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\section{Invariants of spaces over some metric space}\label{invariants}
We consider the category of locally connected topological spaces over
some complete metric space \(M\), whose objects are continuous functions
to \(M\) and whose morphisms between two given functions
\(f \colon X \rightarrow M\) and \(g \colon Y \rightarrow M\) are the
continuous maps \(\varphi \colon X \rightarrow Y\) such that \[
\xymatrix{
X \ar[rr]^{\varphi} \ar[dr]_f & & Y \ar[dl]^g \\
& M
}
\] commutes. In the following we will consider several invariants
(mostly given as functors to other categories and mostly in the special
case where \(M = \R\)) under isomorphisms of objects in this category.
\subsection{Spatial Invariants}\label{spatial-invariants}
\newcommand{\dis}{\operatorname{dis}}
\hypertarget{display-space}{\subsubsection{The Display
Space}\label{display-space}}
We summarize some concluding results from (Funk 1995). We assume that
\(M\) is a complete metric space.
\begin{itemize}
\tightlist
\item
\textbf{Definition.} For a locally connected topological space \(X\),
we denote by \(\Lambda(X)\) it's set of connected components. Given an
open subset \(U \subset X\) we denote by \(\Lambda(U)\) the set of
connected components of \(U\), where we augment \(U\) with the
subspace topology. This defines a cosheaf on \(X\) with values in the
category of sets. Given a continuous map \(f \colon X \rightarrow M\)
we denote by \(\lambda_f\) the pushforward \(f_* \Lambda\) and obtain
the functor \(\lambda\) from the category of locally connected
topological spaces over \(M\) to the category of set-valued cosheaves
on \(M\).
\end{itemize}
For a set-valued (pre)cosheaf Funk (1995) provides a construction
similar to the étalé space of a (pre)sheaf.
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Given a cosheaf \(D\) on \(M\) we define it's
\emph{display space} \(\dis D\) as the disjoint union of costalks
\(S_p := \big\{ x \in \prod_{p \in U \in \mathcal{O}(M)} D U ~ \big| ~ D_{U, V} \big( \pi_U (x) \big) = \pi_V (x) ~~ \text{for all} ~~ U \subseteq V \big\}\)
over all \(p \in M\) where \(\mathcal{O} (M)\) is the set of open
subsets of \(M\). Further we define
\(\gamma_D \colon \dis D \rightarrow M\) by \(\gamma_D (x) = p\) for
all \(x \in S_p\). To specify a topology on \(\dis D\) we provide as a
basis \(\{(U, b)\}_{U \in \mathcal{O}(M),~ b \in D U}\) where
\((U, b) := \{x \in \gamma_D^{-1} (U) ~|~ \pi_U (x) = b\}\) for
\(U \in \mathcal{O}(M)\) and \(b \in D U\).
\end{itemize}
By Funk (1995 Theorem 6.1) \(\dis D\) is locally connected for any
cosheaf \(D\) on \(M\).
\begin{itemize}
\tightlist
\item
\textbf{Assumption.} From this point on we assume all topological
spaces to be locally connected.
\end{itemize}
We continue to specify a natural transformation \(\eta\) from \(\id\) to
\(\gamma \circ \lambda\).
\begin{itemize}
\item
\textbf{Definition.} Given a continuous map
\(f \colon X \rightarrow M\), \(x \in X\) and \(U \in \mathcal{O}(M)\)
with \(f(x) \in U\) let
\([x]_U \in \lambda_f (U) = \Lambda (f^{-1} (U))\) be the connected
component of \(x\). Now we define
\(\eta_f \colon X \rightarrow \dis \lambda_f, x \mapsto ([x]_U)_{f(x) \in U \in \mathcal{O}(M)}\).
\item
\textbf{Lemma.} With \(\eta_f\) defined as above we have
\((\gamma \circ \lambda)_f \circ \eta_f = f\).
\item
\emph{Proof.} Given \(x \in X\) we have by the definition of
\(\eta_f\) that \(\eta_f (x) \in S_{f(x)}\), hence
\((\gamma \circ \lambda)_f (\eta_f (x)) = f(x)\) by the definition of
\((\gamma \circ \lambda)_f\).
\item
\textbf{Lemma.} The map \(\eta_f\) as defined above is continuous.
\item
\emph{Proof.} Given \(U \in \mathcal{O}(M)\) and
\(b \in \lambda_f (U) = \Lambda (f^{-1} (U))\) we need to show that
\(\eta_f^{-1} (\{x \in (\gamma \circ \lambda)_f^{-1} (U) ~|~ \pi_U (x) = b\})\)
is open. To do so we will show that
\(\eta_f^{-1} (\{x \in (\gamma \circ \lambda)_f^{-1} (U) ~|~ \pi_U (x) = b\}) = b\)
which is open, since \(X\) is locally connected. Suppose that
\(x \in X\) is such that
\(\eta_f (x) \in (\gamma \circ \lambda)_f^{-1} (U)\) and
\(\pi_U (\eta_f (x)) = b\), then
\(x \in \eta_f^{-1} ((\gamma \circ \lambda)_f^{-1} (U)) = f^{-1} (U)\)
by the previous lemma. Further we have
\(b = \pi_U (\eta_f (x)) = \pi_U (([x]_V)_{f(x) \in V \in \mathcal{O}(M)} = [x]_U\),
hence \(x \in b\). The converse follows from a similar argument.
\item
\textbf{Definition.} Let \(f \colon X \rightarrow M\) be continuous,
then \(f\) is a \emph{cosheaf space over} \(M\) if \(\eta_f\) is a
homeomorphism.
\end{itemize}
By (Funk 1995, Theorem 5.9 and Remark 5.10) \(\lambda\) and \(\gamma\)
form a pair of adjoint functors \(\lambda \dashv \gamma\) with unit
\(\eta\). Further the counit \(\varepsilon\) for this adjunction is a
natural isomorphism by (Funk 1995, Theorem 6.1). We summarize this as a
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\tightlist
\item
\textbf{Theorem.} \(\lambda\) and \(\gamma\) form a pair of adjoint
functors \(\lambda \dashv \gamma\) with unit \(\eta\) and whose counit
\(\varepsilon\) is an isomorphism.
\end{enumerate}
\begin{itemize}
\item
\textbf{Corollary.} The category of cosheaves on \(M\) is equivalent
to the reflective subcategory of cosheaf spaces over \(M\).
\item
\emph{Proof.} This follows with (Gabriel and Zisman 1967, Proposition
1.3 or
\url{http://ncatlab.org/nlab/show/reflective+subcategory\#characterizations}).
\item
\emph{Remark.} Beyond the above Funk (1995 Theorem 5.17) provides a
topological characterization of cosheaf spaces which hasn't been
mentioned here.
\end{itemize}
\hypertarget{reeb-space}{\subsubsection{The Reeb
Space}\label{reeb-space}}
de Silva, Munch, and Patel (2015) observed that \(\gamma \circ \lambda\)
is closely related to another endofunctor on topological spaces over
\(M\), the Reeb space.
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Given a continuous map
\(f \colon X \rightarrow M\) and \(x \in X\) let \(\pi_f (x)\) be the
connected component of \(x\) in \(f^{-1}(f(x))\). In this way we
obtain a function \(\pi_f \colon X \rightarrow 2^X\) and we endow
\(\pi_f (X)\) with the quotient topology\footnote{This is in line with
the \protect\hyperlink{display-space}{previously made} assumption,
since quotient spaces of locally connected spaces are again locally
connected.}. By the universal property of the quotient space there
is a unique continuous function
\(\tilde{f} \colon \pi_f (X) \rightarrow M\) such that
\(\tilde{f} \circ \pi_f = f\) and we define \(\rho_f = \tilde{f}\).
\end{itemize}
With this definition \(\rho\) forms an endofunctor on topological spaces
over \(M\) and \(\pi\) a natural transformation from \(\id\) to
\(\rho\). Given a continuous map \(f \colon X \rightarrow M\) for a
locally connected topological space \(X\) the universal property of the
quotient space induces a unique map
\(\phi_f \colon \pi_f (X) \rightarrow \dis \lambda_f\) such that
\(\phi_f \circ \pi_f = \eta_f\) and thus in particular
\(\rho_f = (\gamma \circ \lambda)_f \circ \pi_f\), hence we have the
following commutative diagram \[
\xymatrix{
& \id \ar[dl]_{\pi} \ar[dr]^{\eta} \\
\rho \ar[rr]_{\phi} & & \gamma \circ \lambda
}
\] in the category of endofunctors on locally connected topological
spaces over \(M\).
\begin{itemize}
\item
\textbf{Proposition.} The natural transformation
\(\lambda \circ \phi\) from \(\lambda \circ \rho\) to
\(\lambda \circ (\gamma \circ \lambda)\) is an isomorphism.
\item
\emph{Proof.} We apply \(\lambda\) to the previous diagram and obtain
\[
\xymatrix{
& \lambda \ar[dl]_{\lambda \circ \pi} \ar[dr]^{\lambda \circ \eta} \\
\lambda \circ \rho \ar[rr]_{\lambda \circ \phi} & &
\lambda \circ \gamma \circ \lambda .
}
\] Since \(\lambda \circ \pi\) is an isomorphism, it suffices to show
that \(\lambda \circ \eta\) is an isomorphism. Given
\(f \colon X \rightarrow M\) we apply the inverse bijection induced by
the adjunction \((\lambda \dashv \gamma, \eta)\) to the diagram \[
\xymatrix{
f \ar[dr]^{\eta_f} \ar[dd]_{\eta_f} \\
& (\gamma \circ \lambda)_f \\
(\gamma \circ \lambda)_f \ar[ur]_{\id}
}
\] and obtain \[
\xymatrix{
\lambda_f \ar[dr]^{\id} \ar[dd]_{(\lambda \circ \eta)_f} \\
& \lambda_f \\
(\lambda \circ \gamma \circ \lambda)_f ,
\ar[ur]_{(\varepsilon \circ \lambda)_f}
}
\] hence \((\lambda \circ \eta)_f\) is the inverse to
\((\varepsilon \circ \lambda)_f\).
\end{itemize}
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{1}
\tightlist
\item
\textbf{Corollary.} \(\phi\) and \(\eta \circ \rho\) are naturally
ismorphic as functors from the category of topological spaces over
\(\R\) to the category of homomorphisms in the category of topological
spaces over \(\R\).
\end{enumerate}
\begin{itemize}
\tightlist
\item
\emph{Example.} Let \(f \colon X \rightarrow \R\) be a
\href{https://en.wikipedia.org/wiki/Proper_map}{proper}
\href{https://en.wikipedia.org/wiki/Morse_theory\#Fundamental_theorems}{Morse
function}, then the critical points of \(f\) are isolated and since
\(f\) is proper, it's critical values are isolated as well. Hence for
each \(r \in \R\) there is an \(\varepsilon_r > 0\) such that for all
\(0 < \delta \leq \varepsilon_r\) the inclusion of \(f^{-1} (r)\) into
\(f^{-1} ((r - \delta, r + \delta))\) is a homotopy equivalence and
thus \(\phi_f\) is a homeomorphism.
\end{itemize}
de Silva, Munch, and Patel (2015) provide a self-contained treatment of
the above when \(\lambda\) and \(\gamma\) are restricted to full
subcategories of topological spaces over \(\R\) respectively cosheaves
on \(\R\). When \(\phi\) is restricted to this subcategory of
topological spaces over \(\R\) referred to as constructible
\(\R\)-spaces, then \(\phi\) is a natural isomorphism. Further the
authors provide a geometric description of the resulting subcategory of
cosheaf spaces over \(\R\). They refer to this category as \textbf{Reeb}
or as the category of \(\R\)-graphs.
\subsubsection{Ascending Cosheaves}\label{ascending-cosheaves}
In addition to the space \(\R\) we consider the reals augmented with a
coarser topology.
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Let \(\bar{\R}\) be the topological space
\((\R, \{(-\infty, r)\}_{-\infty \leq r \leq \infty})\), then we have
the continuous map
\(\id \colon \R \rightarrow \bar{\R}, x \mapsto x\).
\end{itemize}
We can pushforward cosheaves on \(\R\) to cosheaves on \(\bar{\R}\) via
\(\id\).
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Given a cosheaf \(F\) on \(\R\) and
\(-\infty \leq r \leq \infty\) we define
\(\id_* F ((-\infty, r)) = F ((-\infty, r))\).
\end{itemize}
Similar to defining the pullback for sheaves, we take two steps to
define the pullback of a cosheaf via \(\id\).
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Given a cosheaf \(F\) on \(\bar{\R}\) and an open
subset \(U \subseteq \R\) we define
\(\id^+ F (U) := F ((-\infty, \sup U))\).
\end{itemize}
With this definition \(\id^+ F\) is merely a precosheaf for all we know.
Yet we have the following.
\begin{itemize}
\item
\textbf{Lemma.} Given a cosheaf \(F\) on \(\bar{\R}\) the precosheaf
\(\id^+ F\) is a cosheaf on the poset of open intervals.
\item
\emph{Proof.} Let \((a, b) = \bigcup_{i \in I} (a_i, b_i)\). Without
loss of generality we assume that \(I\) has a linear order such that
\(b_i \leq b_j\) for all \(i \leq j\) and \(a_i \leq a_j\) for all
\(i \leq j\) with \(b_i = b_j\). Since
\(b = \sup_{i \in I} \sup (a_i, b_i) = \sup_{i \in I} b_i\) and since
\(F\) is a cosheaf we have the coequalizer diagram \[
\xymatrix{
\coprod_{i < j} F((-\infty, b_i))
\ar@/^3pt/[r]^{\sigma} \ar@/_3pt/[r]_{\sigma'} &
\coprod_{i \in I} F((-\infty, b_i)) \ar[r] &
F((-\infty, b))
}
\] where \(\sigma_{i,j}\) maps \(F((-\infty, b_i))\) identical to
\(F((-\infty, b_i))\) and \(\sigma'_{i,j}\) maps \(F((-\infty, b_i))\)
to \(F((-\infty, b_j))\) via the induced inclusion. Now suppose we
have \(i < j\) such that \(b_i \leq a_j\), then since \([b_i, a_j]\)
is compact we can find \(i < i_1 < ... < i_k\) such that
\([b_i, a_j] \subseteq \bigcup_{l=1}^k (a_{i_l}, b_{i_l})\) and such
that this cover is minimal. If \(i_k < j\) we set
\(\tau = \sigma_{i_k, j}\), \(\tau' = \sigma'_{i_k, j}\) and if
\(j < i_k\) we set \(\tau = \sigma_{j, i_k}\),
\(\tau' = \sigma'_{j, i_k}\). With this we have
\(\sigma'_{i, j} \circ \sigma_{i, j}^{-1} = \tau' \circ \tau^{-1} \circ \sigma'_{i_{k-1}, i_k} \circ \sigma_{i_{k-1}, i_k}^{-1} \circ ... \circ \sigma'_{i_1, i_2} \circ \sigma_{i_1, i_2}^{-1} \circ \sigma'_{i, i_1} \circ \sigma_{i, i_1}^{-1}\)
and yet at the same time \(b_j > a_{i_k}\), \(b_{i_k} > a_j\),
\(b_{i_{k-1}} > a_{i_k}\), \ldots{}, \(b_{i_1} > a_{i_2}\), and
\(b_i > a_{i_1}\), since \([b_i, a_j]\) is connected and the cover is
minimal. Thus we may omit all terms from the leftmost coproduct in the
above diagram where \(b_i \leq a_j\) without loosing the property of
it being a coequalizer diagram. Now for any \(i < j\) such that
\(b_i > a_j\) we have
\(\id^+ F((a_i, b_i) \cap (a_j, b_j)) = F((-\infty, b_i))\), hence we
may replace the corresponding term in the leftmost coproduct by
\(\id^+ F((a_i, b_i) \cap (a_j, b_j))\). Similarly we may replace the
terms in the middle and the term on the right to arrive at a
coequalizer diagram of the form \[
\xymatrix@C-8pt{
\coprod_{i < j,\, b_i > a_j} \id^+ F((a_i, b_i) \cap (a_j, b_j))
\ar@/^3pt/[r] \ar@/_3pt/[r] &
\coprod_{i \in I} \id^+ F((a_i, b_i)) \ar[r] &
\, \id^+ F((a, b)).
}
\] Now for \(i < j\) such that \(b_i \leq a_j\) we have
\(\id^+ F((a_i, b_i) \cap (a_j, b_j)) = \id^+ F(\emptyset) = F(\emptyset) = \emptyset\)
which does not contribute to the coequalizer and this implies the
claim.
\item
\textbf{Definition.} Given a cosheaf \(F\) on \(\bar{\R}\) and an open
subset \(U \subseteq \R\) we define
\(\id^{-1} F (U) := \varinjlim_{(a, b) \subseteq U} \id^+ F ((a, b))\).
\end{itemize}
By the previous lemma \(\id^{-1} F\) as defined above is a cosheaf.
\begin{itemize}
\item
\emph{Remark.} \(\id^{-1} F\) as defined above is isomorphic to the
cosheafification of \(\id^+ F\) or the cosheaf associated to
\(\id^+ F\), see for example (Funk 1995, Theorem 6.3 and Remark 6.4).
\item
\textbf{Definition.} Given a cosheaf \(F\) on \(\R\) we define a
homomorpism \(\eta'_F\) of cosheaves from \(F\) to
\(\id^{-1} \id_* F\). Since both are cosheaves it suffices to define
\(\eta'_F\) on open intervals. So for
\(-\infty \leq a < b \leq \infty\) we define \(\eta'_F\) from
\(F((a, b))\) to
\(\id^{-1} \id_* F ((a, b)) = \id^+ \id_* F ((a, b)) = F((-\infty, b))\)
to be the map induced by the inclusion
\((a, b) \subseteq (-\infty, b)\).
\item
\textbf{Definition.} Let \(F\) be a cosheaf on \(\R\), then \(F\) is
\emph{ascending} if \(\eta'_F\) is an isomorphism.
\end{itemize}
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{2}
\tightlist
\item
\textbf{Proposition.} \(\id_*\) and \(\id^{-1}\) form a pair of
adjoint functors \(\id_* \dashv \id^{-1}\) with unit \(\eta'\) and
whose counit \(\varepsilon'\) is an isomorphism.
\end{enumerate}
\begin{itemize}
\item
\emph{Proof.} Let \(F\) be a cosheaf on \(\R\), let \(G\) be a cosheaf
on \(\bar{\R}\), and let \(g\) be a homomorphism from \(F\) to
\(\id^{-1} G\). Now suppose we have a morphism \(f\) from \(\id_* F\)
to \(G\) such that \((\id^{-1} f) \circ \eta'_F = g\), then for any
\(r \in \R\) we have
\(g_{(-\infty, r)} = \big((\id^{-1} f) \circ \eta'_G \big)_{(-\infty, r)} = f_{(-\infty, r)}\)
and this determines \(f\). Now suppose \(f\) is defined by
\(g_{(-\infty, r)} = f_{(-\infty, r)}\) for any \(r \in \R\) and we
have \(-\infty \leq a < b \leq \infty\), then \(g_{(a, b)}\) is the
same as \(g_{(-\infty, b)}\) pre-composed with the map induced by
inclusion from \(F((a, b))\) to \(F((-\infty, b))\) by naturality. But
this is the same as \(\big((\id^{-1} f) \circ \eta'_G \big)_{(a, b)}\)
by definition of \(f\), hence \(g\) and \((\id^{-1} f) \circ \eta'_G\)
agree on a basis of \(\R\).
By the above argument \(\varepsilon'_G\) is equal to
\(\id_{\, \id^{-1} G}\) when restricted to
\(\id_* \id^{-1} G ((-\infty, r)) = \id^{-1} G ((-\infty, r)) = G ((-\infty, r))\),
hence \(\varepsilon'_G\) is an isomorphism.
\item
\textbf{Corollary.} The category of cosheaves on \(\bar{\R}\) is
equivalent to the reflective subcategory of ascending cosheaves on
\(\R\).
\item
\emph{Proof.} This follows with (Gabriel and Zisman 1967, Proposition
1.3 or
\url{http://ncatlab.org/nlab/show/reflective+subcategory\#characterizations}).
\end{itemize}
\newcommand{\epi}{\operatorname{epi}}
\subsubsection{Ascending Spaces}\label{ascending-spaces}
Later we will make the ascending cosheaf \(\id^{-1} \id_* \lambda_f\)
for a continuous function \(f\) the cosheaf version of the join tree
associated to \(f\). As an intermediate step we show that we can obtain
this cosheaf not only by post-composing \(\lambda\) with
\(\id^{-1} \id_*\) but also by pre-composing \(\lambda\) with another
functor, the epigraph. This use of the epigraph in defining the join
tree\footnote{Though join trees are referred to as merge trees in the
cited paper. } is due to Morozov, Beketayev, and Weber (2013).
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Let \(f \colon X \rightarrow \R\) be a continuous
map, it's \emph{epigraph} is
\(\epi f := \set{(x, y)}{X \times \R}{y \geq f(x)}\). Further we
define \(\iota_f \colon \epi f \rightarrow \R, (x, y) \mapsto y\) and
\(\kappa_f \colon X \rightarrow \epi f, x \mapsto (x, f(x))\).
\end{itemize}
With these definitions \(\iota\) defines a functor on topological spaces
over \(\R\) with \(\kappa\) a natural transformation from \(\id\) to
\(\iota\).
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{3}
\item
\textbf{Definition.} A function \(f \colon X \rightarrow \R\) is
\emph{ascending} if for all \(r \in \R\) there is a continuous map
\(H_r \colon X \times [0, 1] \rightarrow X\) such that
\(H_r (x, t) = x\) for all \(0 \leq t \leq 1\) and \(x \in X\) with
\(f(x) \geq r\) and such that \(f (H_r (x, t)) = r + t(f(x)-r)\) for
all \(0 \leq t \leq 1\) and \(x \in X\) with \(f(x) \leq r\).
\item
\textbf{Lemma.} For any continuous function
\(f \colon X \rightarrow \R\) the projection
\(\iota_f \colon \epi f \rightarrow \R\) is ascending.
\end{enumerate}
\begin{itemize}
\tightlist
\item
\emph{Proof.} For \(r \in \R\) we set
\(H_r \colon \epi f \times [0, 1] \rightarrow \epi f, ((x, y), t) \mapsto (x, \max \{r + t(y-r), y\})\).
\end{itemize}
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{5}
\tightlist
\item
\textbf{Lemma.} For any ascending function
\(f \colon X \rightarrow \R\) the cosheaf \(\lambda_f\) is ascending
as well.
\end{enumerate}
\begin{itemize}
\item
\emph{Proof.} Given \(-\infty \leq a < r < b \leq \infty\) we proof
that the maps from \(\Lambda(f^{-1} ([r, b)))\) to
\(\Lambda(f^{-1} ((a, b))) = \lambda_f ((a, b))\) respectively
\(\Lambda(f^{-1} ((-\infty, b))) = \lambda_f ((-\infty, b))\) induced
by the inclusions are bijections\footnote{The space
\(f^{-1} ([r, b))\) may not be locally connected. However we won't
need this property.}. From this our claim follows. Since inclusions
as maps of spaces always commute the two bijections commute with
\((\eta' \circ \lambda)_f\) as well, hence \((\eta' \circ \lambda)_f\)
is a bijection as a map from \(\lambda_f ((a, b))\) to
\(\lambda_f ((-\infty, b)) = \id_* \lambda_f ((-\infty, b)) = \id^+ \id_* \lambda_f ((a, b)) = \id^{-1} \id_* \lambda_f ((a, b))\).
And since the open intervals of \(\R\) form a basis, the lemma
follows.
Given any point \(x \in f^{-1} ((a, r))\) the map
\(t \mapsto H_r (x, t)\) defines a continuous path in
\(f^{-1} ((a, b))\) from \(H_r (x, 0) \in f^{-1} ([r, b))\) to \(x\),
hence induced map from \(\Lambda(f^{-1} ([r, b)))\) to
\(\Lambda(f^{-1} ((a, b))) = \lambda_f ((a, b))\) is surjective. Now
suppose \(x, y \in f^{-1} ([r, b))\) lie in the same connected
component \(C\) of \(f^{-1} ((a, b))\), then \(H_r (C, 0)\) is
connected since \(H_r\) is continuous. Further
\(x, y \in H_r (C, 0)\), hence the induced map from
\(\Lambda(f^{-1} ([r, b)))\) to \(\lambda_f ((a, b))\) is injective.
The induced map from \(\Lambda(f^{-1} ([r, b)))\) to
\(\lambda_f ((-\infty, b))\) is a bijection by a similar argument.
\end{itemize}
\begin{itemize}
\tightlist
\item
\emph{Remark.} The previous result remains valid if instead of
\(\lambda\) we consider the pushforward of another cosheaf on \(X\)
that maps inclusions of open sets in \(X\) that are homotopy
equivalences to bijections of sets.
\end{itemize}
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{6}
\tightlist
\item
\textbf{Lemma.} Given a continuous map \(f \colon X \rightarrow \R\)
the homomorphism \(\id_* (\lambda \circ \kappa)_f\) from
\(\id_* \lambda_f\) to \(\id_* (\lambda \circ \iota)_f\) is an
isomorphism.
\end{enumerate}
\begin{itemize}
\tightlist
\item
\emph{Proof.} Given \(b \in \R \cup \{\infty\}\) we show that
\(\kappa_f (f^{-1} ((-\infty, b)) = \{(x, f(x))\}_{\{x \in X | f(x) < b\}}\)
is a strong deformation retract of
\(\iota_f^{-1} ((-\infty, b)) = \set{(x, y)}{X \times (-\infty, b)}{y \geq f(x)}\).
Then the result follows by a similar argument as the previous lemma.
We define
\(R \colon \iota_f^{-1} ((-\infty, b)) \times [0, 1] \rightarrow \iota_f^{-1} ((-\infty, b)), ((x, y), t) \mapsto (x, f(x) + t(y-f(x)))\),
then \(R((x, y), 1) = (x, y)\) and \(R((x, y), 0) = (x, f(x))\) for
all \((x, y) \in \iota_f^{-1} ((-\infty, b))\).
\end{itemize}
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{7}
\tightlist
\item
\textbf{Proposition.} The natural transformations
\(\eta' \circ \lambda\) and \(\lambda \circ \kappa\) are isomorphic as
objects in the category of functors from topological spaces over
\(\R\) to cosheaves on \(\R\) under \(\lambda\). In particular
\(\id^{-1} \id_* \lambda\) and \(\lambda \circ \iota\) are naturally
isomorphic.
\end{enumerate}
\begin{itemize}
\tightlist
\item
\emph{Proof.} Given \(f \colon X \rightarrow \R\) we have the
commutative diagram \[
\xymatrix@C+2pc{
\lambda_f
\ar[r]^{(\lambda \circ \kappa)_f}
\ar[d]|-{(\eta' \circ \lambda)_f} &
(\lambda \circ \iota)_f
\ar[d]|-{(\eta' \circ \lambda \circ \iota)_f} \\
\, \id^{-1} \id_* \lambda_f
\ar[r]_{\id^{-1} \id_* (\lambda \circ \kappa)_f} &
\, \id^{-1} \id_* (\lambda \circ \iota)_f .
}
\] By lemma 5 and lemma 6 the homomorphism
\((\eta' \circ \lambda \circ \iota)_f\) is an isomorphism. And by
lemma 7 we have that \(\id^{-1} \id_* (\lambda \circ \kappa)_f\) is an
isomorphism.
\end{itemize}
\subsubsection{The Join Tree}\label{join-tree}
The following definition\footnote{Though join trees are referred to as
merge trees in the cited paper. } is from (Morozov, Beketayev, and
Weber 2013).
\begin{itemize}
\tightlist
\item
\textbf{Definition.} Let \(f \colon X \rightarrow \R\) be a continuous
map we define it's \emph{join tree} to be the continuous map
\((\rho \circ \iota)_f\) from \((\pi \circ \iota)_f (\epi f)\) to
\(\R\).
\end{itemize}
With this definition \(\rho \circ \iota\) is an endofunctor on
topological spaces over \(\R\). Given a continuous map
\(f \colon X \rightarrow \R\) we have
\((\pi \circ \iota)_f \circ \kappa_f = (\rho \circ \kappa)_f \circ \pi_f\),
so in somewhat sloppy notation
\((\pi \circ \iota) \circ \kappa = (\rho \circ \kappa) \circ \pi\) is a
natural transformation from \(\id\) to \(\rho \circ \iota\). Similarly
we have the function \((\gamma \circ \lambda \circ \iota)_f\) defined on
the display space \(\dis (\lambda \circ \iota)_f\) of
\((\lambda \circ \iota)_f\). And just as with \(\rho\) we have the
natural transformation
\((\eta \circ \iota) \circ \kappa = (\gamma \circ \lambda \circ \kappa) \circ \eta\)
from \(\id\) to \(\gamma \circ \lambda \circ \iota\). The two
constructions are related via the commutative diagram \[
\xymatrix@C+1pc@R-1pc{
&
\rho_f \ar[r]^{(\rho \circ \kappa)_f} \ar[dd]|-{\phi_f} &
(\rho \circ \iota)_f \ar[dd]^{(\phi \circ \iota)_f} \\
f \ar[ur]^{\pi_f} \ar[dr]_{\eta_f} \\
&
(\gamma \circ \lambda)_f
\ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} &
(\gamma \circ \lambda \circ \iota)_f
}
\] given a function \(f \colon X \rightarrow \R\). In the
\protect\hyperlink{reeb-space}{section on the Reeb space} we considered
the left triangle which suggests to replace the Reeb graph functor
\(\rho\) and the natural transformation \(\pi\) by
\(\gamma \circ \lambda\) and \(\eta\) respectively. Now
\(\rho \circ \kappa\) yields a nice and classic map from any Reeb graph
to the corresponding join tree, so our replacement of the Reeb graph
functor \(\rho\) by \(\gamma \circ \lambda\) is only complete, if also
we can replace the join tree functor \(\rho \circ \iota\) and the
natural transformation \(\rho \circ \kappa\) and if we can extend
\(\phi\) to a natural transformation from \(\rho \circ \kappa\) to it's
replacement. And here the commutative square on the right hand side,
suggests we may take \(\lambda \circ \gamma \circ \iota\) as a
replacement for the join tree functor \(\rho \circ \iota\) and to take
\(\gamma \circ \lambda \circ \kappa\) as a replacement for
\(\rho \circ \kappa\), since then we can extend \(\phi\) by
\(\phi \circ \iota\) to a natural transformation from
\(\rho \circ \kappa\) to \(\gamma \circ \lambda \circ \kappa\). We
further note that by corollary 2 \protect\hyperlink{reeb-space}{in that
section} the natural transformation \((\phi, \phi \circ \iota)\) from
\(\rho \circ \kappa\) to \(\gamma \circ \lambda \circ \kappa\) is
isomorphic to the natural transformation
\((\eta \circ \rho, \eta \circ \rho \circ \iota)\) from
\(\rho \circ \kappa\) to
\(\gamma \circ \lambda \circ \rho \circ \kappa\), so our choice of
replacements is the same as if we applied \(\gamma \circ \lambda\) to
the upper row in the diagram. And by proposition 8 we have a natural
isomorphism from \((\gamma \circ \lambda \circ \iota)\) to
\(\gamma \id^{-1} \id_* \lambda\) that commutes with
\(\gamma \circ \lambda \circ \kappa\) and
\(\gamma \circ \eta' \circ \lambda\) sothat we can use the following
\begin{itemize}
\item
\textbf{Proposition.} \(\id_* \lambda\) and \(\gamma \id^{-1}\) form a
pair of adjoint functors \(\id_* \lambda \dashv \gamma \id^{-1}\) with
unit \((\gamma \circ \eta' \circ \lambda) \circ \eta\) and whose
counit is an isomorphism.
\item
\emph{Proof.} The first statement follows from theorem 1, proposition
3 and the general statement that the two pairs of adjoint functors,
when composed in the same way as in our claim, form again a pair of
adjoint functors with the unit described as in the claim, see for
example
\url{https://en.wikipedia.org/wiki/Adjoint_functors\#Composition}. And
for the counit of this composed adjunction we have the formula
\(\id_* \varepsilon \id^{-1} \circ \varepsilon'\). By theorem 1
\(\varepsilon\) is an isomorphism, hence
\(\id_* \varepsilon \id^{-1}\) is an isomorphism and by proposition 3
\(\varepsilon'\) is an isomorphism and thus our claim follows.
\item
\textbf{Corollary.} The category of cosheaves on \(\bar{\R}\) is
equivalent to the reflective subcategory of ascending cosheaf spaces
over \(\R\).
\item
\emph{Proof.} By Gabriel and Zisman (1967 Proposition 1.3 or
\url{http://ncatlab.org/nlab/show/reflective+subcategory\#characterizations})
the category of cosheaves on \(\bar{\R}\) is equivalent to the
reflective subcategory of those spaces \(f \colon X \rightarrow \R\)
over \(\R\) for which
\((\gamma \circ \eta' \circ \lambda)_f \circ \eta_f\) is an
isomorphism. Now suppose this is the case for \(f\), then \(f\) is
isomorphic to \(\gamma \id^{-1} \id_* \lambda\) which is in the image
of \(\gamma\) and thus a cosheaf space, hence \(\eta_f\) is an
isomorphism. From this it follows that
\((\gamma \circ \eta' \circ \lambda)_f\) is an isomorphism as well,
hence by proposition 8 \((\gamma \circ \lambda \circ\kappa)_f\) is an
isomorphism. Now we consider the commutative diagram \[
\xymatrix@C+1pc{
f \ar[r]^{\kappa_f} \ar[d]_{\eta_f} &
\iota_f \ar[d]^{(\eta \circ \iota)_f} \\
(\gamma \circ \lambda)_f
\ar[r]_{(\gamma \circ \lambda \circ \kappa)_f} &
(\gamma \circ \lambda \circ \iota)_f .
}
\] Hence we have the retract\footnote{By a retract we mean a
homomorphism \(R\) in the category of topological spaces over \(\R\)
from \(\iota_f\) to \(f\) such that \(R \circ \kappa_f = \id\).}
\(R := \eta_f^{-1} \circ (\gamma \circ \lambda \circ \kappa)_f^{-1} \circ (\eta \circ \iota)_f\)
from \(\iota_f\) to \(f\). By lemma 5 \(\iota_f\) is ascending, so
given \(r \in \R\) there is a map \(H_r\) as in definition 4. Now let
\(\tilde{H}_r \colon X \times [0, 1] \rightarrow X\) be defined by
\(\tilde{H}_r (x, t) = r(H_r(\kappa_f(x), t))\) then \(\tilde{H}_r\)
inherits the properties needed in order for \(f\) to be ascending.
Conversely if \(f\) is an ascending cosheaf space over \(\R\), then
\(\eta_f\) is an isomorphism since \(f\) is a cosheaf space. And by
lemma 6 \(\lambda_f\) is ascending, hence \((\eta' \circ \lambda)_f\)
is an isomorphism.
\end{itemize}
In conclusion \((\gamma \circ \lambda \circ \iota)_f\) is an ascending
cosheaf space over \(\R\) given a function \(f\). It's cosheaf of
connected components
\((\lambda \circ \gamma \circ \lambda \circ \iota)_f\) is isomorphic to
\((\lambda \circ \iota)_f\) by theorem 1. By lemma 5 and lemma 6
\((\lambda \circ \iota)_f\) is ascending, and thus we have an associated
cosheaf \(\id_* (\lambda \circ \iota)_f\) on \(\bar{\R}\) via the
adjunction \(\id_* \dashv \id^{-1}\) by proposition 3. By lemma 7 this
cosheaf is isomorphic to \(\id_* \lambda_f\) which is the cosheaf on
\(\bar{\R}\) associated to \(\gamma \id^{-1} \id_* \lambda_f\) via the
adjunction \(\id_* \lambda \dashv \gamma \id^{-1}\). Now applying
\(\id^{-1}\) to \(\id_* \lambda_f \cong \id_* (\lambda \circ \iota)_f\)
recovers \((\lambda \circ \iota)_f\), hence
\((\gamma \circ \lambda \circ \iota)_f\) and
\(\gamma \id^{-1} \id_* \lambda_f\) are isomorphic and thus a posteriori
\(\id_* \lambda_f\) is the cosheaf on \(\bar{\R}\) associated to the
ascending cosheaf space \((\gamma \circ \lambda \circ \iota)_f\) via the
adjunction \(\id_* \lambda \dashv \gamma \id^{-1}\). (Here the author
allowed himself some redundance repeating the proof of proposition 8.)
\subsection{From Cosheaves to Sheaves}\label{cosheaves-to-sheaves}
\subsubsection{From Sets to Algebras}\label{sets-to-algebras}
For an integral domain \(A\) we consider the contravariant functor
\(\hom(\_, A)\) from the category of sets to the category of commutative
unital \(A\)-algebras. We note that since \(A\) is an integral domain
the idempotents of \(\hom(L, A)\) for any set \(L\) are precisely the
maps from \(L\) to \(A\) with values in \(\{0, 1\}\).
\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\setcounter{enumi}{8}
\tightlist
\item
\textbf{Lemma.} \(\hom(\_, A)\) is
\href{http://ncatlab.org/nlab/show/pseudomonic+functor}{pseudomonic}.
\end{enumerate}
\begin{itemize}
\item
\emph{Proof.} \(\hom(\_, A)\) is faithful since for any map
\(m \colon L \rightarrow K\) and \(k \in K\) we have
\(m^{-1} (k) = \big(\hom(m, A) (1_k)\big)^{-1} (1)\) where
\(1_k := 1_{\{k\}}\) and \(1_{K'}\) is the
\href{https://en.wikipedia.org/wiki/Indicator_function}{indicator
function} for any subset \(K' \subseteq K\).
Now suppose \(\varphi\) is an isomorphism from \(\hom(K, A)\) to
\(\hom(L, A)\) then \(\varphi\) induces a bijection between the
non-zero
\href{https://en.wikipedia.org/wiki/Idempotent_element\#Types_of_ring_idempotents}{centrally
primitive} idempotents of \(\hom(K, A)\) and \(\hom(L, A)\). Now the
non-zero centrally primitive idempotents of \(\hom(K, A)\) are just
the maps of the form \(1_k\) for some \(k \in K\) and similarly for
\(\hom(L, A)\). Let \(m \colon L \rightarrow K\) be the corresponding
inverse bijection, then for any \(c \in \hom(K, A)\) and \(l \in L\)
we have \[
\begin{split}
\varphi(c) \cdot 1_l & = \varphi(c \cdot 1_{m(l)})
= \varphi(c(m(l)) 1_{m(l)}) \\
& = c(m(l)) \varphi(1_{m(l)})
= c(m(l)) 1_l \\
& = \hom(m, A) (c) \cdot 1_l
\end{split}
\] and thus \(\varphi = \hom(m, A)\).
\item
\textbf{Corollary.} The functor \(\hom(\_, A)\) induces an
anti-equivalence between the category of sets and the replete image of
\(\hom(\_, A)\).
\item
\textbf{Corollary.} For any category \(\mathcal{C}\) the functor
\(\hom(\_, A)\) induces an anti-equivalence between the category of
set-valued precosheaves on \(\mathcal{C}\) and the category of
presheaves with values in the replete image of \(\hom(\_, A)\).
\item
\textbf{Lemma.} \(\hom(\_, A)\) is full when restricted to the
category of finite sets.
\item
\emph{Proof.} Let \(\varphi \colon \hom(K, A) \rightarrow \hom(L, A)\)
be a homomorphism with \(K\) and \(L\) finite, then \(\varphi(1_k)\)
is an idempotent for each \(k \in K\) and thus we have subsets
\(L_k \subseteq L\) such that \(\varphi(1_k) = 1_{L_k}\). Further we
have
\(\sum_{l \in L} 1_l = 1 = \varphi(1) = \varphi\big(\sum_{k \in K} 1_k) = \sum_{k \in K} \varphi(1_k) = \sum_{k \in K} 1_{L_k}\)
and thus \(L = \bigcup_{k \in K} L_k\). Now for any \(k, k' \in K\)
with \(k \neq k'\) we have
\(0 = \varphi(0) = \varphi (1_k \cdot 1_{k'}) = 1_{L_k} \cdot 1_{L_k'}\),
hence \(L_k\) and \(L_{k'}\) are disjoint. Altogether we obtain that
the subsets \(L_k\) with \(k \in K\) form a partition of \(L\) and we
may define a map \(m \colon L \rightarrow K\) such that \(m(l) = k\)
for \(l \in L_k\) for all \(k \in K\). With this definition we have
\(\varphi = \hom(m, A)\) since the two maps agree on a basis of
\(\hom(K, A)\).
\end{itemize}
The following example shows that we cannot assume the unrestricted
functor \(\hom(\_, A)\) to be full, if \(A\) is a general ring.
\begin{itemize}
\item
\emph{Example.} We consider \(\hom (\N, \Z / p \Z)\). Let
\(\mathfrak{a}\) be the ideal of all \(c \in \hom (\N, \Z / p \Z)\)
with \(c^{-1} (0)\) cofinite. By
\href{https://en.wikipedia.org/wiki/Krull's_theorem}{Krull's theorem}
\(\hom (\N, \Z / p \Z)\) has a maximal ideal \(\mathfrak{m}\) with
\(\mathfrak{a} \subset \mathfrak{m}\) and this gives a homomorphism of
fields
\(i \colon \Z / p \Z \rightarrow \hom (\N, \Z / p \Z) / \mathfrak{m}\).
We further have \([c]^p - [c] = [c^p - c] = 0\) for all
\([c] \in \hom (\N, \Z / p \Z) / \mathfrak{m}\) and as \(X^p - X\) is
a polynomial of degree \(p\) it has at most \(p\) roots in
\(\hom (\N, \Z / p \Z) / \mathfrak{m}\) and thus \(i\) is a bijection.
Now the canonical homomorphism from \(\hom (\N, \Z / p \Z)\) to the
quotient \(\hom (\N, \Z / p \Z) / \mathfrak{m}\) yields a homomorphism
\(\varphi \colon \hom (\N, \Z / p \Z) \rightarrow \hom(\{1\}, \Z / p \Z) \cong \Z / p \Z\)
which is not in the image of \(\hom(\_, \Z / p \Z)\), since for any
map \(m \colon \{1\} \rightarrow \N\) the element
\(1_{m(1)} \in \mathfrak{a} \subset \mathfrak{m}\) is mapped to
\(1 \in \Z / p \Z\) under \(\hom(m, \Z / p \Z)\).
\item
\emph{Remark.} From a discussion similar to that of the previous lemma
and example we can conclude that for sets \(K\) and \(L\) with \(L\)
non-empty, the map from \(\hom(L, K)\) to
\(\hom_{A\text{-algebras}} (\hom(K, A), \hom(L, A))\) induced by
\(\hom(\_, A)\) is surjective if and only if all ideals\footnote{which
are prime necessarily} \(\mathfrak{p}\) of \(\hom(K, A)\), with
\(\hom(K, A) / \mathfrak{p} \cong A\) as \(A\)-algebras, are of the
form \(\set{c}{\hom(K, A)}{c(k) = 0}\) for some \(k \in K\).
\item
\textbf{Lemma.} \(\hom(\_, A)\) is continuous as a functor from the
opposed category of sets to the category of \(A\)-algebras.
\item
\emph{Proof.} We argue that \(\hom(\_, A)\) is continuous as a functor
to the category of commutative rings, the lemma then follows by
\href{http://ncatlab.org/nlab/show/overcategory\#LimitsAndColimits}{a
general result about limits in the under category}. We fix a small
category \(D\). For an object \(X\) of any category \(\mathcal{C}\) we
denote by \(\Delta (X)\) the constant functor from \(D\) to
\(\mathcal{C}\) that maps any object of \(D\) to \(X\) and any
morphism of \(D\) to the identity. Let \(F\) be a functor from \(D\)
to the category of sets, then we have the canonical natural
transformation \(t \colon F \rightarrow \Delta( \colim (F) )\). Now
\(\hom(\Delta(\colim(F)), A) = \Delta(\hom(\colim(F), A))\) and by the
universal property of the limit of \(\hom(F(\_), A)\) we have a
homomorphism of rings
\(s \colon \lim(\hom(F(\_), A)) \rightarrow \hom(\colim(F), A)\) such
that \((\hom(\_, A) \circ t) \circ \Delta(s)\) is the canonical
natural transformation from \(\Delta(\lim(\hom(F(\_), A)))\) to
\(\hom(F(\_), A)\). Now the forgetful functor from the category of
commutative rings to the category of sets is continuous as well as
\(\hom(\_, A)\) as a functor to the category of sets, hence in the
category of sets both \((\hom(\_, A) \circ t) \circ \Delta(s)\) and
\(\hom(\_, A) \circ t\) itself satisfy the universal property of the
limit of \(\hom(F(\_), A)\), and thus \(s\) is a bijection.
\item
\textbf{Corollary.} If \(D\) is a set-valued cosheaf, then
\(\hom(D(\_), A)\) defines a sheaf with values in the category of
\(A\)-algebras.
\item
\emph{Example.} For any locally path connected topological space \(X\)
the singular homology \(H_0 (X)\) is naturally isomorphic to the free
abelian group with basis \(\Lambda(X)\) and by the universal property
of the free ablian group the restriction from
\(\hom_{\Z} (H_0 (X), A)\) to \(\hom(\Lambda(X), A)\) is an
isomorphism of \(A\)-modules. Further we have a natural isomorphism of
\(A\)-modules from \(H^0 (X, A)\) to \(\hom_{\Z} (H_0 (X), A)\) by the
universal coefficient theorem and since for any \(x \in X\) and
\(\alpha, \beta \in H^0 (X, A)\) we have \[
\begin{split}
\langle \alpha \cup \beta, [x] \rangle
& = \langle H^0 (d, A) (\alpha \times \beta), [x] \rangle
= \langle \alpha \times \beta, H_0 (d) ([x]) \rangle \\
& = \langle \alpha \times \beta, [(x, x)] \rangle
= \langle \alpha \times \beta, [x] \times [x] \rangle \\
& = \langle \alpha, [x] \rangle \langle \beta, [x] \rangle,
\end{split}
\] where \(d \colon X \rightarrow X \times X, x \mapsto (x, x)\) is
the diagonal map, the composition of these two isomorphisms is an
ismorphism of \(A\)-algebras. Since the above identifications are
natural in \(X\), the functors \(\hom(\Lambda(\_), A)\) and
\(H^0 (\_, A)\) define isomorphic sheaves on any locally path
connected topological space.
Given a continuous function \(f \colon X \rightarrow M\) from a
locally path connected topological space \(X\) to \(M\), the sheaves
\(f_* \hom(\Lambda(\_), A) \cong f_* H^0 (\_, A)\) and
\(\hom(\lambda_f (\_), A)\) are identical. Bubenik, de Silva, and
Scott (2014) define a generalized persistence module on the poset of
open sets of \(M\) to be a functor to another category, thus
\(\lambda_f\) is a generalized persistence module with values in the
opposed category of sets and \(f_* H^0 (\_, A)\) is a persistence
module with values in the category of \(A\)-algebras. A functor from a
category \(\mathcal{C}\) to a category \(\mathcal{D}\) then gives rise
to a map from the generalized persistence modules with values in
\(\mathcal{C}\) to persistence modules with values in \(\mathcal{D}\),
so in their language \(f_* H^0 (\_, A)\) is the image of \(\lambda_f\)
under the map induced by \(\hom(\_, A)\) and thus their theory can be
used to relate these two constructions in the context of topological
persistence.
\end{itemize}
\section*{References}\label{references}
\addcontentsline{toc}{section}{References}
\hypertarget{refs}{}
\hypertarget{ref-bubenik2014}{}
Bubenik, Peter, Vin de Silva, and Jonathan Scott. 2014. ``Metrics for
Generalized Persistence Modules.'' \emph{Foundations of Computational
Mathematics}. Springer US, 1--31.
doi:\href{https://doi.org/10.1007/s10208-014-9229-5}{10.1007/s10208-014-9229-5}.
\hypertarget{ref-deSilva2015}{}
de Silva, Vin, Elizabeth Munch, and Amit Patel. 2015. ``Categorification
of Reeb Graphs.'' \emph{ArXiv:1501.04147}.
\url{http://arxiv.org/abs/1501.04147}.
\hypertarget{ref-MR1322801}{}
Funk, J. 1995. ``The Display Locale of a Cosheaf.'' \emph{Cahiers
Topologie Géom. Différentielle Catég.} 36 (1): 53--93.
\hypertarget{ref-MR0210125}{}
Gabriel, P., and M. Zisman. 1967. \emph{Calculus of Fractions and
Homotopy Theory}. Ergebnisse Der Mathematik Und Ihrer Grenzgebiete, Band
35. Springer-Verlag New York, Inc., New York.
\hypertarget{ref-morozov2013}{}
Morozov, Dmitriy, Kenes Beketayev, and Gunther Weber. 2013.
``Interleaving Distance Between Merge Trees.'' In \emph{Proceedings of
TopoInVis}.
\url{http://www.mrzv.org/publications/interleaving-distance-merge-trees/}.
\end{document}