{k}\hookrightarrow\Delta^{\mathcal{J}}$, for flags
$\mathcal{J}$ in $P$. Hence, it suffices to show that, for such $\mathcal{J}$,
$\mathcal{J}^{\prime},$ $k$,

$\Delta^{\mathcal{J}}\otimes\partial\Delta^{\mathcal{J}^{\prime}}\cup_{\Lambda^{\mathcal{J}}_{k}\otimes\partial\Delta^{\mathcal{J}^{\prime}}}\Lambda_{k}^{\mathcal{J}}\otimes\Delta^{\mathcal{J}^{\prime}}\hookrightarrow\Delta^{\mathcal{J}}\otimes\Delta^{\mathcal{J}^{\prime}}$

is an FSAE.

The proof of the first step is analogous in both arguments, so we only do the
one with fixed $i$. First note that the exterior product $\otimes$ does
commute with colimits in both arguments (by the analogous statement in Set).
From this (i) immediately follows. For (ii), note that for an arrow $A\to
A^{\prime}$ and $B^{\prime}$ the pushout of $s$ along this arrow, the
following diagram is a pushout diagram.

${B\otimes X\cup_{A\otimes X}A\otimes Y}$${B\otimes Y}$${B^{\prime}\otimes
X\cup_{A^{\prime}\otimes X}A^{\prime}\otimes Y}$${B^{\prime}\otimes Y}$

An easy way to verify this statement is to note that by the alternative
construction of $\otimes$ via the perspective of categories of Set-valued
presheaves (1.1.25) we may just as well check this in Set, replacing $\otimes$
by $\times$. This is an easy exercise of manipulations of pushouts in a closed
monoidal category. It can be found in the appendix (Lemma A.0.2). We are
lacking (iii). So, let $A:\kappa\to\textnormal{sSet}_{P\times P^{\prime}}$ be
a transfinite composition diagram for some ordinal $\kappa$. By transfinite
induction, let $\kappa$ be minimal such that we have not shown closedness
under transfinite composition (the induction start is obvious). If
$\kappa=\kappa^{\prime}+1$ is a successor ordinal then
$A^{\kappa}=A^{\kappa^{\prime}}$. If $\kappa^{\prime}$ is a successor ordinal,
then the transfinite composition is just the one given by the restriction of
$A$ to $\kappa^{\prime}$. So the result follows by the induction hypothesis.
If $\kappa^{\prime}=\kappa^{\prime\prime}+1$ is a successor ordinal, then
consider the diagram below:

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}\pgfsys<EMAIL_ADDRESS>(2.3)

Similarly to the previous diagram, one checks in Set that the square in this
diagram is a pushout diagram. By assumption, the upper vertical and the lower
right morphism are FSAEs. By stability under pushouts and composition, the map
on the bottom is an FSAE, but this is precisely the map, corresponding to the
composition $A^{0}\to A^{\kappa^{\prime\prime}}\to
A^{\kappa^{\prime}}=A^{\kappa}$ i.e. to the transfinite composition we
considered. Now, let $\kappa$ be a limit ordinal. Just as before, consider the
pushout square in (2.3) but replace (formally) $\kappa^{\prime}$ with
$\kappa$, $\kappa^{\prime\prime}$ with $\kappa^{\prime}$, and $0$ with
$\kappa^{\prime\prime}$. We obtain a diagram
$\kappa\to\textnormal{sSet}_{P\times P^{\prime}}$ given by

$\kappa^{\prime\prime}<\kappa^{\prime}\mapsto\\{A^{\kappa}\otimes
X\cup_{A^{\kappa^{\prime\prime}}\otimes X}A^{\kappa^{\prime\prime}}\otimes
Y\hookrightarrow A^{\kappa}\otimes X\cup_{A^{k^{\prime}}\otimes
X}A^{\kappa^{\prime}}\otimes Y\\}.$

By the induction hypothesis and stability of FSAEs under pushout, all the
arrows in this diagram are FSAEs. Further, by commutativity of colimits and
the fact that $\otimes$ commutes with colimits, this fulfils the requirements
of a transfinite composition diagram, for each limit ordinal
$\kappa^{\prime}<\kappa$. The colimit of this diagram is $A^{\kappa}\otimes Y$
(again, this is easily checked in Set by the same argument as before). In
particular,

$A^{\kappa}\otimes X\cup_{A^{0}\otimes X}A^{0}\otimes Y\hookrightarrow
A^{\kappa}\otimes Y$

is a transfinite composition of FSAEs, making it an FSAE also.

For the second part of the proof, consider a d-flag
$\mathcal{J}=(p_{0}\leq...\leq p_{m})$ in $P$, a d-flag
$\mathcal{J}^{\prime}=(p^{\prime}_{0}\leq...\leq p^{\prime}_{m})$ in
$P^{\prime}$ and some $k\leq n$ such that
$\Lambda^{\mathcal{J}}_{k}\hookrightarrow\Delta^{\mathcal{J}}$ is admissible.
We use the pairing in the proof of [Mos19, Prop. 16] and Lemma 2.2.19 to show
that the latter still works in the filtered setting. We give a sketch of this
construction. Let
$A:=\Delta^{\mathcal{J}}\otimes\partial\Delta^{\mathcal{J}^{\prime}}\cup_{\Lambda^{\mathcal{J}}_{k}\otimes\partial\Delta^{\mathcal{J}^{\prime}}}\Lambda_{k}^{\mathcal{J}}\otimes\Delta^{\mathcal{J}^{\prime}}$
and $B:=\Delta^{\mathcal{J}}\otimes\Delta^{\mathcal{J}^{\prime}}.$

We are now in the setting where all the simplicial sets involved come from
simplicial complexes. On the underlying simplicial sets $\otimes$ is just
given by the product of ordered simplicial complexes. Every $N$-simplex in $B$
is given by a finite sequence

$(\mu_{0},\nu_{0})\leq(\mu_{1},\nu_{1})\leq...\leq(\mu_{N},\nu_{N})$

in $[m]\times[n]$ with respect to the induced preorder on the product. The
simplex is non-degenerate if at each step the $\mu$\- or the $\nu$-entry
strictly increases. The $i$-th face is obtained, by leaving out the $i$-th
entry. Thus, $i$ is admissible with respect to the product filtration of this
simplex if and only if
$(p_{\mu_{i}},p^{\prime}_{\nu_{i}})=(p_{\mu_{i+1}},p^{\prime}_{\nu_{i+1}})$ or
$(p_{\mu_{i}},p^{\prime}_{\nu_{i}})=(p_{\mu_{i-1}},p^{\prime}_{\nu_{i-1}}).$ A
non-degenerate simplex in $B$ belongs to $B_{n.d.}\setminus A_{n.d.}$ if and
only if in the corresponding sequence the the $\nu_{i}$ skip no value in $[n]$
and the $\mu_{i}$ skip no value in $[m]$ other than $k$. Pictorially, we may
think of such a sequence as a walk in the $[m]\times[n]$ grid as in Figure
2.9.

Figure 2.9: Walk in $[m]\times[n]$ where $P,P^{\prime}=[1]$ and
$\mathcal{J}=(0\leq 0\leq 0\leq 0\leq 0\leq 1\leq 1\leq 1\leq 1\leq 1)$,
$\mathcal{J}^{\prime}=(0\leq 0\leq 0\leq 1\leq 1\leq 1\leq 1)$ indicated by
blue and red.

At each time $i$ in the walk, the next step is of type
$(+1,+0),(+0,+1),(+1,+0)$, or in addition to this possibly of type
$(+2,+0),(+2,+1)$ if the next column is the $k$-th and it is skipped. For
$k<m$, [Mos19] constructs a pairing on $A\hookrightarrow B$ by setting $B_{I}$
to consist of those simplices where the $k$-th column is not skipped and the
step that leaves the column is of type $(+1,0)$. This is then paired with the
simplex described by the path where the last point on the $k$-th column is
removed (see Figure 2.10 for an example).

Figure 2.10: The walks corresponding to a pair of simplices in
$B_{n.d.}\setminus A_{n.d.}$. The removed vertex is marked.

For $0<k$, one obtains a pairing by taking $B_{I}$ to consist of those
simplices where the corresponding walk does not skip the $k$-th column and the
prior step is of type $(+1,+0)$. One then pairs such a simplex with the one
given removing the first point on the $k$-th column. In both cases Moss shows
in [Mos19] that this gives a regular, proper pairing on the underlying
simplicial sets. Now, if $p_{k}=p_{k+1}$ we take the prior pairing, if
$p_{k}=p_{k-1}$ we take the latter pairing on the underlying simplicial map
$A\hookrightarrow B$. To see this gives a proper, regular pairing on
$A\hookrightarrow B$ we just need to verify that in both cases the
requirements of Lemma 2.2.19 are met. That is, we need to check that, for each
pair given by $T$, if $i$ is the index of the removed point, then $i$ is
admissible. If $p_{k}=p_{k+1}$, then the removed point corresponds to a vertex
of stratum $(p_{k},p_{j}^{\prime})$, for some $0\leq j\leq n$. As the next
point is reached through a $(+1,+0)$, it corresponds to a
$(p_{k+1},p^{\prime}_{j})=(p_{k},p^{\prime}_{j})$ vertex. In particular, $i$
is admissible. The case $p_{k}=p_{k-1}$ works analogously. ∎

We are mostly interested in the case where either $P$ or $P^{\prime}$ is the
partially ordered set with one element, i.e. the corresponding category of
filtered simplicial sets is just the category of simplicial sets. We obtain
the following immediate corollary of Proposition 2.2.33.

######  Corollary 2.2.34.

Let $X\xhookrightarrow{i}Y$ be a cofibration in $\textnormal{sSet}_{P}$ and
$X^{\prime}\xhookrightarrow{i^{\prime}}Y^{\prime}$ be a cofibration in sSet.
Then if either of the two is an FSAE, so is the morphism in
$\textnormal{s{Set}}_{P}\cong\textnormal{s{Set}}_{P\times\star}$:

$Y\otimes X^{\prime}\cup_{X\otimes X^{\prime}}X\otimes
Y^{\prime}\hookrightarrow Y\otimes Y^{\prime}$

This result has a series of immediate consequences for the interactions of
FSAEs with filtered simplicial mapping cylinders (see Proposition 2.2.35).
These can be thought of as filtered, simplicial set analogues to the results
found for example in ([Coh73, Ch. 2,]), [Whi39, Sec. 6], [KP86, Ch. 6.3]).
They are the decisive arguments when it comes to verifying the Eckmann-
Siebenmann axioms (Lemma 2.1.13) for the construction of a Whitehead group.

######  Proposition 2.2.35.

Let $A\xhookrightarrow{s}B$ be an FSAE, $B^{\prime}\xhookrightarrow{i}B$ be a
cofibration and $B\xrightarrow{f}Z$ be a morphism in
$\textnormal{s{Set}}_{P}$. Then all of the following morphisms in
$\textnormal{s{Set}}_{P}$ (illustrated in Figure 2.11) are FSAEs.

  1. (i)

$i_{0},i_{1}:Z\hookrightarrow Z\otimes\Delta^{1}$

  2. (ii)

$Z\hookrightarrow M_{f}$

  3. (iii)

$B\cup_{A\hookrightarrow M_{f\circ s}}M_{f\circ s}\hookrightarrow M_{f}$

  4. (iv)

$M_{f\circ i}\hookrightarrow M_{f}$

In particular, for any elementary stratum preserving simplicial homotopy
$Y_{0}\otimes\Delta^{1}\xrightarrow{H}Z$ with $f:=H_{0}$ and $g:=H_{1}$,
denote by $\hat{M}_{H}$ the pushout given by

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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-12.897pt}{-23.66277pt}\pgfsys@lineto{22.0478pt}{-23.66277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.24779pt}{-23.66277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\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<EMAIL_ADDRESS>
Then the compositions

  1. (v)

$M_{f},M_{g}\hookrightarrow M_{H}\to\hat{M}_{H}$

are both FSAEs.

Figure 2.11: An illustration of some examples of the FSAEs (i) to (v) in
Proposition 2.2.35.

###### Proof.

For (i) take $Y=Z$,$X=\emptyset$, $X^{\prime}=\Delta^{0}$ and
$i^{\prime}=i_{0},i_{1}$ in Corollary 2.2.34. For (ii) note that
$Z\hookrightarrow M_{f}$ is given by the pushout diagram

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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.57535pt}{-23.4883pt}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
By (i), $i_{0}$ is an FSAE, hence, so is $Y\hookrightarrow M_{f}$. For (iii)
first consider the case of Corollary 2.2.34, where $X=A,Y=B$ and
$X^{\prime}=\Delta^{0}\sqcup\Delta^{0},Y^{\prime}=\Delta^{1},i^{\prime}=i_{0}\sqcup
i_{1}$. We obtain that

$B\underset{A\xhookrightarrow{i_{1}}A\otimes\Delta^{1}}{\cup}A\otimes\Delta^{1}\underset{{A\xhookrightarrow{i_{0}}A\otimes\Delta^{1}}}{\cup}B\hookrightarrow
B\otimes\Delta^{1}$

is an FSAE. Now, take the inclusion of $B$ at $0$ into the previous homotopy
pushout $j_{0}$ and consider the following commutative diagram. We omit some
of the pushout subscript, to simplify notation from here on out.

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The upper horizontal composition is just the inclusion of $B$ into
$B\otimes\Delta^{1}$ at $0$ and the lower horizontal composition is
$Z\hookrightarrow M_{f}$. In particular, the inner left and the outer square
are cocartesian. Hence, so is the inner right square. Thus, as we have just
seen that the upper right horizontal is an FSAE, the same holds for the
morphism in (iii), by stability under pushouts. For (iv) take
$X=B^{\prime},Y=B$ and
$X^{\prime}=\Delta^{0},Y^{\prime}=\Delta^{1},i^{\prime}=i_{0}$. Then the
result follows by a similar pushout argument as we used for (iii).

We prove (v) for $f$. The case of $g$ works analogously. First note that by
(i), (iii) and (iv) the cofibrations $M_{f}\hookrightarrow
Y_{0}\otimes\Delta^{1}\cup M_{f}\hookrightarrow M_{H}$ as well as their
composition are FSAEs. The $\cup$ stands for the pushout of
$Y_{0}\xhookrightarrow{i_{0}}Y_{0}\otimes\Delta^{1}$ along
$Y_{0}\hookrightarrow M_{f}$ here. Now, consider the commutative diagram

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The outer square is cocartesian by definition. The inner left square fits into

${Y_{0}}$${Y_{0}\otimes\Delta^{1}}$${Y_{0}}$${M_{f}}$${Y_{0}\otimes\Delta^{1}\underset{...}{\cup}M_{f}}$${M_{f}}$$\scriptstyle{i_{1}}$

The horizontal compositions of 2.2.4 are the identity making the outer square
cocartesian. The left inner square is cocartesian by definition. In
particular, the right inner square, which is the left inner square in (2.4) is
cocartesian. Hence, so is the inner right square in (2.4). By stability under
pushouts, this shows that $M_{f}\hookrightarrow\hat{M}_{H}$ is an FSAE. ∎

In particular, Proposition 2.2.35 shows that with our specification of
elementary collapses we do not run into problems such as we illustrated in
Example 2.2.8.

###  2.3 The Whitehead group of a filtered simplicial set

In Section 2.1 we have reviewed a general setting that Whitehead groups can be
constructed in. Then, in in Section 2.2 we begun a detailed analysis of a
class of expansions of $P$-filtered simplicial sets, FSAEs, which work as a
filtered analogue to the expansion in the classical simple homotopy theory. We
now show that (finite) FSAEs fit into this general setting (Section 2.1) and
thus obtain a Whitehead group and torsion for filtered simplicial sets. We
then study these invariants in detail, showing that they behave much like the
classical ones (Section 2.3.2). Finally, we obtain the analogue to the result
that every finite CW-complex is simple homotopy equivalent to a finite
simplicial complex for our setting (Section 2.3.3).

####  2.3.1 The Eckmann-Siebenmann approach for filtered simplicial sets

Recall the setting of Section 2.1. Given a category $\mathcal{C}$ contained in
a larger category $\hat{\mathcal{C}}$ and a class of morphisms $\Sigma$ in
$\mathcal{C}$ fulfilling certain axioms, it allows us to construct a Whitehead
group and a Whitehead torsion. In this subsection, we apply this theory to the
case where $\hat{\mathcal{C}}$ is the category of finite filtered simplicial
sets in $\textnormal{s{Set}}_{P}$, $\mathcal{C}$ is the subcategory of
$\hat{\mathcal{C}}$ with the same objects morphisms taken to be only
cofibrations in the Douteau model structure - i.e. inclusions of filtered
simplicial subsets - and $\Sigma$ is the class of (finite) FSAEs.

Recall that we denote by
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ the full subcategory
of $\mathcal{H}\textnormal{s{Set}}_{P}$ given by finite filtered simplicial
sets. Further, let $\mathcal{C}$ ($\mathcal{C}^{\prime}$) be the subcategory
of $\textnormal{s{Set}}_{P}$ given by finite (arbitrary) filtered simplicial
sets and morphisms cofibrations and $\Sigma$ $(\Sigma^{\prime})$ the class of
morphism in $\mathcal{C}$ ($\mathcal{C}^{\prime}$)that are FSAEs. As every
FSAE is in particular an anodyne extension and these are precisely the trivial
cofibrations in the Douteau-Henrique model structure on
$\textnormal{s{Set}}_{P}$ [Dou18, Thm. 2.14], by the universal property of the
localization of a category we obtain a commutative (on the nose) diagram of
functors

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All of the functors above are given by the identity on objects. It turns out,
all of them are in fact fully faithful. Before we prove this, we start by
proving two lemmata. Denote by $Q$ and $Q^{\prime}$ the structure functors
$\mathcal{C}\to\mathcal{C}(\Sigma^{-1})$ and
$\mathcal{C}^{\prime}\to\mathcal{C}^{\prime}(\Sigma^{\prime-1}$ respectively.

######  Lemma 2.3.1.

Let $X\xhookrightarrow{f,g}Y$ be two cofibrations in $\textnormal{s{Set}}_{P}$
that are homotopic through a (elementary, stratum preserving) simplicial
homotopy. Then

$Q^{\prime}(f)=Q^{\prime}(g).$

If both $X$ and $Y$ are finite, then also

$Q(f)=Q(g).$

###### Proof.

The argument both in the finite and in the infinite case are identical, so we
prove the finite one. We fist show the case where the two are homotopic
through a cofibration $H:X\otimes\Delta^{1}\hookrightarrow Y$. We then show an
additional lemma to complete the proof of this one. Consider the two
cofibrations $X\xhookrightarrow{i_{0},i_{1}}X\otimes\Delta^{1}$. If we can
show that $Q(i_{0})=Q(i_{1})$, we are done since then

$Q(f)=Q(H)\circ Q(i_{0})=Q(H)\circ Q(i_{1})=Q(g).$

Now, note that $\textnormal{sd}(\Delta)=\Lambda^{2}_{2}$. Consider the
following diagram in sSet.

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for $i$ the inclusion at $0$ or $1$. All of the maps in the diagram are FSAEs.
In particular, if we apply $X\otimes(-)$ by Proposition 2.2.33, we obtain
commutative diagrams of FSAEs

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