}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.518pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
19.82355pt\hfil&\hfil\hskip 44.51797pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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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{$\scriptstyle{i_{j}^{\prime}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{31.82353pt}{-13.57863pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-11.05453pt}{-23.77608pt}\pgfsys@lineto{10.70557pt}{-23.77608pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-11.05453pt}{-23.77608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.90555pt}{-23.77608pt}\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}},$

$j=0,1$. By commutativity and the fact that all arrows involved become
isomorphisms under $Q$, it suffices to show that

$Q(i_{0}^{\prime})=Q(i_{1}^{\prime}).$

Let
$t:X\otimes\textnormal{sd}(\Delta^{1})\xrightarrow{\sim}X\otimes\textnormal{sd}(\Delta^{1})$
be the isomorphism induced by the swap of endpoints
$\textnormal{sd}(\Delta^{1})\to\textnormal{sd}(\Delta^{1})$. Denote by
$i_{m}:X\hookrightarrow X\otimes\textnormal{sd}(\Delta^{1})$ the inclusion
induced by sending $\Delta^{0}$ to the middle vertex in
$\textnormal{sd}(\Delta^{1})$. The diagram in $\mathcal{C}$

${X}$${X\otimes\textnormal{sd}(\Delta^{1})}$${X\otimes\textnormal{sd}(\Delta^{1})}$${X\otimes\textnormal{sd}(\Delta^{1})}$$\scriptstyle{i_{m}}$$\scriptstyle{i_{m}}$$\scriptstyle{t}$$\scriptstyle{1}$

commutes. By the same usage of Proposition 2.2.33 as before, one shows that
$i_{m}$ is an FSAE. In particular, by commutativity of the last diagram,

$Q(t)=Q(1_{X\otimes\textnormal{sd}(\Delta^{1})})$

and it follows

$Q(i^{\prime}_{0})=Q(t)Q(i^{\prime}_{1})=Q(1)Q(i^{\prime}_{1})=Q(i_{1}^{\prime})$

completing the first part of this proof. ∎

######  Lemma 2.3.2.

Let $X\xhookrightarrow{f}Y$ be a cofibration in $\textnormal{s{Set}}_{P}$.
Then the mapping cylinder splitting

${X}$${Y}$${M_{f}}$$\scriptstyle{f}$

commutes in $\mathcal{C}^{\prime}(\Sigma^{\prime-1})$. If $X$ and $Y$ are
finite, it also commutes in $\mathcal{C}(\Sigma^{-1}))$.

###### Proof.

Denote by $H$ the structure map $X\otimes\Delta^{1}\to M_{f}$. Since $f$ is a
cofibration, so is this (by stability under pushouts).
$X\xhookrightarrow{i_{1}}X\otimes\Delta^{1}\xrightarrow{H}M_{f}$ is the lower
diagonal map in the statement.
$X\xhookrightarrow{i_{0}}X\otimes\Delta^{1}\xrightarrow{H}M_{f}$ is the
composition of the upper right with the vertical. In particular, by the first
part of the proof of Lemma 2.3.1, the result follows. ∎

###### proof of Lemma 2.3.1; part two.

We still have to show that $Q(f)=Q(g)$ if the two are homotopic through a
simplicial homotopy $H:X\otimes\Delta^{1}\to Y$ that is not necessarily a
cofibration. To show this, consider the following diagram in
$\textnormal{s{Set}}_{P}$.

${M_{f}}$${X}$${Y}$${\hat{M}_{H}}$${M_{g}}$$\scriptstyle{g}$$\scriptstyle{f}$

Here $\hat{M}_{H}$ and the arrows into it are the ones constructed in
Proposition 2.2.35 (i.e. $\hat{M}_{H}$ is the mapping cylinder of $H$ where
the $X\otimes\Delta^{1}$ at $1$ was collapsed to $X$.) By Proposition 2.2.35
Item (v), all arrows in the right part of the diagram are FSAEs and the upper
right and lower right triangles are commutative by construction. By Lemma
2.3.2 the upper left and lower left triangle commute in
$\mathcal{C}(\Sigma^{-1})$. Passing to $\mathcal{C}(\Sigma^{-1})$ we now have
a diagram where:

  * •

The outer square commutes.

  * •

The upper left and lower left triangle commute.

  * •

The upper right and lower right triangle commute and consist only of
isomorphisms.

In particular, a quick diagram chase shows that in $\mathcal{C}(\Sigma^{-1})$
we have $Q(f)=Q(g)$. ∎

######  Proposition 2.3.3.

All of the functors in (2.3.1) are fully faithful. In particular, the
horizontals are isomorphisms of categories.

###### Proof.

We prove the finite case as the infinite one is basically just a simpler
version of this one.

We start by showing fullness. By Proposition 2.2.30 it suffices to show this
that morphisms in $\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ of
shape $[f]$ for some morphism $f:X\to Y$ in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. Using mapping cylinder
factorization we obtain the diagram

$\leavevmode\hbox to64.43pt{\vbox
to49.34pt{\pgfpicture\makeatletter\hbox{\hskip
32.21384pt\lower-24.66942pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.21384pt}{-24.66942pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.31943pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.3736pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
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}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{20.84024pt}{2.18375pt}\pgfsys@lineto{20.84024pt}{-13.57643pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

which is commutative in $\mathcal{H}\textnormal{s{Set}}_{P}$. As, $M_{f}$ is
again finite, by Proposition 2.2.35, $Y\hookrightarrow M_{f}$ is a (finite)
FSAE, hence induces an isomorphism in $\mathcal{C}(\Sigma^{-1})$. So it
suffices to show that the class of $X\hookrightarrow M_{f}$ in
$\mathcal{H}\textnormal{s{Set}}_{P}$ comes from an arrow in
$\mathcal{C}(\Sigma^{-1})$. But this is obvious, as $X\hookrightarrow M_{f}$
is a cofibration of finite filtered simplicial sets.

It remains to show the faithfulness. First note that by the same argument as
in Section 2.1 (i.e. the fact that we can always change direction in a zigzag
by using pushouts) every morphism in $\mathcal{C}(\Sigma^{-1})$ is of the
shape $Q(s)^{-1}Q(f)$ where $s$ is a (finite) FSAE and $f$ a morphism in
$\mathcal{C}$. So suppose we are given two such morphisms
$Q(s)^{-1}Q(f),Q(s^{\prime})^{-1}Q(g)$ with
$[s]^{-1}[f]=[s^{\prime}]^{-1}[g]$. We need to show
$Q(s)^{-1}Q(f)=Q(s^{\prime})^{-1}Q(g)$. By invertibility of
$Q(s),Q(s^{\prime})$ we may without loss of generality assume that $f,g$ have
the same source $X$ and target $Y$ and that $s=s^{\prime}=1_{Y}$. In
particular, we then have

$[f]=[g].$

By Proposition 2.2.30 there then exists an $s\in\Sigma$ such that $s\circ f$
and $s\circ g$ are (stratum preserving, elementarily) simplicially homotopic.
By Lemma 2.3.1 this implies $Q(s\circ f)=Q(s\circ g)$. As $Q(s)$ is
invertible, we obtain

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

∎

As a consequence of Proposition 2.3.3, our choices of $\mathcal{C}$ and
$\Sigma$ are promising candidates for constructing a Whitehead group (with
respect to FSAEs i.e. with respect to elementary expansions) on
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$
($\mathcal{H}\textnormal{sSet}$). For $X\in\mathcal{C}$,
$(\mathcal{C^{\prime}})$ and $\Sigma$, $\Sigma^{\prime}$ as before recall the
definitions of a simple morphism and of $A(X),E(X)$ (Definition 2.1.1). For
the remainder of this section, we only be focus on the finite setting i.e. on
$\mathcal{C},\Sigma$. Quite clearly, all of the arguments below also work in
the infinite one, with the exception that $A(X)$ is in general probably not of
set size in this case. If one is willing to work with groups of class size,
this might produce a fruitful theory as well.

In the finite setting however, $E(X)$ is clearly of set size, as the
equivalence relation on $E(X)$ is up to isomorphism. We now show that the
triple $\mathcal{C}\subset\hat{\mathcal{C}},\Sigma$ satisfies the axioms in
Definition 2.1.2, i.e. admits a Whitehead group.

######  Theorem 2.3.4.

Let $\hat{\mathcal{C}}$ be the full subcategory of $\textnormal{s{Set}}_{P}$
given by finite filtered simplicial sets. Let $\mathcal{C}$ be the subcategory
which has the same objects as $\mathcal{C}$ but only cofibrations in
$\textnormal{s{Set}}_{P}$ as morphisms. Let $\Sigma$ be the class of FSAEs in
$\hat{\mathcal{C}}$. Then $\mathcal{C}\subset\hat{\mathcal{C}}$, $\Sigma$
admits a Whitehead group (see Definition 2.1.2).

###### Proof.

(A0) of Definition 2.1.2 is obvious as FSAEs contain all isomorphism and are
closed under composition by definition. Similiarly, as cofibrations are closed
under pushouts and FSAEs are closed under pushouts by definition, we
immediately obtain (A1). It remains to show (A2). So let $f,g:X\to Y$ be two
cofibrations of simplicial sets, such that $Q(f)=Q(g)$. Then, in particular,
$[f]=[g]$. By Proposition 2.2.30 and the fact that (finite) FSAEs are stable
under composition we may hence assume without loss of generality that $f$ and
$g$ are elementary simplicially homotopic through a simplicial homotopy $H$.
By Proposition 2.2.35 (v), we then have a commutative diagram

${X}$${M_{f}}$${M_{g}}$${\hat{M}_{H}}$$\scriptstyle{s_{f}}$$\scriptstyle{s_{g}}$

in $\mathcal{C}$, with $s_{f}$ and $s_{g}$ FSAEs. Now, extend this diagram as
follows:

${X}$${Y}$${M_{f},}$${M_{g}}$${Y}$${Y\otimes\Delta^{1}}$${\hat{M}_{H}}$${Y\otimes\Delta^{1}}$${R_{f}}$${R_{g}}$${R}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{i_{1}}$$\scriptstyle{s_{f}}$$\scriptstyle{s^{\prime}_{f}}$$\scriptstyle{s^{\prime}_{g}}$$\scriptstyle{s_{g}}$$\scriptstyle{i_{1}}$

All the squares ending in $R$s are obtained by pushing out. The upper left and
upper right square commute by definition. Now, as both $f$ and $g$ are
cofibrations, we have that $s^{\prime}_{f}$ and $s^{\prime}_{g}$ are FSAEs, by
Proposition 2.2.35 (iv). Furher $i_{1}$ is an FSAE by Proposition 2.2.35 (iv).
In particular, by stability under pushouts, every arrow, but the ones in the
first row is an FSAE. Hence, by stability under composition and commutativity
of the diagram, we have shown (A2). ∎

Thus Theorem 2.3.4, we can apply Theorem 2.1.3 together with Proposition 2.3.3
to obtain two functors

$\displaystyle\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}\cong\mathcal{C}(\Sigma^{-1})$
$\displaystyle\xrightarrow{A}\textnormal{{MonAb}};$ (2.5)
$\displaystyle\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}\cong\mathcal{C}(\Sigma^{-1})$
$\displaystyle\xrightarrow{E}\textnormal{{Ab}}.$ (2.6)

We give a more detailed analysis of these two functors in Section 2.3.2. It
can also be helpful to go back to Theorem 2.1.3, where the basic constructions
underlying $A$ and $E$ are listed. We are now in shape to finally define the
Whitehead group of a finite filtered simplicial set.

######  Definition 2.3.5.

Let Let $X\in\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.

  * •

Denote by $A_{P}$ the functor in (2.5).
$A_{P}(X)$ is called the Whitehead monoid of the filtered simplicial set $X$.

  * •

Denote by $Wh_{P}$ the functor in (2.6).
$Wh_{P}(X)$ is called the Whitehead group of the filtered simplicial set $X$.

  * •

For a morphism $\alpha:X\to Y$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$, let
$\tau_{P}(\alpha)\in A_{P}(X)$ ($...\in Wh_{P}(X)$ if $\alpha$ is an
isomorphism) be the simple morphism class of the morphism in
$\mathcal{C}(\Sigma^{-1})$ corresponding to $\alpha$.
$\tau_{P}(\alpha)$ is called the Whitehead torsion of $\alpha$.

  * •

For a morphism $f:X\to Y$ in $\textnormal{s{Set}}_{P}$, define
$\tau_{P}(f):=\tau_{P}([f])\in A_{P}(X)$ ($...\in Wh_{P}(X)$ if $f$ is a weak
equivalence).
$\tau_{P}(f)$ is called the Whitehead torsion of $f$.

In the following subsection, we investigate some of the properties of
$A_{P},Wh_{P}$ and $\tau_{P}$. In particular, we give a more explicit
description of their functoriality for morphisms in $\textnormal{s{Set}}_{P}$.
It turns out, in many ways they behave just like the classical theory. This of
course stems from the fact, that most of what we have done here is a
formalization of the arguments used there. More generally, it seems that if
one keeps really good track of what we are actually using, our argument should
generalize to any cofibrantly generated combinatorial simplicial model
category, fulfilling an analogue of Proposition 2.2.30 for compact objects,
and also the analogue of Proposition 2.2.33.

####  2.3.2 Properties of $Wh_{P}$, $A_{P}$, $\tau_{P}$ and of simple
equivalences

The astute reader will have noticed that, while we have spent a great deal of
time talking about FSAEs - the filtered simplicial analogue to expansions in
the classical setting - we have yet to define what a simple equivalence is, in
the setting of finite filtered simplicial sets. This is because to really say
anything interesting about these objects, it is helpful to already have the
theory of Theorem 2.1.3 in place. Showing that one can use FSAEs of finite
filtered simplicial sets to construct a Whitehead group (a Whitehead monoid)
and a Whitehead torsion has been the content of the last subsection, of
Proposition 2.3.3, Theorem 2.3.4 and Definition 2.3.5 to be more precise. Now,
that these are set up, we can start our investigation of simple equivalences.
It can be of interest to compare the results in this section to the ones found
for example in [Whi39] for a simplicial complex setting, [Whi50], [Coh73] for
a CW setting and [KP86] for an abstract setting, which they and their proofs
are clearly inspired by.

First, we use Proposition 2.3.3 together with Definition 2.1.1 to define
simple equivalences of filtered simplicial sets. Recall that we are in the
setting of Section 2.1 with
$\hat{\mathcal{C}}=\textnormal{s{Set}}_{P}^{\textnormal{fin}}$, $\mathcal{C}$
the subcategory given by cofibrations in $\hat{\mathcal{C}}$ and $\Sigma$ the
class of FSAEs in $\hat{\mathcal{C}}$ and that we have a (canonical)
isomorphism of categories
$\mathcal{C}(\Sigma^{-1})\cong\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$,
by Proposition 2.3.3.

######  Definition 2.3.6.

  * •

A morphism in $\alpha$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is called a simple
equivalence if and only if the morphism in $\mathcal{C}(\Sigma^{-1})$
corresponding to $\alpha$ is a simple morphism. That is, if it is a finite
composition of morphisms of shape $[s]$ or $[s]^{-1}$, where $s$ is an FSAE in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.

  * •

Two objects in $X,Y\in\mathcal{H}\textnormal{s{Set}}_{P}$ are said to have the
same simple homotopy type or also to be simply equivalent if there is a simple
equivalence $X\xrightarrow{\alpha}Y$.

  * •

A morphism in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is called a simple
equivalence if $[f]$ is a simple equivalence.

  * •

We say two morphisms $\alpha,\beta$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ with the same source
have the same simple morphism class if their corresponding morphism in
$\mathcal{C}(\Sigma^{-1})$ have the same simple morphism class. That is, if
there is a simple equivalence $\sigma$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ such that
$\alpha=\sigma\circ\beta$. The equivalence class generated by this relation is
called the simple morphism class of $\alpha$, denoted by
$\langle\alpha\rangle$.

  * •

The simple morphism class of a morphism $f$ in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is defined to be the simple
morphism class of $[f]$, denoted by $\langle f\rangle$.

  * •

A zigzag in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$

$X=X^{0}\leftrightarrow...\leftrightarrow X^{n}=Y$

with all arrows FSAEs is called a deformation between $X$ and $Y$. We also say
$X$ deforms into $Y$ and vice versa.

  * •

A commutative diagram in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$

$\leavevmode\hbox to107.72pt{\vbox
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where the verticals are cofibrations, and the lower horizontal is a
deformation from $Y_{0}$ to $Y_{n}$ is called a deformation from $a_{0}$ to
$a_{n}$. We also say $a_{0}$ deforms into $a_{n}$ and vice versa.

Clearly, every finite FSAE is a simple equivalence. Also, by definition, any
deformation induces a simple equivalence, by taking the zigzag as a morphism
in $\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. In particular, we
obtain a vast class of examples of simple equivalences, by using the results
in Proposition 2.2.35 together with the fact that simple equivalences fulfill
a two out of three rule, by definition.

######  Example 2.3.7.

Let $X\xrightarrow{f}X^{\prime}$ be a morphism and $X\xhookrightarrow{a}Y$ be
a cofibration in $\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.
Denote by $i_{X},i_{X^{\prime}}$ the respective inclusions into the mapping
cylinder $M_{f}$. Then the following arrows are simple equivalences.

  1. (i)

$X\otimes\Delta^{1}\xrightarrow{p_{X}}X$

  2. (ii)

$M_{f}\xrightarrow{p_{X}^{\prime}}X^{\prime}$

  3. (iii)

$M_{f}\cup_{i_{X^{\prime}},a}Y\xrightarrow{p_{X^{\prime}}\cup
a}X^{\prime}\cup_{f,a}Y$

We have illustrated an example of the third in Figure 2.12.

Figure 2.12: An example of the simple equivalence in (iii) for $P=\\{0,1\\}$
with the $0$-stratum marked in blue and the $1$-stratum marked in red. The
pushout square is to the left, the simple equivalence to the right.

The first two are an immediate consequence of the results in Proposition
2.2.35 and the two out of three law. For the third one …

###### Proof.

…, consider first the pushout square:

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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.68791pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
19.99718pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\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}}&\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\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{$\scriptstyle{i_{0}}$}
}}\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}{31.99716pt}{-15.16862pt}\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}{34.34993pt}{-8.31857pt}\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{$\scriptstyle{\phi}$}
}}\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{-27.37718pt}{-25.36607pt}\pgfsys@lineto{-4.17717pt}{-25.36607pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.97719pt}{-25.36607pt}\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>
As the left vertical is an FSAE, so is the right. Secondly, consider the
pushout square

${X\otimes\Delta^{1}\cup_{g,i_{1}}Y}$${M_{f}\cup_{i_{X^{\prime}},a}Y}$${Y\otimes\Delta^{1}}$${X^{\prime}\cup_{i_{0}\circ
f,a}Y\otimes\Delta^{1}}$$\scriptstyle{\psi}$

Again, as by $g$ is a cofibration, by Proposition 2.2.33 the left vertical and
hence also the right vertical is an FSAE. Now, consider the triangle diagram:

$\leavevmode\hbox to208.11pt{\vbox
to55.99pt{\pgfpicture\makeatletter\hbox{\hskip
104.05272pt\lower-27.9972pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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23.7464pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${M_{f}\cup_{i_{X^{\prime}},a}Y}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
23.7464pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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}\hbox{${X^{\prime}\cup_{f,a}Y}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.73198pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 59.57434pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{${X^{\prime}\cup_{i_{0}\circ f,a}Y\otimes\Delta^{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
35.57437pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-56.35992pt}{11.7556pt}\pgfsys@lineto{61.9888pt}{11.7556pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.18379pt}{15.91948pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{p_{X^{\prime}}\cup a}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.94151pt}{-13.91356pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\psi}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{61.76924pt}{1.75673pt}\pgfsys@lineto{24.92604pt}{-15.33145pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.90717}{0.42075}{-0.42075}{0.90717}{61.76924pt}{1.75673pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.90717}{-0.42075}{0.42075}{-0.90717}{24.74463pt}{-15.41557pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.17212pt}{-13.78244pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\phi}$}
}}\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>(2.7)

It is clearly not commutative in $\textnormal{s{Set}}_{P}$. However, a left
inverse to $\phi$ and hence an inverse in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is given by the
collapsing the cylinder map, $X^{\prime}\cup_{i_{0}\circ
f,a}Y\otimes\Delta^{1}\to X^{\prime}\cup_{f,a}Y$. The latter fits into a
commutative diagram:

$\leavevmode\hbox to208.02pt{\vbox
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104.00864pt\lower-27.9972pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{${M_{f}\cup_{i_{X^{\prime}},a}Y}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
23.7464pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
35.53029pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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0.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
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In particular, (2.7) commutes in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. Hence, by the two out
of three property, we obtain that
$M_{f}\cup_{i_{X^{\prime}},a}Y\xrightarrow{p_{X^{\prime}}\cup
a}X^{\prime}\cup_{f,a}Y$ is a simple equivalence. ∎

We begin with a few statements that are immediate consequences of the
isomorphism of categories
$\mathcal{C}(\Sigma^{-1})\cong\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$
and the results in Section 2.1. For the following, let
$X\in\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.

######  Remark 2.3.8.

$\tau_{P}$ induces a one to one correspondence between the simple morphism
classes with source $X$ and $A_{P}(X)$. In particular, $\alpha:X\to Y$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is a simple
equivalence if and only if $\tau_{P}(\alpha)=0$. Due to this, we do not
distinguish between elements of $A_{P}(X)$ and simple morphism classes (of
morphism with source $X$) and for the sake of notational brevity, we write
$\langle f\rangle$ instead of $\tau_{P}(f)$ for the remainder of this
subsection.

By Lemma 2.1.5 every morphism in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is of the shape
$[s]^{-1}\circ[a]$ where $a$ is a cofibration and $s$ is a FSAE. Hence, every
simple morphism class in $A_{P}(X)$ is of the shape $\langle a\rangle$ for
some cofibration $a:X\xhookrightarrow{}Y$ in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. By Lemma 2.1.7, two such
cofibrations have the same simple morphism class if and only if they differ up
to a deformation. In particular, we can think also think of $A_{P}(X)$ as
cofibrations in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ with source $X$
modulo deformation.

By the same argument as we used for the proof of Lemma 2.1.5 (i.e. stability
of FSAEs under pushout and composition), every deformation between object or
morphism can be replaced by one that is of the shape $s:X\to
S\xleftarrow{}Y:t$ with $s$ and $t$ FSAEs.

######  Remark 2.3.9.

As a morphism in a model category is a weak equivalence if and only if it
induces an isomorphism in the homotopy category, every simple equivalence in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is a weak equivalence. In
particular, $f:X\to Y$ in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ is a
weak equivalence if and only if $\tau_{P}(f)$ ends up in $Wh_{P}(X)$. Hence,
we can think of $Wh_{P}(X)$ as the simple morphism classes of weak
equivalences, with source $X$ or alternatively by Remark 2.3.8 as acyclic
cofibrations with source $X$ modulo deformation.

Just as in the classical scenario, one then has the following
characterizations of simple equivalences.

######  Proposition 2.3.10.

Let $f:X\to Y$ be a morphism in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.
Then the following are equivalent:

  1. (i)

$f$ is a simple equivalence.

  2. (ii)

$[f]=[t]^{-1}\circ[s]$ for $s,t$ finite FSAEs.

  3. (iii)

$\tau_{P}(f)=0$

  4. (iv)

$X\hookrightarrow M_{f}$ is a simple equivalence.

###### Proof.

The first three statements are equivalent by Remark 2.3.8 and Lemma 2.1.6.
Clearly, weak equivalences fulfill a two out of three property by definition.
Hence, by mapping cylinder factorization and Proposition 2.2.35 (the fact that
$Y\hookrightarrow M_{f}$ is an FSAE), we obtain that (i) is equivalent to
(iv). ∎

Next up, we study the functoriality of $A_{P}$ and of $Wh_{P}$. By a slight
abuse of notation, we denote any functoriality of the above by a $(-)_{*}$.
That is, we use $f_{*}$ for $A_{P}([f]),Wh_{P}([f])$, for $f$ in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ and $\alpha_{*}$ for
$A_{P}(\alpha),Wh_{P}(\alpha)$ for $\alpha$ in
$\mathcal{H}\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. What is meant, will
usually be obvious from context. So far, we really only have an explicit
description of $\alpha_{*}\langle\beta\rangle$ after a choice of cofibrations
$a$ and $b$ such that $[a]=\alpha$ ,$[b]=\beta$. By Theorem 2.1.3,
$\alpha_{*}(\langle\beta\rangle$) is then given by $\langle
b^{\prime}\rangle$, where $b^{\prime}$ is a pushout of $b$ along $a$ in
$\textnormal{s{Set}}_{P}$. As is usually the case, for pushouts to interact
well with homotopy properties, it suffices for one of the two maps to be a
cofibration (and the objects to be cofibrant). This is reflected in
Proposition 2.3.12. It is very reminiscent of the setting of homotopy pushouts
(see for ex. [KP86]), and also reflects in (iii) of Example 2.3.7. Before we
prove this proposition, it is useful to know the following.

######  Lemma 2.3.11.

Let $X^{\prime}\xrightarrow{s}X$ be an FSAE in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ and $\langle\alpha\rangle\in
A_{P}(X)$. Then

$(s_{*})^{-1}\langle\alpha\rangle=\langle\alpha\circ[s]\rangle$

###### Proof.

By Remark 2.3.8, we may without loss of generality assume that $\alpha=[a]$,
for a cofibration $a$. Now, use Lemma 2.1.9 to obtain:

$s_{*}\langle a\circ s\rangle=s_{*}\langle s\rangle+\langle a\rangle=\langle
a\rangle.$

In particular, as $s_{*}$ is a bijection this shows the result. ∎

######  Proposition 2.3.12.

Let $X\in\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ and $f,g$ morphisms in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ with source $f$. Consider a
pushout square in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$

${X}$${X^{\prime}}$${Y}$${Y^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{g^{\prime}}$

and denote by the diagonal by $d$. If either $f$ or $g$ is a cofibration,
then:

$f_{*}\langle g\rangle=\langle g^{\prime}\rangle,$

and

$\langle f\rangle+\langle g\rangle=\langle d\rangle.$

###### Proof.

We start with the case where $g$ is a cofibration. Denote by $i_{X^{\prime}}$
and $i_{X}$ the respective inclusions into $M_{f}$. Then
$[f]=[i_{X}^{\prime}]^{-1}\circ[i_{X}]$. Now, consider the pushout composition
diagram

${X}$${M_{f}}$${X^{\prime}}$${Y}$${M_{f}\cup_{i_{X},g}X^{\prime}}$${Y^{\prime}}$$\scriptstyle{i_{X}}$$\scriptstyle{g}$$\scriptstyle{p_{X^{\prime}}}$$\scriptstyle{g^{\prime}}$
(2.8)

Then as $[p_{X^{\prime}}]=[i_{X^{\prime}}]^{-1}$, by Lemma 2.3.11

$f_{*}\langle g\rangle=\langle
X^{\prime}\xhookrightarrow{i_{X^{\prime}}}M_{f}\hookrightarrow
M_{f}\cup_{i_{X},g}Y\rangle.$

In Example 2.3.7 we have shown that since $g$ is a cofibration

$M_{f}\cup_{i_{X},g}Y\to Y^{\prime}=X^{\prime}\cup_{g,f}Y$

is a simple morphism. Hence, by commutativity of (2.8), we obtain:

$\displaystyle f_{*}\langle g\rangle$ $\displaystyle=\langle
X^{\prime}\xhookrightarrow{i_{X}^{\prime}}M_{f}\hookrightarrow
M_{f}\cup_{i_{X},g}Y\rangle$ $\displaystyle=\langle
X^{\prime}\xrightarrow{[p_{X^{\prime}}]^{-1}}M_{f}\to M_{f}\cup_{i_{X},g}Y\to
X^{\prime}\cup_{g,f}Y\rangle$ $\displaystyle=\langle g^{\prime}\rangle.$

For the proof of the case where $f$ is a cofibration, again we refer to (2.8)
but with the roles of $f$ and $g$ swapped. In this case, the result follows
from composability of pushouts and again the fact that if $f$ is a
cofibration, then

$M_{g}\cup_{i_{X},f}X^{\prime}\to Y^{\prime}=Y\cup_{f,g}X^{\prime}$

is a simple equivalence (Example 2.3.7). By the same fact, we also have that
the diagonal of the left hand square in (2.8) only differs from the diagonal
of the rectangle by a simple equivalence, showing the addition formula. ∎

As two immediate corollaries of this and Proposition 2.3.10 we obtain:

######  Corollary 2.3.13.

Simple equivalences are stable under pushout along cofibrations in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. Simple equivalences that are
also cofibrations are stable under pushouts along arbitrary morphisms in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$.

######  Corollary 2.3.14.

Let $f:X\to Y$ and $g:Y\to Z$ be morphisms in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$. Then

$f_{*}\langle g\circ f\rangle=f_{*}\langle f\rangle+\langle g\rangle.$

###### Proof.

Take a mapping cylinder factorization of $f$, $f=p_{Y}\circ i_{X}$, and let
$i_{Y}$ be the inclusion of $Y$ into $M_{f}$. Then set $\hat{g}:=g\circ
p_{Y}$. Now, using Proposition 2.3.12, just as in Lemma 2.1.9 one obtains:

$(i_{X})_{*}\langle\hat{g}\circ i_{X}\rangle=(i_{X})_{*}\langle
i_{X}\rangle+\langle\hat{g}\rangle.$

Finally, applying $(p_{Y})_{*}=(i_{Y})_{*}^{-1}$ to the last equation gives:

$\displaystyle f_{*}\langle g\circ f\rangle=f_{*}\langle\hat{g}\circ
i_{X}\rangle$ $\displaystyle=f_{*}\langle
i_{X}\rangle+(p_{Y})_{*}\langle\hat{g}\rangle$ $\displaystyle=f_{*}\langle
p_{Y}\circ i_{X}\rangle+(i_{Y})_{*}^{-1}\langle g\circ p_{Y}\rangle$
$\displaystyle=f_{*}\langle f\rangle+\langle g\circ p_{Y}\circ i_{Y}\rangle$
$\displaystyle=f_{*}\langle f\rangle+\langle g\rangle,$

where the second equality follows from the fact that $p_{Y}$ is a simple
equivalence, and the third equality follows from Lemma 2.3.11. ∎

######  Corollary 2.3.15.

Let $X^{\prime}\xrightarrow{s}X$ be a simple equivalence in
$\textnormal{s{Set}}_{P}^{\textnormal{fin}}$ and $\langle\alpha\rangle\in
A_{P}(X)$. Then

$(s_{*})^{-1}\langle\alpha\rangle=\langle\alpha\circ[s]\rangle$

###### Proof.

The proof is identical to the one of Lemma 2.3.11, but this time using
Corollary 2.3.14. ∎

The results in Corollary 2.3.13 is of course reminiscent of the interaction of
cofibrations and weak equivalences in a cofibrant model category. This makes
on hope for analogue to the cube lemma (see for example [Hov07, Lem. 5.2.6]).
In the classical setting, such a result follows from the sum theorem [Coh73,
Prop. 23.1]. In the filtered case one can argue similarly. In fact, we now
show the following result:

######  Proposition 2.3.16.

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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.73195pt}{-10.40123pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\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@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (2.9)

in $\textnormal{s{Set}}_{P}^{\textnormal{fin}}$, where the left and the right
face are cocartesian, the upper morphisms in these squares are cofibrations
and $h_{0}$ is a weak homotopy equivalence. Denote by $f_{0}$ the diagonal in
the left face. Then:

$\langle h\rangle=f_{1*}\langle h_{1}\rangle+f_{2*}\langle h_{2}\rangle-
f_{0*}\langle h_{0}\rangle.$

###### Proof.

We first reduce to the case where all maps in the left cocartesian face are
cofibrations. Using mapping cylinder factorization on the left vertical and
factoring this pushout face we obtain a commutative diagram:

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}\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}{-27.9326pt}{-51.54651pt}\pgfsys@invoke{
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}\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}},$ (2.10)

where $\tilde{f}_{1},\tilde{f}_{2}$ are cofibrations, as they are pushouts of
cofibrations, and $s$ is a simple equivalence. By Corollary 2.3.13,
$s^{\prime}$ is also a simple equivalence. Denote by $\tilde{f}_{0}$ the
diagonal going from $X_{0}$ to $M^{\prime}$. Assuming we have already shown
the formula in the two cofibration case, we then obtain:

$\langle\tilde{h}\rangle=\tilde{f}_{1*}\langle
h_{1}\rangle+\tilde{f}_{2*}\langle\tilde{h}_{2}\rangle-\tilde{f}_{0*}\langle
h_{0}\rangle$ (2.11)

Now, by Corollary 2.3.15 and commutativity of the diagram, we have

$\displaystyle\langle\tilde{h}\rangle$
$\displaystyle=(s^{\prime})^{-1}_{*}\langle h\rangle;$
$\displaystyle\langle\tilde{h}_{2}\rangle$ $\displaystyle=(s)^{-1}_{*}\langle
h_{2}\rangle.$

Together with (2.11) we obtain:

$\displaystyle\langle h\rangle$
$\displaystyle=s^{\prime}_{*}(\tilde{f}_{1*}\langle
h_{1}\rangle+\tilde{f}_{2*}\langle\tilde{h}_{2}\rangle-\tilde{f}_{0*}\langle
h_{0}\rangle)$ $\displaystyle=(s^{\prime}\circ\tilde{f}_{1})_{*}\langle
h_{1}\rangle+(s^{\prime}\circ\tilde{f}_{2})_{*}\langle\tilde{h}_{2}\rangle-(s^{\prime}\circ\tilde{f}_{0})_{*}\langle
h_{0}\rangle$ $\displaystyle=f_{1*}\langle
h_{1}\rangle+f_{2*}(s_{*}\langle\tilde{h}_{2}\rangle)-f_{0*}\langle
h_{0}\rangle$ $\displaystyle=f_{1*}\langle h_{1}\rangle+f_{2*}\langle
h_{2}\rangle-f_{0*}\langle h_{0}\rangle.$

So it remains to show the case where all the maps in the left square are
cofibrations. We proceed similarly to the proof of [Coh73, Prop. 23.1]. To
make simplify notation we omit the subscript of the pushout cups in the
remainder of the proof, wherever possible. Denote by $\phi$ the morphism
$X\hookrightarrow X\cup_{X^{0},i_{1}}M_{h_{0}}=X\cup
X^{0}\otimes\Delta^{1}\cup Y^{0}$. $\phi$ is the bottom horizontal composition
in the commutative diagam:

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