$f^{0}:\textnormal{sd}^{\Sigma}(K\textnormal{ rel }A)^{(0)}\longrightarrow
L^{(0)}$

such that the following conditions hold. Let
$K^{\prime}:=\textnormal{sd}^{\Sigma}(K\textnormal{ rel }A)$. Then:

  1. (i)

For $x\in K^{\prime(0)}$, we have $p_{K^{\prime}}(x)\leq p_{L}(f^{0}(x))$.

  2. (ii)

For $p\in P$ and $x\in K^{\prime(0)}_{\leq p}\setminus A^{(0)}$, we have
$\phi(|\overline{\textnormal{star}}_{K^{\prime}}(x)\cap
B_{A,p}|)\subset|\textnormal{star}_{L}(f^{0}(x))|$.

  3. (iii)

$f^{0}$ agrees with $f_{A}$ on $A^{(0)}$.

Before we prove this proposition, we show that it implies Theorem 1.3.25 to
motivate the conditions a little more.

######  Corollary 1.3.29.

$f^{0}$ as in Proposition 1.3.28 extends to a morphism of filtered simplicial
complexes $f:K^{\prime}\to L$ fulfilling:

  1. (i)

For $p\in P$ and $\sigma\in(K^{\prime}-A)_{\leq p}$,
$\phi(|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap
B_{A,p}|)\subset|\textnormal{star}_{L}(f(\sigma))|$.

  2. (ii)

$f|_{A}=f_{A}$.

In particular, Theorem 1.3.25 holds.

###### Proof of Corollary 1.3.29.

What we have to show is that $f^{0}$ extends to a map of simplicial complexes;
that is, that for $\sigma\in K^{\prime}$ $f^{0}(\sigma)$ is a simplex in $L$
and that furthermore (by Remark 1.3.16) for each $x\in K^{\prime(0)}$ we also
have

$p_{K^{\prime}}(x)\geq p_{L}{(f(x))}.$

For the first condition, note that, by the equivalent characterization of when
a join of simplices is a simplex in $K$ (Remark 1.3.6), it suffices to show
that certain intersections of stars in $L$ are non-empty. We start with the
cases where $\sigma\in A$ or $\sigma\in K^{\prime}-A$. For the former, this is
obvious as we already know that $f^{0}$ extends to a simplicial map on $A$.
For the latter case, let $p=\textnormal{max}(p_{K^{\prime}}(\sigma))$. Then
$\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap B_{A,p}$ is non-empty.
Hence, (using from Remark 1.3.6 (i) and (ii)), together with (i) we also
obtain that $\bigcap_{x_{i}\in\sigma}|\textnormal{star}(f^{0}(x_{i}))|$ is
non-empty, hence by Remark 1.3.6 that $f^{0}(\sigma)$ is a simplex of $L$. It
remains to show the case where the simplex lies in neither of the two. Since
$A$ is a full subcomplex of $K^{\prime}$, such a simplex is of the shape
$\sigma\star_{K^{\prime}}\tau$, with $\tau\in K^{\prime}-A$ and $\sigma\in A$.
Now, let $x\in\tau$ be a vertex, maximal with respect to $p_{K^{\prime}}$ and
$p:=p_{K^{\prime}}(x)$. As we have just seen before,

$\phi(|x|)\in|\textnormal{star}_{L}(f^{0}(\tau))|.$

At the same time

$|x|\in|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)|=|\overline{\textnormal{star}}_{K}(\sigma)|.$

Therefore, by the assumption (b) of Theorem 1.3.25,

$\phi(|x|)\in|\overline{\textnormal{star}}_{L}(f^{0}(\sigma))|.$

In particular, $|{\textnormal{star}}_{L}(f^{0}(\sigma))|$ and
$|\overline{\textnormal{star}}_{L}(f^{0}(\tau))|$ have non-empty intersection.
Hence,
$f^{0}(\sigma)\star_{L}f^{0}(\sigma)=f^{0}(\sigma\star_{K^{\prime}}\tau)$ is a
simplex of $L$ and thus $f^{0}$ extends to a map of simplicial complexes.

Now, preservation of strata is an immediate consequence of Lemma 1.3.8 and
(ii) of Proposition 1.3.28 and the fact that $\phi$ preserves strata. ∎

###### Proof of Proposition 1.3.28.

We define $\Sigma$ and $f$ inductively. As only values of $p^{\prime}$ lesser
or equal to $p$ occur in the definition of $\textnormal{sd}^{\Sigma}(-)$ it
makes sense to talk about the latter, even if $\Sigma$ is only defined up to
$p$. Assume inductively that we already have defined $\Sigma$ up to $p\in P$
and set $K^{p}:=\textnormal{sd}^{\Sigma_{\leq q}}(K\textnormal{ rel }A)$.
Denote by $B^{p}_{r,A}$ the subcomplex of $K^{p}$ given by $K^{p}-(K^{p}_{\leq
r}\cup A)$, for $r,p\in P$. Further, assume we have already defined

$g^{r}:K^{r,{(0)}}\longrightarrow L^{(0)}$

, for $0\leq r\leq p$ such that $g^{0}$ agrees with $f_{A}$ on $A^{(0)}$, and
that the following hold:

  1. (i)r

For $x\in\big{(}K_{r}^{r}\big{)}^{(0)}$, we have $p_{K^{r}}(x)\leq
p_{L}(g^{p}(x)).$

  2. (ii)r

For $x\in\big{(}K^{r}-A\big{)}^{(0)}$, we have
$\phi(|\overline{\textnormal{star}}_{K^{r}}(x)\cap
B^{r}_{A,r}|)\subset|\textnormal{star}_{L}(g^{r}(x))|.$

  3. (iii)r

$g^{r}$ agrees with $g^{r-1}$ on $K^{r,(0)}_{\leq r-1}\cup
A^{(0)}=K^{r-1,(0)}_{\leq r-1}\cup A^{(0)}$, for $r\geq 1$.

We be drop the $(0)$ superscript for the $0$ skeleton from here on out as it
will be clear when we mean a vertex or a positively dimensional simplex from
usage of $x,y$ vs $\sigma,\tau$ respectively. For $p=0$ consider the open
covering of $|K-A|$ given by pulling back the covering of $|L|$ by the open
stars $|\textnormal{star}_{L}(y)|$, $y\in L$. By Remark 1.3.20, for some
sufficiently large $\Sigma_{0}$ we have that for each
$x\in\textnormal{sd}^{\Sigma_{0}}(K-A)$ there exists a $y\in L$ such that

$\phi(|\overline{\textnormal{star}}_{\textnormal{sd}^{\Sigma_{0}}(K-A)}(x)|)\subset\textnormal{star}_{L}(y).$

Set $g^{0}(x)$ to such a $y$, for $x\in\textnormal{sd}^{\Sigma_{0}}(K-A)$ and
to $f_{A}(x)$ for $x\in A$. Then clearly the conditions (i)r, (ii)r and
(iii)r, for $r=0$, are satisfied.

For the following $p$, no new vertices are added to $|\mathring{N}(K,A)|$ and
by (iii)r we keep $g^{p}=f_{A}$ on $A^{(0)}$. Hence, on $A$ the condition (i)r
is satisfied by assumption. As (ii)r does not depend on what is happening on
$|A|$, we may as well replace $K$ by $K-A$ and hence assume $A=\emptyset$ to
make notation a little cleaner.

For the inductive step $p$ to $p+1$, let $C$ be the subcomplex given by such
simplices $\sigma\in K^{p}$ where
$\phi(|\sigma|)\subset|\mathring{N}(L,L_{\leq p})|-|L_{\leq p}|$. Since $\phi$
is a stratum preserving map, $C\subset B^{p}_{p+1}$. Then, by Remark 1.3.13,
specifically (iv) of the latter, $|C|$ is covered by the pulled back cover
given by open sets of shape $|\textnormal{star}_{L}(\\{y_{p},y\\})|$, for a
$1$-simplex $\\{y_{p},y\\}\in L$ with $y_{p}\in L_{\leq p}$ and
$y\in\partial(N(L,L_{\leq p})$. Hence, for some sufficiently large
$\Sigma_{p+1}$, we have that for every
$x\in\textnormal{sd}^{\Sigma_{p+1}}(C)$, there are $y_{p}\in L_{p}$ and a
$y\in\partial N(L,L_{p})$ such that $\\{y_{p},y\\}\in L$ and

$\displaystyle\phi(|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap\textnormal{sd}^{\Sigma_{p+1}}(C)|)\subset\textnormal{star}_{L}(\\{y_{p},y\\})\subset\textnormal{star}_{L}(y).$
(1.11)

This defines $K^{p+1}$. Now, for $g^{p+1}$, consider the following cases:

  1. (a)

$x\in K^{p+1}_{\leq p}=K^{p}_{\leq p}$:

$g^{p+1}(x):=g^{p}(x).$

  2. (b)

$x\in B^{p+1}_{p+1}$, $|x|\in\mathring{\sigma}$ for $\sigma\in K^{p}$, and
$\sigma$ has a least one vertex, $x_{0}\in\sigma$, such that
$p_{L}\big{(}g^{p}(x_{0})\big{)}>p$:

$g^{p+1}(x):=g^{p}(x_{0}),$

for such a $x_{0}$.

  3. (c)

Neither are the case. Hence, $x\in B^{p+1}_{p+1}$. Take the $\sigma\in K^{p}$
such that $|x|\in\mathring{\sigma}$. Then $|\sigma|\subset|B^{p+1}_{p+1}|.$
and $g^{p}(\sigma)\subset L_{\leq p}$. As no new vertices were added to
$|K^{p}|$ outside of $|B^{p+1}_{p+1}|$, we have that
$|B^{p+1}_{p+1}|=|B^{p}_{p+1}|$. In particular, $\sigma\in B^{p}_{p+1}\subset
B^{p}_{p}$. Hence, by Remark 1.3.6, specifically (i) and (ii) there together
with (ii)r for $r=p$:

$\phi(|\sigma|)\subset\phi(|\overline{\textnormal{star}}_{K^{p}}(\sigma)\cap
B^{p}_{p}|)\subset\bigcap_{x_{i}\in\sigma}|\textnormal{star}_{L}(g^{p}(x_{i}))|\subset|\mathring{N}(L,L_{p})|.$

Here, the last inclusion comes from $g^{p}(\sigma)\subset L_{\leq p}$. In
other words, as $\phi$ preserves strata, $\sigma\in C$. Now, set

$g^{p+1}(x):=y$

for a $y$ given by (1.11).

We need to see that this satisfies the inductive conditions ((i)r, (ii)r and
(iii)r), for $r=p+1$. The first condition is immediate by construction. We
have not changed $g^{p+1}$ on vertices in $K^{p}_{p}$ and outside of it we
have systematically set $g^{p+1}(x)$ to vertices in strata of index higher
than $p$. The third condition is also obviously fulfilled.

It remains to show (ii)r. For $x\in K^{p+1}_{\leq p}$, first note that as we
have not added any new vertices to $|\mathring{N}(K^{p},K^{p}_{p})|$,
$|\overline{\textnormal{star}}_{K^{p}}(x)|=|\overline{\textnormal{star}}_{K^{p+1}}(x)|$
and $|B^{p+1}_{p+1}|=|B^{p}_{p+1}|$. Hence, in this case (ii)r for $r=p$ gives
the result. We are left with the case $x\in B^{p+1}_{p+1}$. So, $g^{p+1}(x)$
is either given as in (b) on in (c).

In case (b), let $\sigma$ be the simplex with $|x|\in|\mathring{\sigma}|$ and
$x_{0}\in\sigma$ with $g^{p}(x_{0})\notin L_{\leq p}$. Then
$|\overline{\textnormal{star}}_{K^{p+1}}(x)|\subset|\overline{\textnormal{star}}_{K^{p}}(x_{0})|$
and $|B^{p+1}_{p+1}|=|B^{p}_{p+1}|\subset|B^{p}_{p}|$. Hence,

$\phi(|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap
B^{p+1}_{p+1}|)\subset\phi(|\overline{\textnormal{star}}_{K^{p}}(x_{0})\cap
B^{p}_{p}|)\subset|\textnormal{star}_{L}(g^{p}(x_{0}))|=|\textnormal{star}_{L}(g^{p+1}(x))|.$

In case (c), consider $\sigma$ as in (c). For any vertex $x_{i}$ of $\sigma$
we have $g^{p}(x_{i})\in L_{\leq p}$. In particular, by (ii)r , for $r=p$, and
again Remark 1.3.6 we obtain:

$\phi(|\overline{\textnormal{star}}_{K^{p}}(\sigma)\cap
B^{p}_{p}|)\subset\bigcap_{x_{i}\in\sigma}\textnormal{star}_{L}(x_{i})\subset|\mathring{N}(L,L_{\leq
p})|.$

Hence, as $\phi$ preserves strata,
$|\overline{\textnormal{star}}_{K^{p}}(\sigma)\cap B^{p}_{p+1}|$ is mapped
into $|\mathring{N}(L,L_{\leq p})|\setminus|L_{\leq p}|$. In other words,

$\overline{\textnormal{star}}_{K^{p}}(\sigma)\cap B^{p}_{p+1}\subset C.$

In particular,

$|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap
B^{p+1}_{p+1}|\subset|\overline{\textnormal{star}}_{K^{p}}(\sigma)\cap
B^{p}_{p+1}|\subset|C|$

and thereby

$|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap
B^{p+1}_{p+1}|=|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap\textnormal{sd}^{\Sigma_{\leq
p+1}}C|.$

Hence, by construction of $g^{p+1}(K)$, we have

$\phi(|\overline{\textnormal{star}}_{K^{p+1}}(x)\cap
B^{p+1}_{p+1}|)\subset\textnormal{star}_{L}(\\{y_{p},y\\})\subset\textnormal{star}_{L}(y)=\textnormal{star}_{L}(g^{p+1}(x)),$

for $y$ and $y_{p}$ as in (c).

Next we show that if we set $f^{0}:=g^{q}$ the conditions in the statement of
the proposition are fulfilled. (iii) is obvious by (iii)r and the start of the
induction. For (i), note that by construction for $p\leq q$ we have

$|B^{p}_{A,p}|=|B^{p+1}_{A,p}|=...=|B^{q}_{A,p}|=|B_{A,p}|.$

and for $x\in K^{\prime}_{\leq p}=K^{p}_{\leq p}$:

$|\overline{\textnormal{star}}_{K^{p}}(x)|=|\overline{\textnormal{star}}_{K^{p+1}}(x)|=...=|\overline{\textnormal{star}}_{K^{q}}(x)|=|\overline{\textnormal{star}}_{K^{\prime}}(x)|.$

Hence, (ii) follows by (iii)r together with (ii)r. We are lacking that $f^{0}$
fulfills (i). Using (iii)r and (i)r we immediately obtain that, for each $x\in
K^{\prime}$,

$p_{K^{\prime}}(x)\leq p_{L}(g^{q}(x))=p_{L}(f^{0}(x)).$

This finishes the proof.

∎

To prove Theorem 1.3.24 we are left with showing the following.

######  Proposition 1.3.30.

The stratum preserving simplicial map $K^{\prime}\to L$ from Corollary 1.3.29
is stratified homotopic to $\phi$.

###### Proof.

First, note that by Lemma 1.3.11, whenever the straight line homotopy between
to stratum preserving maps can be constructed it is automatically stratum
preserving. For the straight line homotopy between a continuous map between
realizations of simplicial complexes $\psi:|\tilde{K}|\to|\tilde{L}|$ and the
realization of a simplicial map $g:\tilde{K}\to\tilde{L}$ to exist it suffices
that, for any simplex $\sigma\in\tilde{K}$, the condition

$\psi(\mathring{\sigma})\subset|\overline{\textnormal{star}}_{L}(g(\sigma))|$

is fulfilled. In particular, by (i) of Corollary 1.3.29, we can use the
straight line homotopy on $|K^{\prime}_{q}\sqcup...\sqcup K^{\prime}_{0}|$. We
now inductively extend this homotopy over

$\mathcal{K}^{p}:=|B_{p}\sqcup K^{\prime}_{p-1}\sqcup
K^{\prime}_{p-2}\sqcup...\sqcup K^{\prime}_{0}|,$

going from $p=q$ to $p=0$. We induce over the following assumption. There
exists a stratum preserving homotopy

$F^{p}:\mathcal{K}^{p}\otimes\Delta^{1}\longrightarrow|L|$

between the restrictions of $\phi$ and $|f|$, fulfilling additionally the
following smallness condition. For $r\leq p$, $\sigma\in K^{\prime}_{r}$ and
$\xi\in|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap
B_{r}|\cap\mathcal{K}^{p}$ we also have

$F^{p}(\\{\xi\\}\times[0,1))\subset|\textnormal{star}_{L}(f(\sigma)|.$ (1.12)

Note how, by Lemma 1.3.10 and (i) of Corollary 1.3.29, this condition always
holds on $\mathcal{K}^{q}$ for any $p$ as then

$F^{p}(\\{\xi\\}\times[0,1))=[\phi(\xi),|f|(\xi)),$

which is a half open line segment starting in
$|\textnormal{star}_{L}(f(\sigma)|$. This gives the start of the induction.
Now, from $p+1$ to $p$, note that

$\displaystyle\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}$
$\displaystyle=(|\mathring{N}(K^{\prime},K^{\prime}_{\leq
p})|\setminus|K^{\prime}_{\leq p}|)\cap|B_{p}|$
$\displaystyle=|\mathring{N}(B_{p},K^{\prime}_{p})|\setminus|K^{\prime}_{p}|.$

In particular, by Remark 1.3.13, $\mathcal{K}^{p}-\mathcal{K}^{p+1}$ is given
by the disjoint union of rays $(\alpha,\beta)$ of $N(B_{p},K^{\prime}_{p})$,
with $\alpha\in|K^{\prime}_{p}|$ and
$\beta\in|B_{p}-K^{\prime}_{p}|=|B_{p+1}|$. Now, consider the maps

$\displaystyle\alpha_{K^{\prime}_{p}}:|\mathring{N}(B_{p},K^{\prime}_{p})|\longrightarrow|K^{\prime}_{p}|;$
$\displaystyle\beta_{K^{\prime}_{p}}:|N(B_{p},K^{\prime}_{p})|\setminus|K^{\prime}_{p}|\longrightarrow|B_{p+1}|$

from 1.3.14, sending a $\xi\in\mathcal{K}^{p}-\mathcal{K}^{p+1}$ to the
respective end points of the ray it is contained in. We omit the subscript
$K^{\prime}_{p}$ from here on out. Now, define:

$\displaystyle\lambda:(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times\left[0,2\right]$
$\displaystyle\longrightarrow|L|$ $\displaystyle(\xi,t)$
$\displaystyle\longmapsto\phi((1-t)\alpha(\xi)+t\beta(\xi))\textnormal{ for
$t\leq 1$},$ $\displaystyle(\xi,t)$ $\displaystyle\longmapsto
F^{p+1}(\beta(\xi),t-1)\textnormal{ for $t\geq 1$}.$

Note that for the radial parameter map

$s_{K^{\prime}_{p}}:|\mathring{N}(B_{p},K^{\prime}_{p})|\longrightarrow[0,1]$

from 1.3.14 (where we again omit the subscript from here on out) we have

$\lambda(\xi,s(\xi))=\phi(\xi).$

Now, for a fixed $\xi\in\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}$, let $r\leq
p$ and $\sigma\in K^{\prime}_{p}$ be such that
$\xi\in|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap B_{r}|.$ Then,
as $\xi\in(\alpha(\xi),\beta(\xi))$, we also have that

$(\alpha(\xi),\beta(\xi))\subset|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap
B_{r}|$

and hence that

$[\alpha(\xi),\beta(\xi)]\subset|\overline{\textnormal{star}}_{K^{\prime}}(\sigma)\cap
B_{r}|.$

Consequently, by (i) of Corollary 1.3.29, we obtain

$\phi([\alpha(\xi),\beta(\xi)])\subset\textnormal{star}_{L}(f(\sigma)).$

By the induction hypothesis, we also have

$F^{p+1}(\\{\beta(\xi)\\}\times[0,1))\subset|\textnormal{star}_{L}(f(\sigma))|.$

In particular, we obtain

$\lambda(\\{\xi\\}\times[0,2))\subset|\textnormal{star}_{L}(f(\sigma))|$
(1.13)

and

$\lambda(\\{\xi\\}\times[0,2])\subset|\overline{\textnormal{star}}_{L}(f(\sigma))|.$

Such a $\sigma$ and $r$ always exist as $\xi\in|N(B_{p},K^{\prime}_{p})|$.
$\sigma$ can be taken to be the simplex in $K^{\prime}_{p}$ such that
$\alpha(\xi)\in\mathring{\sigma}$. In this case, we also have

$|f|(\alpha(\xi))\in\mathring{f(\sigma)}.$

In particular, the following map defined by linearly interpolating between
$\lambda(\xi,t)$ and $|f|(\alpha(\xi))$ is well-defined.

$\displaystyle\Lambda:(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times\left[0,2\right]\times[0,1]$
$\displaystyle\longrightarrow|L|$ $\displaystyle(\xi,t,s)$
$\displaystyle\longmapsto(1-s)|f|(\alpha(\xi))+s\lambda(\xi,t)$

Then, by Lemma 1.3.10 and (1.13), we obtain that, for a $\sigma$ as in (1.12),
we always have

$\Lambda(\\{\xi\\}\times[0,2)\times(0,1])\subset|\textnormal{star}_{L}(f(\sigma))|.$
(1.14)

Furthermore, by Lemma 1.3.11, for
$\xi\in\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}$, $(\alpha(\xi),\beta(\xi)]$
lies in the same stratum as $\beta(\xi)$. Denote its index by $p^{\prime}$.
Then, again by Lemma 1.3.11, the fact that $\Lambda$ is constructed by linear
interpolation and the assumption that $F^{p+1}$ is stratum preserving, we
obtain that

$\Lambda(\\{\xi\\}\times(0,2]\times(0,1])\subset|L|_{p^{\prime}}$ (1.15)

Now, let

$\kappa:\nicefrac{{[0,2]\times[0,1]}}{{[0,2]\times\\{0\\}}}\longrightarrow[0,1]\times[0,1],$

such that $\kappa$ restricts to the linear homeomorphisms (where we indicate
the orientation by the ordering of the interval):

$\displaystyle[0,1]\times\\{1\\}$
$\displaystyle\longrightarrow\\{0\\}\times[0,1],$
$\displaystyle[1,2]\times\\{1\\}$
$\displaystyle\longrightarrow[0,1]\times\\{1\\},$
$\displaystyle\\{2\\}\times[1,0]$
$\displaystyle\longrightarrow\\{1\\}\times[1,0],$
$\displaystyle\\{0\\}\times[0,1]$
$\displaystyle\longrightarrow[1,0]\times\\{0\\}.$

As $\Lambda$ is constant along $\\{\xi\\}\times[0,2]\times\\{0\\}$ and
$(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})$ is locally compact (hence
quotients commute with products with the latter) $\Lambda$ descends to a map

$\bar{\Lambda}:(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times\big{(}\nicefrac{{[0,2]\times[0,1]}}{{[0,2]\times\\{0\\}}}\big{)}\longrightarrow|L|$

and then by composing with an inverse to
$1_{\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}}\times\kappa$ induces a map:

$\tilde{\Lambda}:(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times
I^{2}\longrightarrow|L|$

such that:

$\tilde{\Lambda}(\xi,t,s)=\begin{cases}\phi((1-s)\alpha(\xi)+s\beta(\xi))&\text{,
if }t=0\\\ F^{p+1}(\beta(\xi),t)&\text{, if }s=1\\\
(1-s)|f|(\alpha(\xi))+s|f|(\beta(\xi))&\text{, if }t=1\\\
F^{p+1}(\alpha(\xi),t)&\text{, if }s=0\end{cases}$ (1.16)

Now, set $F^{p}$ on $(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times I$ to

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By (1.10) of (1.3.14) and the piecewise linearity of $|f|$, we obtain that
this in fact gives a homotopy between
$\phi|_{\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}}$ and
$|f||_{\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}}$. (For an illustration, see
Figure 1.18). If $\xi$ converges to $\xi_{0}$ in the boundary of
$\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}$, then $s(\xi)$ converges either to
$1$ or to $0$ and $\xi$ to either $\beta(\xi_{0})$ or $\alpha(\xi_{0})$
respectively. Hence, again by (1.16), we obtain a continuous extension of
$F^{p+1}$ to $\mathcal{K}^{p}$. We need to check that this homotopy is in fact
stratum preserving, and that the inductive condition (1.12) holds. Let
$\xi\in\mathcal{K}^{p}\setminus\mathcal{K}^{p+1}$. Under the map

$(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times\Delta^{1}\longrightarrow(\mathcal{K}^{p}\setminus\mathcal{K}^{p+1})\times
I^{2}$

${\xi}\times[0,1)$ maps into $\\{\xi\\}\times[0,1)\times(0,1).$ Under the
inverse of $1\times\kappa$ the latter maps into
$\\{\xi\\}\times(0,2)\times(0,1)\cup[0,1]\times\\{1\\}.$ Hence, the inductive
smallness condition follows by (1.14). Analogously, $\\{\xi\\}\times[0,1]$ is
mapped into $(0,1]\times[0,1]$ which in turn is mapped into
$(0,2]\times(0,1]$. Hence, by (1.15), we conclude that $F^{p}$ is stratum
preserving. ∎

Figure 1.18: An illustration of $\lambda$, $\Lambda$ and $F$, marked
respectively in different colours.

####  1.3.4 Filtered ordered simplicial complexes

Recall that an ordered simplicial complex is a simplicial complex $K$ together
with a linear ordering on every simplex $\sigma\in K$, compatible in the sense
that if $\tau\subset\sigma$, then the ordering on $\tau$ is the restriction of
the one in $\sigma$. We usually write simplices in an ordered simplicial
complex in the shape $\\{x_{0}\leq...\leq x_{k}\\}$. Further, recall that a
map of ordered simplicial complexes is a map of simplicial complexes
restricting to a monotonous map on each simplex. In Example 1.0.2 we defined
the category of $P$-filtered ordered simplicial complexes,
$\textnormal{{sCplx}}^{\operatorname{o}}_{P}$. As we did for the several other
categories of filtered objects we introduced, we begin this subsection by
giving some alternative descriptions of this category.

######  Remark 1.3.31.

Similarly to Remark 1.3.1, a $P$-filtered ordered simplicial complex is
alternatively described as a simplicial complex $K$ together with a map
$p_{K}^{0}:K^{(0)}\to P$ fulfilling some condition. In the non-ordered case,
this condition was that the image of every simplex is a flag in $P$. In the
ordered case, we need to reflect the fact that $p_{K}:K\to\textnormal{sd}(P)$
is a map of ordered complexes. This is equivalently specified by:

$\displaystyle(x<y)\in K\implies p^{0}_{K}(x)\leq p^{0}_{K}(y).$

In other words, this requires that the ordering on $K$ is such that vertices
in lower strata always come before vertices in higher strata. A morphism of
ordered filtered simplicial complexes $f:K\to L$ is equivalently a map of the
underlying ordered simplicial complexes, fulfilling
$p_{K}(x)=p_{L}\big{(}f(x)\big{)}.$ This justifies calling such simplicial
maps stratum preserving.

The advantage of the ordered over the unordered category is that it embeds
fully faithfully into $\textnormal{s{Set}}_{P}$. We use this to apply the
simplicial approximation theorems Theorem 1.3.24 and Theorem 1.3.25 in the
setting of filtered simplicial sets.

######  Construction 1.3.32.

Consider the functor

$\displaystyle S^{o}:\textnormal{{sCplx}}^{\operatorname{o}}$
$\displaystyle\longrightarrow\textnormal{s{Set}}$ $\displaystyle K$
$\displaystyle\longmapsto\big{\\{}[n]\mapsto\textnormal{Hom}_{{\textbf{sCplx}}^{o}}(\Delta^{n},K)\big{\\}}$

with the obvious functoriality on morphisms and $\Delta^{n}$ thought of as the
ordered simplicial complex given by flags of $[n]$.
Simplicial sets of the form $S^{o}(K)$ have the property that every simplex
$\sigma\in X([k])$ is uniquely determined by its multiset of vertices, that is
by $(x_{i})_{i\in[k]}$ allowing for permutations. This follows immediately
from the definition of a map of (ordered) simplicial complexes. It turns out
that this property completely describes simplicial sets that are isomorphic to
$S(K)$, for some $K\in\textnormal{{sCplx}}$.
To see this, denote by $\mathcal{C}$ the full subcategory of simplicial sets
fulfilling the latter property. Consider the functor from sSet to sCplx
constructed as follows. For $X\in\textnormal{s{Set}}$ define a simplicial
complex with vertices $X([0])$, by taking the simplices to be such subsets
$\\{x_{0},...,x_{k}\\}\subset X_{0}$ that are the set of vertices of a simplex
$\sigma\in X([k])$. This construction becomes functorial by checking that for
a map of simplicial set $f:X\to Y$ the induced map $f([0]):X([0])\to Y([0])$
induces a map of simplicial complexes. We obtain a functor

$\displaystyle C:\textnormal{s{Set}}\longrightarrow\textnormal{{sCplx}}.$

In general, the image of this functor does not naturally carry the structure
of an ordered simplicial complex. This is due to the fact that two vertices in
a simplicial set might be connected by two $1$-simplices with different
orientation. However, in the case where $X\in\mathcal{C}$, the relation given
on $X([0])$ by

$x\leq y\iff x=d_{0}(\sigma),y=d_{1}(\sigma)$

for some $\sigma\in X([1])$, turns $C(X)$ into an ordered simplicial complex
as then no two conflicting orders occur on any simplex. For $f:X\to
Y\in\mathcal{C}$, $C(f)$ is then a map of ordered simplicial complexes, with
respect to this additional structure. Hence, we obtain a functor:

$\displaystyle
C^{o}:\textnormal{s{Set}}\longrightarrow\textnormal{{sCplx}}^{\operatorname{o}}.$

We then have the following proposition.

######  Proposition 1.3.33.

In the setting of 1.3.32, the functor $S$ is a fully faithful embedding of
categories. An inverse of its restriction in the image to $\mathcal{C}$ is
given by $C^{o}$.

###### Proof.

By a slight abuse of notation, we denote the factorization of $S$ into
$\mathcal{C}$ also by $S$. It is a straightforward verification that
$1_{\textnormal{{sCplx}}^{\operatorname{o}}}\cong C\circ S$. For the other
composition, we obtain a natural transformation to the identity by the map

$\textnormal{Hom}_{\textnormal{{sCplx}}^{\operatorname{o}}}(\Delta^{n},C^{o}(X))\to
X([n]),$

sending a map $\sigma$ on the left to the unique simplex in $X$ given by the
multiset given by $(\sigma(\\{i\\})_{i\in[n]}$. It suffices to show, this is a
bijection. This is immediate, from the fact that $X\in\mathcal{C}$. ∎

Further, note that $S^{o}(\textnormal{sd}(P))$ is naturally isomorphic to
$N(P)$. In particular, as $\textnormal{{sCplx}}^{\operatorname{o}}_{P}$ and
$\textnormal{s{Set}}_{P}$ are given by over-categories of these respectively,
we obtain:

######  Corollary 1.3.34.

$S^{o}$ from 1.3.32 induces a fully faithful embedding

$\textnormal{{sCplx}}^{\operatorname{o}}_{P}\hookrightarrow\textnormal{s{Set}}_{P}.$

Filtered simplicial sets isomorphic to spaces in the image of this embedding
are precisely those $X\in\textnormal{s{Set}}_{P}$ where every $n$-dimensional
simplex $\sigma:\Delta^{n}\to X$ is uniquely determined by the multiset of its
vertices, that is by the family $(\sigma(\\{i\\}))_{i\in[n]}$ modulo
permutation.

Using this fact, we will often think of filtered ordered simplicial complexes
as filtered simplicial sets. In particular, the following definition makes
sense:

######  Definition 1.3.35.

Let $X\in\textnormal{s{Set}}_{P}$ be such that every $n$-dimensional simplex
$\sigma:\Delta^{n}\to X$ by its’ multiset of vertices. Then, we call $X$ a
$P$-filtered ordered simplicial complex, or ($P$-)FOS-complex for short. We
nearly always omit the $P$.

It is easy to see that this definition is also compatible with realization
functors, i.e. that there are natural isomorphism between

$\mathcal{C}\xrightarrow{C^{o}}\textnormal{{sCplx}}^{\operatorname{o}}_{P}\to\textnormal{{sCplx}}_{P}\xrightarrow{|-|_{P}}\textnormal{{Top}}_{P}$

and

$|-|_{P}:\textnormal{s{Set}}_{P}\to\textnormal{{Top}}_{P}.$

Where the functor
$\textnormal{{sCplx}}^{\operatorname{o}}_{P}\to\textnormal{{sCplx}}_{P}$ is
the forgetful functor forgetting about the orderings on simplices. Hence, we
do not distinguish between these two ways of realizing FOS-complexes, and
denote them by $|-|_{P}$, or just $|-|$, for the sake of notational brevity.

It can also be useful to have the following alternative characterization of an
FOS-complex.

######  Lemma 1.3.36.

A $P$-filtered simplicial set is an FOS-complex if and only if every non-
degenerate simplex is uniquely determined by its set of vertices.

###### Proof.

This has nothing to do with the filtrations, so let us assume they are
trivial. First, note that an FOS-complex has no non-degenerate simplices with
degenerate faces. In particular, every non-degenerate simplex of dimension
$n$-has precisely $n$ different vertices. Thus, being an FOS-complex implies
the other property. Now, let $X$ be a simplicial set satisfying the latter.
Note that for such an $X$ every nondegenerate simplex of dimension $n$ has
precisely $n$ different vertices. We can always take a maximal face of such a
simplex without repetitions in the vertices. Such a face is non-degenerate and
has the same underlying vertex set as the original simplex. Hence, it is the
simplex. Next, let $\sigma$ and $\tau$ be two possibly degenerate simplices
with the same multiset of vertices. Then, by assumption, the two simplices
they degenerate from agree and have no repeating vertices. The respective
degeneracy maps giving $\sigma$ and $\tau$ are already completely specified by
the multiplicities in their corresponding multisets. Hence, as the latter
agree, $\sigma$ and $\tau$ come from the same nondegenerate simplex and
through the same degeneracy map, in other words $\sigma=\tau$. ∎

We finish this subsection with a type of “dense up to homotopy equivalence”
result, for the embedding $S^{o}$. It is an immediate consequence of Theorem
2.3.23. The reader worried about circularity may be assured that we only use
this result in Section 1.4. The largest part of the second chapter up to
Section 2.4 is completely independent from this result.

######  Proposition 1.3.37.

Every finite filtered simplicial set $X\in\textnormal{s{Set}}_{P}^{fin}$ is
isomorphic in $\mathcal{H}\textnormal{s{Set}}_{P}$ to a finite FOS-complex of
the same dimension.

In particular, many of the homotopy theoretic questions on filtered simplicial
sets can be reduced to questions on FOS-complexes.

####  1.3.5 Last vertex maps of relative subdivisions

Another advantage, of FOS-complexes over nonordered ones, is that their
subdivisions admit last vertex maps. This provides a way to represent
subdivision homeomorphisms up to stratum preserving homotopy. We have already
seen such constructions for filtered simplicial sets in Section 1.1.2. The
fully faithful inclusions of categories

$S^{o}:\textnormal{{sCplx}}^{\operatorname{o}}_{P}\hookrightarrow\textnormal{s{Set}}_{P}$

is compatible with subdivisions in the following sense.

######  Construction 1.3.38.

Let $K$ be any filtered simplicial complex. Then $\textnormal{sd}(K)$ is
naturally an FOS-complex, with the order on vertices given by inclusion.
Hence, we can also think of sd as an endofunctor of
$\textnormal{{sCplx}}^{\operatorname{o}}_{P}$, as well as as a functor

$\textnormal{{sCplx}}_{P}\longrightarrow\textnormal{{sCplx}}^{\operatorname{o}}_{P}.$

It is not hard to see that there is a natural isomorphism

$\textnormal{sd}(S^{o}(K))\cong S^{o}(\textnormal{sd}(K)).$

Indeed, first note that the two definitions are clearly compatible on
simplices and then extend by the colimit definition of sd for simplicial sets.
Hence, this definition is also compatible with simplicial set version of
subdivision. Finally, consider the last vertex map

$\textnormal{sd}(K)\longrightarrow K$

given on vertices by

$\\{x_{0}\leq...\leq x_{k}\\}\mapsto x_{k}.$

This induces a natural transformation:

$\textnormal{sd}\longrightarrow
1_{\textnormal{{sCplx}}^{\operatorname{o}}_{P}}.$

Again, it is easy to see that this is compatible with the definition of the
last vertex transformation $sd\longrightarrow 1_{\textnormal{s{Set}}_{P}}$
under the inclusion of categories given by $S^{o}$.

In the ordered setting, we can also equip the relative versions of
subdivision, 1.3.21, with last vertex maps.

######  Construction 1.3.39.

Let $K$ be an FOS-complex and $A$ a full subcomplex such that whenever two
vertices $x\in K-A$ and $a\in A$ lie in a common simplex we have $a<x$. Then
the relative subdivision $\textnormal{sd}(K\textnormal{ rel }A)$ becomes an
FOS-complex (over $P$), by taking the induced ordering on
$\textnormal{sd}(K-A)\sqcup A$ and setting $a<\sigma$, whenever two vertices
$a\in A^{(0)}$ and $\sigma\in\textnormal{sd}(K-A)^{(0)}$ lie in a common
simplex. Now, the last vertex map $\textnormal{l.v.}:\textnormal{sd}(K)\to K$
factors into two maps of FOS-complexes,

$\textnormal{sd}(K)\xrightarrow{l_{0}}\textnormal{sd}(K\textnormal{ rel
}A)\xrightarrow{l_{1}}K,$

fitting into the commutative diagram:

${\textnormal{sd}(A)}$${A}$${A}$${\textnormal{sd}(K)}$${\textnormal{sd}(K\textnormal{
rel
}A)}$${K}$${\textnormal{sd}(K-A)}$${\textnormal{sd}(K-A)}$${K-A}$l.v.1$\scriptstyle{l_{0}}$$\scriptstyle{l_{1}}$$\scriptstyle{1}$l.v.

The maps are constructed as follows.
$l_{0}$ is defined on vertices via

$\displaystyle\textnormal{sd}(K-A)^{(0)}$ $\displaystyle\ni$
$\displaystyle\tau$ $\displaystyle\longmapsto\textnormal{ }\tau$
$\displaystyle\big{(}\textnormal{sd}(N(X,A))-\textnormal{sd}(A)\big{)}^{(0)}$
$\displaystyle\ni$ $\displaystyle\sigma\star\tau$
$\displaystyle\longmapsto\textnormal{ }\tau$
$\displaystyle\textnormal{sd}(A)^{(0)}$ $\displaystyle\ni$
$\displaystyle\sigma$ $\displaystyle\longmapsto\textnormal{
}\textnormal{l.v.}(\sigma).$

This maps a simplex
$\\{\sigma_{0}\subset...\subset\sigma_{k}\subset\sigma_{k+1}\star\tau_{1}\subset...\subset\sigma_{k+l}\star\tau_{l}\\}$,
with $\sigma_{i}\in A$ and $\tau_{j}\in K-A$, to

$\\{\textnormal{l.v.}(\sigma_{0})\leq...\leq\textnormal{l.v.}(\sigma_{k})\leq\tau_{1}\leq...\leq\tau_{l}\\}=\\{\textnormal{l.v.}(\sigma_{0})\leq...\leq\textnormal{l.v.}(\sigma_{k})\\}\star\\{\tau_{1}\leq...\leq\tau_{l}\\}.$

As
$\\{\textnormal{l.v.}(\sigma_{0})\leq...\leq\textnormal{l.v.}(\sigma_{k})\\}\subset\sigma_{k+l}\star\tau_{k}$
this actually defines an (ordered) simplex in $\textnormal{sd}(K\textnormal{
rel }A)$. It is clearly filtered, as we have assumed that vertices in $A$
precede those in $K$ in the ordering. $l_{1}$ is given on vertices by:

$\displaystyle\textnormal{sd}(K-A)^{(0)}\ni\tau$
$\displaystyle\longmapsto\textnormal{ }\textnormal{l.v.}(\tau)$ $\displaystyle
A^{(0)}\ni\sigma$ $\displaystyle\longmapsto\sigma,$

which clearly preserves strata. It maps a simplex
$\sigma\star\\{\tau_{0}\leq...\leq\tau_{k}\\}$ to

$\sigma\star\\{\textnormal{l.v.}(\tau_{0})\leq...\leq\textnormal{l.v.}(\tau_{k})\\}.$

Again, by construction of the FOS-complex $\textnormal{sd}(K\textnormal{ rel
}A)$ and the ordering assumption on $A\subset K$, this defines an ordered
simplex.

######  Proposition 1.3.40.

In the setting of 1.3.39, the realizations of the two arrows $l_{0}$ and
$l_{1}$ are stratified homotopic to the filtered subdivision homeomorphisms
from Proposition 1.3.22.

###### Proof.

This is immediate through a use of straight line homotopies. ∎

We now want to iterate this construction. For this we need:

######  Lemma 1.3.41.

In the setting of 1.3.39 let $A^{\prime}\subset K$ be another full subcomplex
fulfilling the ordering condition and containing $A$. Then the subcomplex
$\textnormal{sd}(A^{\prime}\textnormal{ rel
}A)\subset\textnormal{sd}(K\textnormal{ rel }A)$ satisfies the requirements of
1.3.39 as well.

###### Proof.

We have already seen in Lemma 1.3.18 that fullness is a property that is
preserved under subdivision. Therefore, it suffices to check the ordering
condition. Let $a^{\prime}\in\textnormal{sd}(A^{\prime}\textnormal{ rel }A)$
and $x\in\textnormal{sd}(K^{\prime}\textnormal{ rel
}A)-\textnormal{sd}(A^{\prime}\textnormal{ rel }A)$ be two vertices contained
in a common simplex. In case $a\in A$, the result is immediate from the
construction of the ordering on $\textnormal{sd}(K\textnormal{ rel }A)$, as
$A\subset\textnormal{sd}(A^{\prime}\textnormal{ rel }A).$ If not, then both
lie in $\textnormal{sd}(K-A)$. But here, the ordering relation holds for any,
subcomplex $\textnormal{sd}(A^{\prime\prime})$, $A^{\prime\prime}\subset K$.
This follows, by the definition of the ordering on $\textnormal{sd}(K)$ via
inclusions of simplices, as no simplex not contained in $A^{\prime\prime}$ can
contain a simplex contained in $A^{\prime\prime}$. ∎

We can now use this to study an iterative version of relative subdivisions of
FOS-complexes.

######  Construction 1.3.42.

Let $K$ be an FOS-complex and $\mathcal{A}:=(A^{1}\subset...\subset A^{n})$ be
a family of full subcomplexes of $K$ each satisfying the ordering condition of
<EMAIL_ADDRESS>Now, define
$\textnormal{sd}^{\mathcal{A}}(K)$ by induction over $n$ as follows. For
$n=1$, just set
$\textnormal{sd}^{\mathcal{A}}(K):=\textnormal{sd}(K\textnormal{ rel }A_{1})$.
For $n$ to $n+1$, let $\tilde{\mathcal{A}}$ be the subfamily of $\mathcal{A}$
where $A^{n+1}$ was removed. By another inductive use of Lemma 1.3.41 we set
$\textnormal{sd}^{\mathcal{A}}(K)$ to the FOS-complex defined by

$\textnormal{sd}^{\mathcal{A}}(K):=\textnormal{sd}\Big{(}\textnormal{sd}^{\tilde{\mathcal{A}}}(K)\textnormal{
rel }\textnormal{sd}^{\tilde{\mathcal{A}}}(A^{n+1})\Big{)}.$

Then, inductively by Proposition 1.3.22, we have subdivisions

$\textnormal{sd}^{n}(K)\vartriangleleft\textnormal{sd}^{\mathcal{A}}(K)\vartriangleleft
K,$

where the latter is so that no new vertices are added to
$|{\mathring{N}}(K,A_{1})|$.

######  Corollary 1.3.43.

Let $K$ be an FOS-complex together with a family $\mathcal{A}$ as in 1.3.42 of
length $n$. Then the ($n$-times iterated) last vertex map
$\textnormal{l.v.}^{n}:\textnormal{sd}^{n}(K)\to K$ factors into two maps of
FOS-complees

$\textnormal{sd}^{n}(K)\xrightarrow{l_{0}^{\mathcal{A}}}\textnormal{sd}^{\mathcal{A}}(K)\xrightarrow{l_{1}^{\mathcal{A}}}K$

fitting into the commutative diagram:

${\textnormal{sd}^{n}(A^{1})}$${A^{1}}$${A_{1}}$${\textnormal{sd}^{\mathcal{A}}(K)}$${\textnormal{sd}^{\mathcal{A}}(K)}$${K}$$\scriptstyle{\textnormal{l.v.}^{n}}$1$\scriptstyle{l_{0}^{\mathcal{A}}}$$\scriptstyle{l_{1}^{\mathcal{A}}}$

.

Furthermore, $|l_{0}^{\mathcal{A}}|_{P}$ and $|l_{1}^{\mathcal{A}}|_{P}$ are
respectively stratified homotopic to the subdivision homeomorphisms
$|\textnormal{sd}^{n}(K)|_{P}\xrightarrow{\sim}|\textnormal{sd}^{\mathcal{A}}(K)|_{P}$
and $|\textnormal{sd}^{\mathcal{A}}(K)|_{P}\xrightarrow{\sim}|K|_{P}$.

###### Proof.

For $l_{1}^{\mathcal{A}}$, just take the iterative composition of the
respective $l_{1}$ from 1.3.39. This clearly make the right hand side of the
diagram commute and is stratified homotopic to the subdivision homeomorphism
by an inductive use of Proposition 1.3.40.
$l_{0}^{n}$ is constructed inductively as follows. For $n=1$, take $l_{0}$
from 1.3.39 for $A=A^{1}$. For the inductive step from $n$ to $n+1$, set
$l_{0}^{n+1}$ to the composition:

$\displaystyle\textnormal{sd}^{n+1}(K)=\textnormal{sd}\big{(}\textnormal{sd}^{n}(K)\big{)}\xrightarrow{\textnormal{sd}(l_{0}^{\tilde{\mathcal{A}}})}\textnormal{sd}\big{(}\textnormal{sd}^{\tilde{\mathcal{A}}}(K)\big{)}$
$\displaystyle\xrightarrow{l_{0}}\textnormal{sd}\big{(}\textnormal{sd}^{\tilde{\mathcal{A}}}(K)\textnormal{
rel }\textnormal{sd}^{\tilde{\mathcal{A}}}(A^{n+1})\big{)}$
$\displaystyle=\textnormal{sd}^{\mathcal{A}}(K),$

with $\tilde{\mathcal{A}}$ as in 1.3.42. The right hand side $l_{0}$ is the
one corresponding to the FOS-pair
$\textnormal{sd}^{\tilde{\mathcal{A}}}(A^{n+1})\subset\textnormal{sd}^{\tilde{\mathcal{A}}}(K)$.
Now, consider the diagram:

${\textnormal{sd}^{n+1}(A^{1})}$${\textnormal{sd}(A)}$${A}$${\textnormal{sd}^{n+1}(K)}$${\textnormal{sd}\big{(}\textnormal{sd}^{\tilde{\mathcal{A}}}(K)\big{)}}$${\textnormal{sd}^{\mathcal{A}}(K)}$$\scriptstyle{\textnormal{sd}(\textnormal{l.v.}^{n})}$l.v.$\scriptstyle{\textnormal{sd}(l_{0}^{\tilde{\mathcal{A}}})}$$\scriptstyle{l_{0}}$

.

This commutes by the induction hypothesis and functoriality of sd.
Furthermore, by naturality of l.v., the upper horizontal composition equals
$\textnormal{l.v.}^{n+1}:\textnormal{sd}^{n+1}(A)\to A$. This shows the
commutativity assertion. We then have

$\displaystyle l^{\mathcal{A}}_{1}\circ l^{\mathcal{A}}_{0}$
$\displaystyle=l^{\tilde{\mathcal{A}}}_{1}\circ l_{1}\circ
l_{0}\circ\textnormal{sd}(l_{0}^{\tilde{\mathcal{A}}})$
$\displaystyle=l^{\tilde{\mathcal{A}}}_{1}\circ\textnormal{l.v.}\circ\textnormal{sd}(l_{0}^{\tilde{\mathcal{A}}})$
$\displaystyle=l^{\tilde{\mathcal{A}}}_{1}\circ
l_{0}^{\tilde{\mathcal{A}}}\circ\textnormal{l.v.}$
$\displaystyle=\textnormal{l.v.}^{n+1}$

For the homotopy claim, note that it suffices to construct such a homotopy for
the compositions with
$|\textnormal{sd}^{\mathcal{A}}(K)|_{P}\xrightarrow{\sim}|K|_{P}$ as the
latter is a stratum preserving homeomorphism. The composition

$|\textnormal{sd}^{n}(K)|_{P}\xrightarrow{\sim}|\textnormal{sd}^{\mathcal{A}}(K)|_{P}\xrightarrow{\sim}|K|_{P}$

is the subdivision homeomorphism corresponding to
$\textnormal{sd}^{n}(K)\vartriangleleft K$, by Proposition 1.3.22. The latter
is stratified homotopic to the realization of $\textnormal{l.v.}^{n}$ by
Proposition 1.1.37. On the other hand, we have already seen that
$|\textnormal{sd}^{\mathcal{A}}(K)|_{P}\xrightarrow{\sim}|K|_{P}.$ is
homotopic to $|l_{1}^{\mathcal{A}}|_{P}$. Hence, its composition with
$|l_{0}^{\mathcal{A}}|_{P}$ is also homotopic to
$|\textnormal{l.v.}^{n}|_{P}$. This finishes the proof. ∎

In the next subsection, we use these relative last vertex maps to connect
stratified topological and stratified simplicial homotopy classes.

####  1.3.6 Filtered simplicial approximation, part two

In this subsection, we are going to prove the following alternative version of
the filtered simplicial approximation theorem.

######  Theorem 1.3.44 (Filtered simplicial approximation B).

Let $K,L$ be $P$-filtered ordered simplicial complexes over $P$, where $K$ is
finite. Let

$|K|_{P}\xrightarrow{\phi}|L|_{P}$

be a stratum preserving map. Then there exists an $n\in\mathbb{N}$ and a
stratum preserving (ordered) simplicial map
$\textnormal{sd}^{n}(K)\xrightarrow{f}L$ such that

$|f|_{P}\simeq_{P}\Big{(}|\textnormal{sd}^{n}(K)|_{P}\xrightarrow{|\textnormal{l.v.}^{n}|_{P}}|K|_{P}\xrightarrow{\phi}|L|_{P}\Big{)}\simeq_{P}\Big{(}|\textnormal{sd}^{n}(K)|_{P}\xrightarrow{\sim}|K|_{P}\xrightarrow{\phi}|L|_{P}\Big{)}.$

Conversely, if two stratum preserving (ordered) simplicial maps
$f_{0},f_{1}:K\to L$ are such that

$|f_{0}|_{P}\simeq_{P}|f_{1}|_{P},$

then there exists an $n\in\mathbb{N}$ and a stratum preserving (ordered)
simplicial map

$H:\textnormal{sd}^{n}(K\otimes\Delta^{1})\to L$

such that the following diagrams commutes.

${\textnormal{sd}^{n}(K)}$${K}$${\textnormal{sd}^{n}(K\otimes\Delta^{1})}$${L}$$\scriptstyle{\textnormal{l.v.}^{n}}$$\scriptstyle{\textnormal{sd}^{n}(i_{j})}$$\scriptstyle{f_{j}}$$\scriptstyle{H}$

.

Here $i_{j}$ denotes inclusion of $K$ into $K\otimes\Delta^{1}$ at the
respective endpoint, for $j=0$, $1$.

###### Proof.

We start by showing the results for a linearly ordered set $P=[q]$. Note that,
for a sequence $\Sigma=(\Sigma_{i})_{i\in[q]}$ in $\mathbb{N}$,
$\textnormal{sd}^{\Sigma}(K\textnormal{ rel }A)$ from Section 1.3.3 is a
special case of the iterative relative subdivision defined in 1.3.42, where we
take

$\mathcal{A}:=\\{\underbrace{A\subset...\subset A}_{\Sigma_{0}}\subset
A\underbrace{\cup K_{0}\subset...\subset A\cup K_{0}}_{\Sigma_{1}}\subset
A\cup K_{1}\subset...\subset\underbrace{A\cup K_{q-1}\subset...\subset A\cup
K_{q-1}}_{\Sigma_{q}}\\}.$

Let $\Sigma$ and $f^{\prime}:\textnormal{sd}^{\Sigma}(K)\to L$ be a filtered
simplicial approximation of $\phi$ given by Theorem 1.3.24. Denote
$|\Sigma|:=\sum_{p\in[q]}\Sigma_{p}$. Then note that as the filtration map is
ordered, the subcomplexes $K_{q}\subset K$ fulfill the ordering condition of
Corollary 1.3.43. Hence, by the latter, we can pull back $f^{\prime}$ with
$l_{0}^{\mathcal{A}}$ (using the notation of Corollary 1.3.43) to obtain a
stratum preserving simplicial map

$f^{\prime\prime}:\textnormal{sd}^{|\Sigma|}\xrightarrow{l_{0}^{\mathcal{A}}}\textnormal{sd}^{\Sigma}(K)\to
L$

such that

$|f^{\prime\prime}|_{P}\simeq_{P}\Big{(}|\textnormal{sd}^{|\Sigma|}(K)|_{P}\xrightarrow{|\textnormal{l.v.}^{n}|_{P}}|K|_{P}\xrightarrow{\phi}|L|_{P}\Big{)}\simeq_{P}\Big{(}|\textnormal{sd}^{|\Sigma|}(K)|_{P}\xrightarrow{\sim}|K|_{P}\xrightarrow{\phi}|L|_{P}\Big{)}$
(1.17)

Note however that this is not necessarily an ordered map. Now, set
$n=|\Sigma|+1$ and

$f:\textnormal{sd}^{n}(K)\xrightarrow{\textnormal{sd}(f^{\prime\prime})}\textnormal{sd}{L}\xrightarrow{\textnormal{l.v.}}L.$

Then all of the maps involved are maps of FOS-complexes and by naturality of
l.v. and (1.17) we obtain:

$\displaystyle|f|_{P}$ $\displaystyle=_{\textnormal{
}}|\textnormal{l.v.}\circ\textnormal{sd}(f^{\prime\prime})|_{P}$
$\displaystyle=_{\textnormal{ }}|f^{\prime\prime}\circ\textnormal{l.v.}|_{P}$
$\displaystyle\simeq_{P}\phi\circ|\textnormal{l.v.}\circ\textnormal{l.v.}^{|\Sigma|}|_{P}$
$\displaystyle\simeq_{P}\phi\circ|\textnormal{l.v.}^{n}|_{P}$

finishing the first part of the proof.

For the proof of the homotopy statement, let $f_{0},f_{1}:K\to L$ be maps of
FOS-complexes over $P$ and
$H:|K\otimes\Delta^{1}|_{P}\cong|K|_{P}\otimes\Delta^{1}\to|L|_{P}$ be a
stratum preserving homotopy between their realizations. Then by Example
1.3.27, we can apply Theorem 1.2.8 to the induced homotopy constructed in this
example

$\tilde{H}:|\textnormal{sd}\big{(}K\otimes\textnormal{sd}^{2}(\Delta^{1})\big{)}|_{P}\longrightarrow|\textnormal{sd}(L)|_{P}$

between $|\textnormal{sd}(f_{0})|_{P}$ and $|\textnormal{sd}(f_{1})|_{P}$.
Then, arguing just as we did for the previous statement but using Theorem
1.3.25 instead, we obtain an $n^{\prime}\in\mathbb{N}$ and a morphism of
filtered simplicial complexes

$H^{\prime}:\textnormal{sd}^{n^{\prime}}\Big{(}\textnormal{sd}\big{(}K\otimes\textnormal{sd}^{2}(\Delta^{1})\big{)}\Big{)}\longrightarrow\textnormal{sd}(L).$

By the commutative diagram in Corollary 1.3.43, this is given by
$\textnormal{sd}(f_{j})\circ\textnormal{l.v.}^{n^{\prime}}$ on the respective
inclusions of $\textnormal{sd}^{n^{\prime}+1}(K)$. Now, consider the map of
FOS-complex

$l:\textnormal{sd}^{2}(K\otimes\Delta^{1})\xrightarrow{\big{(}\textnormal{l.v.}^{2}\circ\textnormal{
sd}^{2}(\pi_{K}),\textnormal{
sd}^{2}(\pi_{\Delta^{1}})\big{)}}K\otimes\textnormal{sd}^{2}(\Delta^{1}).$

$l$ restricts to $\textnormal{l.v.}^{2}$ followed by the inclusions
$i^{\prime}_{j}:K\hookrightarrow K\otimes\textnormal{sd}^{2}(\Delta^{1})$ ,at
the respective endpoints, $j=0,1$. Now, set $n=n^{\prime}+4$ and $H$ to the
composition of maps of FOS-complexes

$H:\textnormal{sd}^{n}(K\otimes\Delta^{1})\xrightarrow{\textnormal{sd}^{n^{\prime}+2}(l)}\textnormal{sd}\Big{(}\textnormal{sd}^{n^{\prime}+1}\big{(}K\otimes\textnormal{sd}^{2}(\Delta^{1})\big{)}\Big{)}\xrightarrow{sd(H^{\prime})}\textnormal{sd}^{2}(L)\xrightarrow{\textnormal{l.v.}^{2}}L.$

Pulling this back with $\textnormal{sd}^{n}(i_{j})$ and using the restriction
statements on $H^{\prime}$ and $l$ we obtain the commutative diagram:

${\textnormal{sd}^{n}(K)}$${\textnormal{sd}^{n^{\prime}+2}(K)}$${\textnormal{sd}^{2}(K)}$${K}$${\textnormal{sd}^{n}(K\otimes\Delta^{1})}$${\textnormal{sd}^{n^{\prime}+2}\big{(}K\otimes\textnormal{sd}^{2}(\Delta^{1})\big{)}}$${\textnormal{sd}^{2}(L)}$${L}$$\scriptstyle{\textnormal{sd}^{n}(i_{j})}$$\scriptstyle{\textnormal{sd}^{n^{\prime}+2}(\textnormal{l.v.}^{2})}$$\scriptstyle{\textnormal{sd}^{n^{\prime}+2}(i^{\prime}_{j})}$$\scriptstyle{\textnormal{sd}(\textnormal{l.v.}^{n^{\prime}})}$$\scriptstyle{\textnormal{sd}^{2}(f_{j})}$$\scriptstyle{\textnormal{l.v.}^{2}}$$\scriptstyle{f_{j}}$$\scriptstyle{\textnormal{sd}^{n^{\prime}+2}(l)}$$\scriptstyle{\textnormal{sd}(H^{\prime})}$$\scriptstyle{\textnormal{l.v.}^{2}}$

.

By definition, the bottom horizontal composition is $H$. Furthermore, by
naturality of l.v., the top horizontal composition is $\textnormal{l.v.}^{n}$.
Hence, this provides the diagram in the statement of the theorem. We are left
with showing the statements in the case where $P$ is not a finite linear
partially ordered set. Note that all of the simplicial complexes involved are
finite. Thus, can without loss of generality restrict to the case where $P$ is
finite. (Rigorously, this is done by pulling back to the finite subset
(subcomplex spanned by) the elements of $P$ in the image of the filtration,
approximating, and then composing with the inclusion into $P$ again.) The
finite, not linearly ordered case now is immediately reduced to the linear
order one by refining the order on $P$ to a linear one and using the following
lemma. ∎

######  Lemma 1.3.45.

Let $P$ be a partially ordered set. Let $P^{\prime}$ be another partially
ordered set together with a bijective map of partially ordered set $\rho:P\to
P^{\prime}$. Denote by

$\displaystyle
N(\rho)_{*}:\textnormal{s{Set}}_{P}\longrightarrow\textnormal{s{Set}}_{P^{\prime}}$
$\displaystyle\rho_{*}:\textnormal{{Top}}_{P}\longrightarrow\textnormal{{Top}}_{P^{\prime}}$

the functors obtained by postcomposing filtrations with $N(\rho)$ and (the map
induced on the Alexandroff spaces by) $\rho$ respectively. Then these two
functors are fully faithful, and fit into a diagram (commutative up to natural
isomorphism):

${\textnormal{s{Set}}_{P}}$${\textnormal{s{Set}}_{P^{\prime}}}$${\textnormal{{Top}}_{P}}$${\textnormal{{Top}}_{P^{\prime}}}$$\scriptstyle{|-|_{P}}$$\scriptstyle{N(\rho)_{*}}$$\scriptstyle{|-|_{P^{\prime}}}$$\scriptstyle{\rho_{*}}$

.

Furthermore, $N(\rho)_{*}$ commutes with $\otimes(-)$, sd and l.v..

###### Proof.

The commutativity statements are all immediate from the definitions of the
respective functors and natural transformations. It remains to show that
$\rho_{*}$ and $N(\rho)$ are fully faithful. Both maps we post-compose with,
$\rho$ and $N(\rho)$, are monomorphisms in their respective categories.
Indeed, it is an elementary and easily verified fact on over-categories that
postcomposing with a monomorphism is a fully faithful functor. ∎

This finishes our investigation into filtered simplicial complexes. We now
return to investigating the relationship between $\textnormal{{Top}}_{P}$ and
$\textnormal{s{Set}}_{P}$.

###  1.4 A more detailed analysis of $\mathcal{H}\textnormal{{Top}}_{P}$

We now have several tools available to start a detailed analysis of
$\mathcal{H}\textnormal{{Top}}_{P}$. Our main tool is the filtered simplicial
approximation theorem, Theorem 1.3.44. However, by Proposition 1.3.37, this
does not necessarily mean we need to restrict ourselfs only to FOS-complexes.

####  1.4.1
$|-|_{P}:\mathcal{H}\textnormal{s{Set}}^{fin}_{P}\to\mathcal{H}\textnormal{{Top}}_{P}$
is fully faithful

In the case where $P=\star$ is a one-point set, the Douteau model structure on
$\textnormal{s{Set}}_{P}$ agrees with the classical Kan-Quillen one under the
obvious isomorphism of categories
$\textnormal{s{Set}}_{P}\cong\textnormal{s{Set}}$. In this setting, it is of
course well known that the realization functor defines an equivalence of
categories

$|-|:\mathcal{H}\textnormal{s{Set}}\xrightarrow{\sim}\mathcal{H}\textnormal{{Top}}.$

By Theorem 1.2.23, the filtered realization functor also induces a functor
$|-|_{P}:\mathcal{H}\textnormal{s{Set}}_{P}\to\mathcal{H}\textnormal{{Top}}_{P}$
on homotopy categories. To the best of our knowledge it is not yet known
whether this defines an equivalence of categories. However, we are going to
show in this section (Corollary 1.4.4) that if one restricts to the finite
setting this functor is fully faithful. This shows that at least for filtered
topological spaces that are weakly equivalent to the realizations of finite
filtered simplicial sets most question on homotopy theory can be reduced to
the combinatorial setting. In particular, this allows us to define the
Whitehead group and Whitehead torsion in the topological setting in Section
2.4.

As the title of the subsection states, we are now going to prove Theorem
1.4.1. Our proof is somewhat unorthodox, at least from a model theoretical
perspective, as we first to reduce to FOS-complexes and then use simplicial
approximation theorems. This is certainly more of a classical than a modern
approach to the problem. It comes at the price of having to restrict to the
finite setting. We conjecture that this is just due to the insufficiency of
our method of proof, and in fact
$|-|_{P}:\mathcal{H}\textnormal{s{Set}}_{P}\longrightarrow\mathcal{H}\textnormal{{Top}}_{P}$
is an equivalence of categories (induced by a Quillen equivalence even, see
also Remark 1.1.20). However, as this work was written with simple homotopy
theory in mind, some compactness assumptions are really not that big of a
price to pay.

######  Theorem 1.4.1.

The functor

$|-|_{P}:\mathcal{H}\textnormal{s{Set}}_{P}^{fin}\longrightarrow\mathcal{H}\textnormal{{Top}}_{P}$

induced by Theorem 1.2.23 is fully faithful.

###### Proof.

Let $X,Y\in\textnormal{s{Set}}_{P}^{fin}$. We want to show that

$\textnormal{Hom}_{\mathcal{H}\textnormal{s{Set}}_{P}}(X,Y)\xrightarrow{|-|_{P}}\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|X|_{P},|Y|_{P})$

is a bijection. First, note that, by Proposition 1.3.37, we may assume without
loss of generality that $X$ and $Y$ are finite FOS-complexes. We now want to
reduce to the case, where the right hand side is given by actual stratified
homotopy classes. By Proposition 1.1.27 and the fact that by definition every
object in $\textnormal{{Top}}_{P}$ is fibrant, we have for cofibrant $T$ and
$T^{\prime}$ arbitrary:

$\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(T,T^{\prime})\cong[T,T^{\prime}]_{P}.$
(1.18)

Now, consider the last vertex map

$\textnormal{l.v.}_{P}:\textnormal{sd}_{P}(X)\longrightarrow X.$

By [Dou19b, A.3] or alternatively Proposition 2.3.19, the latter is a weak
equivalence. In particular, the induced map

$|\textnormal{l.v.}_{P}|_{P}:|\textnormal{sd}_{P}(X)|_{P}\longrightarrow|X|_{P}$

induces an isomorphism in $\mathcal{H}\textnormal{{Top}}_{P}$. By Proposition
1.1.39, $|\textnormal{sd}_{P}(X)|_{P}$ is cofibrant. Furthermore, similarly to
the case of sd one easily checks that $\textnormal{sd}_{P}(X)$ is still an
FOS-complex. Hence, we assume without loss of generality that both $X$ and $Y$
are finite FOS-complex, and that $|X|_{P}$ is cofibrant. Thus, using Equation
1.18, we are left with showing that

$|-|_{P}:\textnormal{Hom}_{\mathcal{H}\textnormal{s{Set}}_{P}}(X,Y)\longrightarrow\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|X|_{P},|Y|_{P})\cong[|X|_{P},|Y|_{P}]_{P}$

is a bijection. Now, let $[\phi]$ be any arrow on the right hand side,
represented stratum preserving map $\phi:|X|_{P}\to|Y|_{P}$. By the second
version of the filtered simplicial approximation theorem (Theorem 1.3.44), for
some sufficiently large $n$ and some map of FOS-complexes
$f:\textnormal{sd}^{n}(X)\xrightarrow{f}Y$ we have

$[\phi]\circ[|\textnormal{l.v.}|_{P}^{n}]=[|f|_{P}].$

$[\textnormal{l.v.}^{n}]$ is an isomorphism in
$\mathcal{H}\textnormal{s{Set}}_{P}$. In particular,

$[\phi]=|[f]\circ[\textnormal{l.v.}^{n}]^{-1}|_{P}$

showing surjectivity. Now, conversely let $\varphi_{0},\varphi_{1}$ be two
arrows on the left hand side of the equation. By Proposition 1.1.46, both fit
into a commutative diagram

${\textnormal{sd}^{n}_{P}(X)}$${X}$${Y}$$\scriptstyle{[\textnormal{l.v.}^{n}_{P}]}$$\scriptstyle{[f_{i}]}$$\scriptstyle{\varphi_{i}}$

For appropriate morphisms in $\textnormal{s{Set}}_{P}^{fin}$ $f_{i}$. Without
loss of generality, we can assume $n_{0}=n_{1}$. Furthermore, inverting
$[\textnormal{l.v.}^{n_{i}}]$, we can then assume that $n_{i}=0$. Hence, we
are now in the setting where $[f_{i}]=\varphi_{i}$ and
$|f_{0}|_{P}\simeq_{P}|f_{1}|_{P}$. By the homotopy part of Theorem 1.3.44,
there is a commutative diagram in $\textnormal{s{Set}}_{P}$:

${\textnormal{sd}^{n}(X)}$${X}$${\textnormal{sd}^{n}(X\otimes\Delta^{1})}$${Y}$$\scriptstyle{\textnormal{l.v.}^{n}}$$\scriptstyle{\textnormal{sd}^{n}(i_{j})}$$\scriptstyle{f_{j}}$$\scriptstyle{H}$

.

Corollary 2.3.20 here, states that $\textnormal{l.v.}:\textnormal{sd}(-)$
preserves weak equivalences (for finite arguments, but this is easily
generalized). Again, we assure that there is no risk of circularity involved
here. Thus,
$\textnormal{sd}^{n}(X)\sqcup\textnormal{sd}^{n}(X)\hookrightarrow\textnormal{sd}^{n}(X\otimes\Delta^{1})$
gives a cylinder object for $\textnormal{sd}^{n}(X)$ that witnesses a left
homotopy between $f_{0}\circ\textnormal{l.v.}^{n}$ and
$f_{1}\circ\textnormal{l.v.}^{n}$. Hence,

$[f_{0}\circ\textnormal{l.v.}^{n}]=[f_{1}\circ\textnormal{l.v.}^{n}]$

and, by the invertibility of $[\textnormal{l.v.}]$ in
$\mathcal{H}\textnormal{s{Set}}_{P}$, we obtain

$[f_{0}]=[f_{1}].$

∎

In particular, we obtain the following immediate corollary, using the fact
that a morphism in the model category is a weak equivalence if and only if it
descends to an isomorphism in the homotopy categoy.

######  Corollary 1.4.2.

A morphism $f:X\to Y$ in $\textnormal{s{Set}}_{P}^{fin}$ is a weak equivalence
with respect to the Douteau model structure if and only if $|f|_{P}$ is a weak
equivalence with respect to the Henrique-Douteau model structure.

Hence, for spaces that are (stratified homotopy equivalent) stratum preserving
homeomorphic to the realization of a finite simplicial set, questions on
homotopies can often be reduced to the combinatorial setting of filtered
simplicial sets. It is useful to have a name for such spaces.

######  Definition 1.4.3.

Let $T\in\textnormal{{Top}}_{P}$.

  * •

A filtered simplicial set $X\in\textnormal{s{Set}}_{P}$ together with a
stratum preserving homeomorphism $|X|_{P}\cong T$ is called a triangulation of
$T$. If $X$ is finite, we call it a finite triangulation.

  * •

A filtered topological space admitting such a (finite) triangulation is called
(finitely) triangulable

In particular, we obtain:

######  Corollary 1.4.4.

Denote by $\mathcal{H}\textnormal{{Top}}^{t-fin}_{P}$ the full subcategory of
$\mathcal{H}\textnormal{{Top}}_{P}$ given by such filtered spaces that are
weakly equivalent to a triangulable filtered space. Then

$|-|_{P}:\mathcal{H}\textnormal{s{Set}}_{P}^{fin}\longrightarrow\mathcal{H}\textnormal{{Top}}_{P}$

induces an equivalence of categories

$\mathcal{H}\textnormal{s{Set}}_{P}^{fin}\xrightarrow{\simeq}\mathcal{H}\textnormal{{Top}}^{t-fin}_{P}.$

######  Corollary 1.4.5.

A morphism of finite filtered simplicial sets $f:X\to Y$ in
$\textnormal{s{Set}}_{P}$ is a weak equivalence, with respect to the Douteau
model structure if and only if $|f|_{P}$ is a weak equivalence, with respect
to the Henrique-Douteau model structure on $\textnormal{{Top}}_{P}$.

####  1.4.2 Homotopy classes into fibrant, finitely triangulable filtered
spaces

As we have already shown in the proof of Theorem 1.4.1, the Hom-sets in
$\mathcal{H\textnormal{{Top}}_{P}}$, with a triangulable filtered space at the
domain, admit fairly explicit descriptions. This is owed to the fact that
cofibrant approximations are given by $\textnormal{sd}_{P}(-)$. More
precisely, in this case we have a natural bijection:

$\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|X|_{P},T)\cong\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|\textnormal{sd}_{P}(X)|_{P},T)=\big{[}|\textnormal{sd}_{P}(X)|_{P},T\big{]}_{P}.$

It would be nice to have such a result, not only for spaces cofibrant in the
Douteau-Henrique model structure, but also for spaces arising more naturally
in the study of spaces with singularities. Such a result would make the
homotopy category of filtered spaces a lot easier to understand. Recall (see
Definition 1.1.12) that a f-stratified space (over $P$) is a filtered space
$T$ that has the right lifting property with respect to admissible horn
inclusions as in the diagram below.

${{|\Lambda_{k}^{\mathcal{J}}|_{P}}}$${T}$${{|\Delta^{\mathcal{J}}|_{P}}}$

As we already mentioned in Example 1.1.14, examples of such spaces include all
conically stratified spaces (in particular all PL pseudomanifolds and PL
stratified spaces in the original sense of Goresky and MacPherson) and all
homotopically stratified spaces in the sense of Quinn. One then has:

######  Lemma 1.4.6.

Let $T\in\textnormal{{Top}}_{P}$ be f-stratified. Then $T$ has the right
lifting property with respect to all realizations of (filtered) anodyne
extensions $A\hookrightarrow B$ in $\textnormal{s{Set}}_{P}$.

###### Proof.

The class of morphisms having the left lifting property with respect to
another class of morphisms, is always saturated. That is, it is closed under
coproducts, pushouts, transfinite compositions (see [nLa20h], for a
definition) and retracts and contains all isomorphisms. The class of anodyne
extensions is the saturated class generated by all admissible horn inclusions.
Hence, the result is immediate from the fact that $|-|_{P}$ preserves
colimits, and hence maps the saturated class of the latter into the saturated
class generated by realizations of horn inclusions. ∎

In Remark 1.1.20, we conjectured that f-stratified spaces can in fact be taken
to be the fibrant objects in an appropriate model structure on
$\textnormal{{Top}}_{P}$ that has the same weak equivalence as the Henrique-
Douteau model structure. A strong hint at such a result is that it turns out
that for f-stratified spaces, which are triangulable by finite FOS-complexes,
the Hom sets in $\mathcal{H}\textnormal{{Top}}_{P}$ are actually given by sets
of stratified homotopy classes.

######  Theorem 1.4.7.

Let $X,Y\in\textnormal{s{Set}}_{P}^{fin}$ such that $X$ is an FOS-complex and
$Y$ such that $|Y|_{P}$ is an f-stratified space. Then the natural map

$[|X|_{P},|Y|_{P}]_{P}\longrightarrow\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|X|_{P},|Y|_{P})$

is a bijection.

###### Proof.

Let $Y\xhookrightarrow{i}F(Y)$ be a cofibrant fibrant replacement for $Y$.
Then, by Lemma 1.4.6 $|i|_{P}$ admits a retract $r$, which is a weak
equivalence as it is a retract of a weak equivalence by Theorem 1.2.23. Now,
consider the following commutative diagram:

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to94.08pt{\pgfpicture\makeatletter\hbox{\hskip 112.74861pt\lower-47.089pt\hbox
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where the right lower vertical is the isomorphism induced by the weak
equivalence $r$. The upper right vertical is an isomorphism by Theorem 1.4.1,
which clearly extends to all $P$-filtered simplicial sets that are weakly
equivalent to a finite one, i.e. also to $F(Y)$. In particular, by
commutativity, the bottom right horizontal is also onto. It remains to show
injectivity. Consider stratified homotopy classes of maps
$\phi_{0},\phi_{1}:|X|_{P}\to|Y|_{P}$ whose images in
$\textnormal{Hom}_{\mathcal{H}\textnormal{{Top}}_{P}}(|X|_{P},|Y|_{P})$ agree.
By pulling back with the subdivision homeomorphism
$|\textnormal{sd}^{n}(X)|_{P}\xrightarrow{\sim}|X|_{p}$ and invoking Theorem
1.3.44, we may without loss of generality assume that both come from homotopy
classes of morphisms of filtered simplicial sets $f_{0},f_{1}:X\to Y$. They
are mapped to by $[i\circ f_{0}],[i\circ f_{1}]\in[X,F(Y)]_{P}$ respectively.
Thus, by commutativity of the diagram and the fact that the composition
”right, down, down” is a bijection, this already implies
$[\phi_{0}]=[\phi_{1}]$. ∎

In particular, this result also holds true for filtered spaces $T_{X},T_{Y}$
which are respectively stratified homotopy equivalent to realizations of such
filtered simplicial sets.

######  Remark 1.4.8.

The additional assumption that $X$ is an FOS-complex and not just a finite
simplicial set should be omittable. To do this, one only needs to extend the
filtered simplicial approximation theorem to all finite $P$-filtered
simplicial sets. One way to approach this, might be to define a model
structure on $\textnormal{s{Set}}_{P}$ that uses sd instead of
$\textnormal{sd}_{P}$. However, one might just as well take the approach we
illuminated in Remark 1.1.20, which should give a more complete understanding
of the situation.

This result gives a rather clear view of what the morphisms in
$\mathcal{H}\textnormal{{Top}}_{P}$ look like for many classical examples of
stratified spaces. To be more precise, we obtain:

######  Corollary 1.4.9.

Let $\textnormal{{Top}}_{P}^{f,fin,PL}$ be the full subcategory of
f-stratified, compact polyhedra in $\textnormal{{Top}}_{P}$, i.e. f-stratified
spaces, triangulable by a finite FOS-complex. Let
$\mathcal{H}^{naive}\textnormal{{Top}}_{P}^{f,fin,PL}$ be the category
obtained from it, by identifying stratified homotopic maps. Then
$\textnormal{{Top}}_{P}^{f,fin,PL}\hookrightarrow\textnormal{{Top}}_{P}$
induces a fully faithful embedding

$\mathcal{H}^{naive}\textnormal{{Top}}_{P}^{f,fin,PL}\hookrightarrow\mathcal{H}\textnormal{{Top}}_{P}.$

In particular, the analogous result holds, for the full subcategories of
$\textnormal{{Top}}_{P}^{f,fin,PL}$ given by such realizations of finite FOS-
complexes that are pseudo-varieties, cs-stratified, conically stratified or
homotopically stratified spaces.

We should note that a similar statement has been conjectured by Douteau in
[Dou19b, 8.1.4]. However, there it is conjectured for the strongly filtered
setting.

##  Chapter 2 Simple Stratified Homotopy Theory

Roughly speaking, the origin of simple homotopy theory is the question of
which homotopy equivalences $f:X\to Y$ between appropriately combinatorial
objects, say CW-complexes, can be represented through a sequence of certain
combinatorial moves (expansions and collapses). As there is already a lot of
excellent introductory work into simple homotopy theory, we refrain from
giving an overview of the classical perspective. The reader completely new to
the subject should find motivation and analogies for our stratified
perspective in [Coh73]. In the classical setting of CW-complexes (or
simplicial complexes) and elementary expansions, it turns out that the answer
to this question is encoded by an element $\tau(f)$ \- the Whitehead torsion -
of a purely algebraic group $Wh(X)$ \- the Whitehead group of $X$. In this
chapter we ask and attempt to answer the analogous question but for filtered
(stratified) spaces. Before one can even dare to attempt giving an answer, one
of course has to be a little bit more specific as to what the correct analogy
should be. In other words:

What are the “appropriately combinatorial objects”?
What are the “homotopy equivalences”?
What are the “combinatorial moves”?

The first of these three questions we have already answered in Section 1.1.2
with the category $\textnormal{s{Set}}_{P}$ of $P$-filtered simplicial sets.
It simply seems the most well explored category of filtered objects of
combinatorial nature that is not too restrictive when it comes to pushout and
mapping cylinder constructions. To answer the second question, we take the
weak equivalences in the Douteau model structure on $\textnormal{s{Set}}_{P}$
and the Henrique-Douteau model structure on $\textnormal{{Top}}_{P}$
respectively. We have seen in various results of Chapter 1 (in particular
Theorem 1.2.23, Theorem 1.4.1 and Corollary 1.4.2) that they interact rather
nicely. In particular, $|-|_{P}$ preserves and reflects weak equivalences as
long as the source and target are finite filtered simplicial sets. The reason
we chose weak equivalences over the more rigid filtered homotopy equivalences
partially lies in the answer to the third question. The combinatorial moves of
our choice are the pushouts of admissible horn inclusions. We illustrate in
detail in Section 2.2.1 why we think that they, together with the weak
equivalences, make for a good candidate for a study of a stratified simple
homotopy theory. Now, that at least the question is well posed, we can
illuminate how we are set out to answer it. The goal is of course to construct
a stratified analogue to the Whitehead group and the Whitehead torsion and
show that these have similar properties as their classical counterparts. To do
so, we make use of a result of Eckmann, Bolthausen and Siebenmann which allows
for the construction of such objects in a very general category theoretical
setting, given that a certain set of axioms is verified. Introducing this
approach (and fixing a minor mistake in the formulation of the axioms) is the
content of Section 2.1. In the following section (2.2) we then define the
analogue to expansions in the classical theory and do an in-depth study of
their behavior. One of our main tools of study is a recent publication on
strong anodyne extensions by Sean Moss ([Mos19]). We use the results obtained
in this section to show that the axioms of Section 2.1 are verified (Theorem
2.3.4). This proof marks the beginning of Section 2.3.1. In particular, doing
so we have then shown the existence of a Whitehead group and torsion for
filtered and in particular stratified spaces. They answer the initial question
at the beginning chapter. The remainder of Section 2.3.1 is spent on analyzing
the behavior of this stratified Whitehead torsion and group in more detail,
showing that it behaves much like the classical theory. For example, we show
that with respect to the relevant notion of simple equivalence every finite
$P$-filtered simplicial set has the simple homotopy type of a $P$-filtered
ordered simplicial complex (Theorem 2.3.23). Finally in Section 2.4, we use
the results of Chapter 1 to give a series of equivalent descriptions of the
stratified Whitehead group and torsion, also generalizing the latter to
continuous stratum preserving maps instead of purely simplicial ones
(Proposition 2.4.4). We use this to prove that the Whitehead group we defined
agrees with the classical one in case where $P=\star$ is a one point set
(Theorem 2.4.5).

###  2.1 The Eckmann-Siebenmann approach

Historically, several authors have been concerned with the question of how to
generalize Whitehead’s simple homotopy theory to other settings such as
locally finite CW-complexes (see [Sie70], [eck]) or even more generally to
more abstract homotopy theoretical settings, for example categories of
chaincomplexes (see [KP86]). Luckily, these approaches have all been
formulated in a very general category theoretical setting, so they are easily
transferred to our question at hand. Here, we present the approach that was
independently taken by Siebenmann as well as Eckmann and his student
Bolthausen (see [Sie70], [eck]). It will turn out, later on in Section 2.1,
that this approach can be used to formulate a simple homotopy theory for
filtered simplicial sets, and ultimately for finitely triangulated filtered
spaces. However, we should note that the way the axioms for this approach were
formulated in [Sie70] and [eck] is slightly flawed. To be more precise, in the
precise way they are phrased in it does not apply to any setting known to us
and certainly not to the settings the authors were interested in (see Remark
2.1.4). However, this is more of a technical difficulty, and easily fixed by a
slight reformulation of the axioms. To make sure that all the results we need
still hold under these new conditions, we give sketches of the proofs in
[eck]. We note that it might certainly be fruitful to translate the work below
to a more model category theoretic setting. This has already been hinted at in
[KP86, Ch.VI.2]. However, the following suffices for the applications we have
in mind.

For the remainder of this section, let $\mathcal{C}$ be some category embedded
in a larger category $\hat{\mathcal{C}}$ that has the same objects as
$\mathcal{C}$ but potentially more morphisms. Further, let $\Sigma$ be a class
of morphisms in $\mathcal{C}$. Now, consider the localized category
$\mathcal{C}(\Sigma^{-1})$ (see for example [GZ12, Ch. I]) and denote by $Q$
the structure functor $Q:\mathcal{C}\to\mathcal{C}(\Sigma^{-1})$.

######  Definition 2.1.1.

A morphism in $\mathcal{C}(\Sigma^{-1})$ is called simple if it is given by a
composition of morphisms of the shape $Q(s),Q(s)^{-1}$ for $s\in\Sigma$. Two
morphisms $\alpha,\beta$ in $\mathcal{C}^{-1}$ with the same source
$X\in\mathcal{C}$ are said to have the same simple morphism class if
$\beta=\gamma\circ\alpha$ for some $\gamma$ that is simple (clearly this
construction gives an equivalence relation, and hence a well-defined notion of
equivalence class). We denote the class of $\alpha$ by $\langle\alpha\rangle$.
Denote by $A(X)$ the class of simple morphism classes of morphisms with source
$X\in\mathcal{C}$. Denote by $E(X)$ the subclass of $A(X)$ given by classes
$\langle\alpha\rangle$ where some (and hence every)
$\alpha^{\prime}\in\langle\alpha\rangle$ is an isomorphism in
$\mathcal{C}(\Sigma^{-1})$. $E(X)$ and hence $A(X)$ has a distinguished
element given by $\langle 1_{X}\rangle$.

Now, suppose we are in a sufficiently nice setting. That is, one where
$\mathcal{C}(\Sigma^{-1})$ is equivalent to some homotopy category we are
interested in studying and one where $A(X)$ is sufficiently small, i.e. of set
size. This is for example the case for $\hat{\mathcal{C}}$ the category of
finite CW-complexes, $\mathcal{C}$ the given by inclusions of subcomplexes and
$\Sigma$ the class of compositions of elementary expansions. We know that in
these cases $E$ in fact defines a functor on $\mathcal{C}(\Sigma^{-1})$ into
monoids and $E(X)$ defines a functor into abelian groups, the Whitehead group.
([eck],[Sie70]). One can ask the general question of what requirements need to
be fulfilled for this to be the case. This is answered in the following
definition and theorem.

######  Definition 2.1.2.

We say that a triple $\mathcal{C}\subset\hat{\mathcal{C}}$, $\Sigma$ as above
(and such that $A(X)$ is of set size for $X\in\mathcal{C}$) admits a Whitehead
group if the following axioms are fulfilled.

  1. (A0)

$\Sigma$ contains all isomorphisms in $\mathcal{\hat{C}}$ and is closed under
composition.

  2. (A1)

Let $f:X\to X^{\prime}$ and $s:X\to Y$ be morphisms in $\mathcal{C}$. Then the
the pushout diagram

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in $\hat{\mathcal{C}}$ exists and furthermore
$f^{\prime},s^{\prime}\in\mathcal{C}$. If further $s\in\Sigma$, then so is
$s^{\prime}$.

  3. (A2)

Let $f,g:X\to Y$ be two morphisms in $\mathcal{C}$ such that $Q(f)=Q(g)$. Then
there exists a commutative diagram in $\mathcal{C}$

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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}},$

where $s,t\in\Sigma$.

######  Theorem 2.1.3.

Let $\mathcal{C}\subset\hat{\mathcal{C}},\Sigma$ admit a Whitehead group.

Then every morphism in $\mathcal{C}(\Sigma^{-1})$ is of the form
$Q(s)^{-1}Q(f)$ for some $s\in\Sigma$, $f\in\mathcal{C}$. In particular, for
each $X\in\mathcal{C}$ and each $\langle\alpha\rangle\in A(X)$ there exists a
$a\in\mathcal{C}$ such that $\langle\alpha\rangle=\langle Q(a)\rangle.$

Let $X\in\mathcal{C}$. For $\langle\alpha\rangle$ and $\langle\beta\rangle$ in
$A(X)$ define $\langle\alpha\rangle+\langle\beta\rangle$ by the simple
isomorphism class of the diagonal in a pushout diagram

${X}$${Y}$${Z}$${W}$$\scriptstyle{a}$$\scriptstyle{b}$

in $\hat{\mathcal{C}}$, where $Q(a)\in\langle\alpha\rangle$ and
$Q(a)\in\langle\beta\rangle$. Furthermore for $\langle\alpha\rangle$ in $A(X)$
and $f:X\to X^{\prime}\in\mathcal{C}$ define $f_{*}\langle\alpha\rangle\in
E(Y)$ as $\langle Q(a^{\prime})\rangle$, where $a^{\prime}$ is the pushout of
a representative of $\alpha$ in $\mathcal{C}$ along $f$ in
$\hat{\mathcal{C}}$. Both of these constructions are well-defined.

Then,

$\displaystyle\mathcal{C}$ $\displaystyle\to\textnormal{{AbMon}}$
$\displaystyle X$ $\displaystyle\mapsto(A(X),+,\langle 1_{X}\rangle)$
$\displaystyle\\{X\xrightarrow{f}X^{\prime}\\}$
$\displaystyle\mapsto\\{\langle\alpha\rangle\mapsto
f_{*}\langle\alpha\rangle\\}$

defines a functor into the category of abelian monoids. Furthermore,

$\displaystyle\mathcal{C}$ $\displaystyle\to\textnormal{{Ab}}$ $\displaystyle
X$ $\displaystyle\mapsto(E(X),+,\langle 1_{X}\rangle)$
$\displaystyle\\{X\xrightarrow{f}X^{\prime}\\}$
$\displaystyle\mapsto\\{\langle\alpha\rangle\mapsto
f_{*}\langle\alpha\rangle\\}$

defines a functor into the category of abelian groups. Both send $s\in\Sigma$
to an isomorphism, i.e. induce functors:

$\displaystyle E:\mathcal{C}(\Sigma^{-1})$
$\displaystyle\to\textnormal{{Ab}},$ $\displaystyle
A:\mathcal{C}(\Sigma^{-1})$ $\displaystyle\to\textnormal{{AbMon}}.$

######  Remark 2.1.4.

Before we move on to a proof, it might be useful to shed a little bit of light
on the usage of the two categories $\mathcal{C}$ and $\hat{\mathcal{C}}$ here.
Note that both the functoriality as well as the group structure are defined by
taking pushouts of representatives of equivalence classes. In general pushouts
interact badly with homotopies and the induced equivalence relation (see for
example [KP86, Ch.V]). For example, the pushout of a homotopy equivalence
might not be a homotopy equivalence anymore, which would interfere with the
functoriality of $E$ above. Similarly glueings of homotopy equivalences might
not be homotopy equivalences, interfering with the definition of addition.
However, there are of course classes of maps that interact more nicely, when
it comes to pushouts: Cofibrations (see for example [KP86, Ch. V Sec. 6, 7,
Sec]). Note for example that the pushout of an acyclic cofibration of
topological spaces is again an acyclic cofibration ([KP86, Ch. I, Cor. 6.14])
and of course this also holds true in any model category [Hir09]. Usually, the
way to circumvent these difficulties is by replacing maps $f:X\to Y$ by the
mapping cylinder inclusions $X\hookrightarrow M_{f}$, i.e. passing to homotopy
pushouts. This is for example how abstract simple homotopy theory is built
purely cylinder based in [KP86, Ch. VI, Sec. 3]. Alternatively, one can
restrict to whatever class of morphisms interact nicely with homotopy and
pushouts in this setting. This is essentially the step of passing from
$\hat{\mathcal{C}}$ to $\mathcal{C}$. In many cases, one can then use an
appropriate factorization system to represent morphisms in $\hat{\mathcal{C}}$
by morphisms in $\mathcal{C}$. Furthermore, one then hopes that
$\mathcal{C}(\Sigma^{-1})$ is still equivalent as a category to the homotopy
category one is interested in studying. This is for example also done
implicitly in [Coh73], where the geometric Whitehead group is constructed from
pairs of CW-complexes that are deformation retractions, as any inclusion of a
subcomplex of CW-complexes is a cofibration.

In the original sources for this chapter ([eck],[Sie70]), however, no mention
of a second category $\hat{\mathcal{C}}$ is made. In fact, the analogue to
(A1) in Definition 2.1.2 is just formulated via pushouts in $\mathcal{C}$.
While both authors are applying the construction precisely as we mentioned in
the above paragraph, they seem to be under the assumption that in the category
with objects CW-complexes and with morphisms inclusions of subcomplexes the
pushout is given by the pushout in the category of CW-complexes ([eck, p.6],
[Sie70, Paragraph 2]). This is false. Yes, the pushout diagram in CW-complexes
does exist and all of its structure maps are still inclusions of subcomplexes,
but it does not fulfil the existence part of the universal property in the
subcategory of inclusions of subcomplexes. Take for example the pushout along
the empty complex of two points, $\star$. Trivially, the constant map into
another point makes the diagram (given by the maps from the empty complex into
the points) commute. But there is no map $\star\sqcup\star\to\star$ that is an
inclusion of a subcomplex.

Nevertheless, this is neither really hurtful to the results nor to the
theoretical impact of their contributions as neither of them really uses the
universal property of the pushout anywhere in their proof. What the authors
are really using the pushout for is to have a canonical way to produce
commutative squares. To be more precise, they need a way to produce
commutative squares, unique up to isomorphism of diagrams and they want this
to be functorial in the sense that these diagrams have the composition
properties of pushout squares. This is precisely what we verify by our usage
of (A1).

We now begin the proof of Theorem 2.1.3. All the following statements are to
be understood in this setting.

######  Lemma 2.1.5.

Every morphism in $\mathcal{C}(\Sigma^{-1})$ is of the form $Q(s)^{-1}Q(f)$,
for some $s\in\Sigma$, $f\in\mathcal{C}$.

###### Proof.

By definition of the localized category [GZ12, Ch. I], every morphism in
$\mathcal{C}(\Sigma^{-1})$ from $X$ to $Y$ is given by a zigzag

$X=X_{0}\leftrightarrow X_{1}\leftrightarrow...\leftrightarrow X_{n}=Y,$

where we allow all arrows in $\mathcal{C}$ in right direction and only arrows
in $\Sigma$ in left direction. We need to show that each such zigzag can be
brought into the shape

$X\xrightarrow{f}Z\xleftarrow{s}Y,$

for $s\in\Sigma$, under the operations inducing the morphism structure in
$\mathcal{C}(\Sigma^{-1})$ [GZ12, Ch. I]. Now, assume by induction that we
have already proven this for zigzags up to length $n-1$. The case $n=1$ is
obvious. Now, by the composition rules in a localized category, we can reduce
a zigzag of length $n$ to the shape

$X\xrightarrow{f}X_{1}\xleftarrow{s}X_{2}\xleftrightarrow{g}Y,$

with $f,g\in\mathcal{C}$ and $s\in\Sigma$. If $g$ points to the left, it is
necessarily in $\Sigma$. Hence, as $\Sigma$ is closed under composition by
(A0), we are done. So let us assume $g$ points to the right. Now, consider the
diagram

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where the right hand square is a pushout diagram in $\hat{\mathcal{C}}$. By