Proof The forward implication is [Reference Adámek, Herrlich and StreckerAHS06, Proposition 11.11]. The converse is straightforward.

Proof Let $e \colon \thinspace A \to B$ be an epimorphism and $m \colon \thinspace X \to Y$ be a monomorphism. Suppose that $f \colon \thinspace A \to X$ and $g \colon \thinspace B \to Y$ are morphisms such that $mf = ge$ . If m is a strong mono, then there exists a unique morphism $d \colon \thinspace B \to X$ such that $f = de$ and $md = g$ ; hence, e is a strong epi. Similarly, if e is a strong epi, then such d exists, and hence m is a strong mono.

Proof By Proposition 2.12, the upper left square in the commutative diagram

is a pushout in $\overline {\boldsymbol {\mathsf {C}} }$ . Identify the underlying set of $X \sqcup _{A \sqcup X} X$ with $X \times \left \{ {0,1} \right \} $ ; thus, there is a glued edge between $(x,0)$ and $(x,1)$ for each $x \in X$ , and a glued edge between $(fa,0)$ and $(ga,1)$ for each $a \in A$ . Define a map $r \colon \thinspace X \sqcup _{A \sqcup X} X \to \mathrm {Coeq}(f,g)$ coinciding with the underlying projection $X \times \left \{ {0,1} \right \} \to X$ . Any pair of points in $X \sqcup _{A \sqcup X} X$ connected by an edge of weight w are mapped by r either to a common point, or to a pair of points connected by an edge of weight w. Thus, r is 1-Lipschitz. Now, q factors as $X \to X \sqcup _{A \sqcup X} X \xrightarrow {r} \mathrm {Coeq}(f,g)$ , where the first arrow is the inclusion to either copy of X. Thus, $\bar r \colon \thinspace X \sqcup _{A \sqcup X} X \to \mathrm {Coeq}(f,g)$ is the canonical morphism in $\overline {\boldsymbol {\mathsf {C}} }$ granted by the universal property of the pushout. We claim that $\bar r$ is an isomorphism in $\overline {\boldsymbol {\mathsf {C}} }$ . It will then follow that $\bar q, \bar q \colon \thinspace X \rightrightarrows \mathrm {Coeq}(f,g)$ yield a pushout; thence, $\bar q$ is the desired coequalizer by Proposition 2.1.

Define $s \colon \thinspace \mathrm {Coeq}(f,g) \to X \sqcup _{A \sqcup X} X$ by $x \mapsto (x,0)$ on underlying sets. If $x,x^{\prime }\in X$ are connected by an internal edge, then $sx, sx^{\prime }$ are connected by an internal edge of the same weight; if there is a glued edge between $x,x^{\prime }\in X$ , then there is a path comprising two glued edges passing through the vertices $sx = (x,0), (x^{\prime },1), (x^{\prime },0) = sx^{\prime }$ . Therefore, s is 2-Lipschitz. Finally, $rs$ and $sr$ are close to the identity maps on their respective domains; hence, $\bar r$ is an isomorphism.

Proof Let $R \subseteq X \times X$ be the equivalence relation, where $(x,x^{\prime }) \in R$ if and only if $Gx = Gx^{\prime }$ . Let $f_1, f_2 \colon \thinspace R \rightrightarrows X$ be the (restrictions of the) factorwise projections. Then, $Z: = \mathrm {Coeq}(f_1,f_2)$ is obtained by coarsely gluing X to itself, where $x,x^{\prime }\in X$ are connected by a glued edge if and only if they lie in the same G-orbit. We claim that the quotient map $X \to X/G$ induces a quasi-isometry $h \colon \thinspace Z \to X/G$ , and thence the desired result will follow. Note that h is 1-Lipschitz and surjective. Observe that for any $x,x^{\prime } \in X$ and $\alpha \in G$ , the paths $x,x^{\prime },\alpha x^{\prime }$ and $x, \alpha x, \alpha x^{\prime }$ each comprise one glued edge and one internal edge of weight $d(x,x^{\prime })$ . Thus, we can replace any given path in Z by a path of the same length where all internal edges appear before all glued edges. It follows that the distance between $x,y\in Z$ is realized by the infimal length of all paths from x to y consisting of at most one internal and at most one glued edge. Since G-orbits are closed, this infimum is realized by a path of the form $x, \alpha y, y$ or $x,y$ (or a trivial path). Therefore, $d_Z(x,y) - 1 \leq d_{X/G}(Gx, Gy) \leq d_Z(x,y)$ .

Proof This follows from Theorem 1.8 and Lemma 2.7.

Proof Assume that $Q_*(f)$ stabilizes at threshold $\sigma _0$ in $\overline {\boldsymbol {\mathsf {Coarse}} }$ . Then, for each $\tau \geq \sigma \geq \sigma _0$ , the map $Q_\sigma (f) \twoheadrightarrow Q_\tau (f)$ admits a lower control. Therefore, for each $\tau \geq \sigma $ , there exists $r(\tau ) \geq 0$ such that for all $x,x^{\prime } \in X$ , whenever $d_{Q_\tau (f)}(x,x^{\prime }) \leq 1$ holds, then $d_{Q_\sigma (f)}(x,x^{\prime }) \leq r(\tau )$ . Consequently,

$$\begin{align*}d_Y(fx,fx^{\prime}) \leq \tau \iff (x,x^{\prime}) \in K_\tau(f) \implies d_{Q_\tau(f)}(x,x^{\prime}) \leq 1 \implies d_{Q_\sigma(f)}(x,x^{\prime}) \leq r(\tau) \end{align*}$$

for all $\tau \geq \sigma $ and $x,x^{\prime } \in X$ , and hence $f_\sigma $ admits a lower control.

Proof The implications (2) $\implies $ (3) and (3) $\implies $ (1) are immediate.

(1) $\implies $ (2). Assume that $\bar f$ is the coequalizer of some parallel pair $\bar g, \bar h \colon \thinspace W \rightrightarrows X$ in $\overline {\boldsymbol {\mathsf {C}} }$ . Choose representatives $g\in \bar g, h\in \bar h$ . Since $\bar f \bar g = \bar f \bar h$ , there exists some $\kappa \geq 0$ such that $fg \approx _\kappa fh$ . Let $\sigma \geq \kappa $ . Then, $(gw, hw) \in K_\kappa (f) \subseteq K_\sigma (f)$ for all $w \in W$ and so $q_\sigma g \approx _1 q_\sigma h$ . Consequently, there exists a unique morphism $\bar \phi _\sigma \colon \thinspace Y \to Q_\sigma (f)$ such that $\bar q_\sigma = \bar \phi _\sigma \bar f$ . We claim that $\bar f_\sigma = \bar \phi _\sigma ^{-1}$ in $\overline {\boldsymbol {\mathsf {C}} }$ .

Observe that $\bar f_\sigma \bar \phi _\sigma \bar f = \bar f_\sigma \bar q_\sigma = \bar f$ satisfies $\bar f_\sigma \bar \phi _\sigma \bar f \bar g = \bar f_\sigma \bar \phi _\sigma \bar f \bar h$ . Appealing to the universal property of $\bar f$ , we deduce that $\bar f_\sigma \bar \phi _\sigma = \bar 1_Y$ and so $\bar \phi _\sigma $ admits a left-inverse $\bar f_\sigma $ . As $\bar q_\sigma $ is epic, so is $\bar \phi _\sigma $ . Since left-invertible epimorphisms are isomorphisms, it follows that $\bar f_\sigma = \bar \phi _\sigma ^{-1}$ .

The, moreover, statement then follows since $\bar f_\tau \bar f_\sigma ^{-1} \colon \thinspace Q_\sigma (f) \to Q_\tau (f)$ is an isomorphism for $\sigma \leq \tau $ large.

Proof The implications (1) $\iff $ (3) $\implies $ (2) follow from Lemma 3.3, while (2) $\implies $ (3) follows from Lemma 3.2 and the assumption that $\bar f$ is an epimorphism.

(3) $\implies $ (4). Choose $\sigma \geq 0$ so that $f_\sigma $ is a coarse equivalence and let $\theta $ be a lower control for $f_\sigma $ . Then, $\theta ^\perp (t) := \sup \{s \geq 0 \,\, \colon \,\, \theta (s) \leq t \}$ is a proper increasing function which satisfies $\theta (s) \leq t \iff s \leq \theta ^\perp (t)$ . If $x,x^{\prime } \in X$ satisfy $t = d_Y(fx,fx^{\prime }) < \infty $ , then $d_{Q_\sigma (f)}(x,x^{\prime }) \leq \theta ^\perp (t)$ . This means there exists a path in $Q_\sigma (f)$ from x to $x^{\prime }$ of length at most $\mu (t) := \theta ^\perp (t) + 1$ ; we may assume that this path does not run over consecutive internal edges. Therefore, this path has $n \in [0, \mu (t)]$ glued edges and at most $n+1$ internal edges. Moreover, each internal edge has weight at most $\mu (t)$ . Choosing the vertices along this path yields a desired sequence of points (allowing for the possibility that $b_i = a_{i+1}$ ).

(4) $\implies $ (3). Choose a representative f and let $\sigma $ be as given by Condition (4) (note that if Condition (4) holds for some $\sigma \geq 0$ , then it also holds for all $\sigma $ large). Suppose $x,x^{\prime } \in X$ satisfy $t = d_Y(fx,fx^{\prime }) < \infty $ . By assumption, there is a path from x to $x^{\prime }$ in $Q_\sigma (f)$ of length at most $(n+1)\mu (t) + n \leq (\mu (t) + 1)^2$ . Consequently, $f_\sigma $ admits a lower control and is hence a coarse equivalence.

Proof By Proposition 2.17, $\mathrm {Eq}_*(f\pi _1, f\pi _2) = K_*(f)$ stabilizes if and only if the equalizer of $\bar f\bar \pi _1, \bar f\bar \pi _2$ exists in $\overline {\boldsymbol {\mathsf {C}} }$ ; in that case, $K_\sigma (f) = \mathrm {Eq}_\sigma (f\pi _1, f\pi _2) \hookrightarrow X \times X$ realizes the equalizer for $\sigma $ large. The result then follows from Proposition 2.2.

Proof Assume that $K_*(f)$ stabilizes at threshold $\sigma \geq 0$ and let $\tau \geq \sigma $ . Then, there exists a natural isomorphism between the pairs $\bar \pi _1 \bar \iota _\sigma , \bar \pi _2 \bar \iota _\sigma \colon \thinspace K_{\sigma }(f) \rightrightarrows X$ and $\bar \pi _1 \bar \iota _\tau , \bar \pi _2 \bar \iota _\tau \colon \thinspace K_{\tau }(f) \rightrightarrows X$ in $\overline {\boldsymbol {\mathsf {C}} }$ whose components are represented by the inclusion $K_\sigma (f) \hookrightarrow K_\tau (f)$ and identity map $1_X \colon \thinspace X \to X$ .

Now, $\bar q_\sigma \colon \thinspace X \to Q_\sigma (f)$ and $\bar q_\tau \colon \thinspace X \to Q_\tau (f)$ are coequalizers of these respective parallel pairs. Therefore, this natural isomorphism induces a canonical morphism $\bar \phi \colon \thinspace Q_\sigma (f) \to Q_\tau (f)$ in $\overline {\boldsymbol {\mathsf {C}} }$ which, moreover, is an isomorphism. By construction, the bonding map $Q_\sigma (f) \to Q_\tau (f)$ (coinciding with the underlying identity of X) from the filtration $Q_*(f)$ makes the right-hand square in the diagram commute and is hence a representative of $\bar \phi $ . Thus, $Q_*(f)$ stabilizes.

Proof The forward implication follows from Lemma 3.7. Conversely, assume that $\bar f$ is an epimorphism and that $K_*(f)$ stabilizes. Then, $Q_*(f)$ stabilizes in $\overline {\boldsymbol {\mathsf {Coarse}} }$ by Lemma 3.8; hence, $\bar f$ is a regular epimorphism in $\overline {\boldsymbol {\mathsf {Coarse}} }$ by Proposition 3.4. By Lemma 3.7, $\bar f$ admits a kernel pair and so it must be an effective epimorphism in $\overline {\boldsymbol {\mathsf {Coarse}} }$ by Lemma 2.6.

Proof Let us regard $Q_\sigma (f)$ as the vertex set of a weighted graph $\Gamma $ with internal and glued edges (as in the coarse gluing construction). Let $x,x^{\prime } \in X$ and $y = fx, y^{\prime }=fx^{\prime }$ . To obtain an upper control, suppose $d_{Q_\sigma (f)}(x,x^{\prime }) < \infty $ . Let P be an edge-path in $\Gamma $ from x to $x^{\prime }$ of length $L \geq 0$ . We may assume that all vertices $x = x_0, \ldots , x_n = x$ appearing along P are distinct and that P has no consecutive pair of internal edges. Thus, $n \leq L$ since glued edges have unit length. Let $y_i = fx_i$ . If the edge of P between $x_i,x_{i+1}$ is

• • a glued edge, then, by definition of $Q_\sigma (f)$ , we have that $d_Y(y_i, y_{i+1}) \leq \sigma $ , and hence $y_i, y_{i+1}$ are connected by an internal edge of weight at most $\Theta (\sigma )$ in $\Gamma ^\Theta _\sigma $ ;

• • an internal edge, then $y_i, y_{i+1}$ are connected by an augmented edge of weight at most $d_X(f^{-1}(y_i), f^{-1}(y_{i+1})) + 1 \leq d_X(x_i,x_{i+1}) + 1$ in $\Gamma ^\Theta _\sigma $ .

Therefore, there exists an edge-path from y to $y^{\prime }$ in $\Gamma ^\Theta _\sigma $ of length at most

$$\begin{align*}\sum_{i=0}^{n-1} \left(d_X(x_i,x_{i+1}) + \Theta(\sigma)\right) = L + \Theta(\sigma) n \leq (\Theta(\sigma) + 1)L. \end{align*}$$

By considering all paths from x to $x^{\prime }$ , we deduce that $Q_\sigma (f) \to U^\Theta _\sigma $ is $(\Theta (\sigma ) + 1)$ -Lipschitz.

For the lower control, suppose that $P^{\prime }$ is an edge-path from y to $y^{\prime }$ in $\Gamma ^\Theta _\sigma $ of length $L \geq 0$ . We may assume all vertices $y = y_0,\ldots , y_n = y$ along $P^{\prime }$ are distinct. Since all edges of $\Gamma ^\Theta _\sigma $ have weight at least 1, we deduce $n \leq L$ . Note that $f^{-1}(y_i)$ is non-empty for all i. If the edge of $P^{\prime }$ between $y_i, y_{i+1}$ forms

• • an augmented edge, then there exist $x^{\prime }_i \in f^{-1}y_i$ and $x_{i+1} \in f^{-1}y_{i+1}$ such that

  $$\begin{align*}d_X(x^{\prime}_i,x_{i+1}) \leq d_X(f^{-1}(y_i), f^{-1}(y_{i+1})) + 1;\end{align*}$$

• • an internal edge, then $d_Y(y_i, y_{i+1}) \leq \sigma $ by construction; in particular, every pair of distinct points in $f^{-1}y_i \cup f^{-1}y_{i+1}$ are connected by a glued edge in $Q_\sigma (f)$ .

In either case, if this edge between $y_i, y_{i+1}$ has weight w, then there exists vertices $x^{\prime }_i \in f^{-1}y_i$ , $x_{i+1} \in f^{-1}y_{i+1}$ connected by an edge of weight at most w in $\Gamma $ . Therefore, there is a path in $\Gamma $ passing through vertices $x, x^{\prime }_0, x_1, \ldots , x^{\prime }_{n-1}, x_n, x^{\prime }$ of length at most $L + n + 1 \leq 2L + 1$ . Consequently, $Q_\sigma (f) \to Y^\Theta _\sigma $ has lower control $t \mapsto \frac {t - 1}{2}$ .

Proof For each $y \in Y$ , choose some $\phi (y) \in f(X)$ such that $d_Y(y, \phi y) \leq r$ ; we shall also assume that $\phi u = u$ for all $u \in f(X)$ . We claim that $\phi \colon \thinspace Y_\sigma ^\Theta \to U_\sigma ^\Theta $ is C-Lipschitz. Since the involved metrics are defined using path metrics, it suffices to check this on edges. Observe that any augmented edge in $\mathrm {Rips}_\sigma ^\Theta (Y; f)$ also belongs to the subgraph $\Gamma _\sigma ^\Theta $ . Now, suppose $y,y^{\prime } \in Y$ are connected by an internal edge of weight w. Then, $d_Y(\phi y, \phi y^{\prime }) \leq d_Y(y,y^{\prime }) + 2r < \infty $ , and hence $\phi y, \phi y^{\prime }$ are connected by an internal edge of weight

$$\begin{align*}\Theta(d_Y(\phi y, \phi y^{\prime})) \leq \Theta(d_Y(y,y^{\prime}) + 2r) \leq C \Theta(d_Y(y,y^{\prime})) = Cw\end{align*}$$

in $\Gamma _\sigma ^\Theta $ , yielding the claim. Now, $\iota $ is 1-Lipschitz and $\phi \iota $ is the identity on $U_\sigma ^\Theta $ , and hence $\iota $ is a bi-Lipschitz embedding. Since $N_r f(X) = Y$ and $\sigma \geq r$ , each $y \in Y$ is connected to some $u \in f(X)$ by an internal edge of weight at most $\Theta (r)$ in $\Gamma _\sigma ^\Theta $ , and so $\iota $ is coarsely surjective .

Proof Let $\rho $ be an (affine) upper control for f. Suppose that $y,y^{\prime } \in Y$ are connected by an edge of weight w in $\mathrm {Rips}^\Theta _\infty (Y;f)$ . If this is an internal edge, then $d_Y(y,y^{\prime }) \leq \Theta (d_Y(y,y^{\prime })) = w$ ; otherwise, this is an augmented edge, and so there exist $x \in f^{-1}(y)$ and $x^{\prime } \in f^{-1}(y^{\prime })$ such that $d_X(x,x^{\prime }) \leq d_X(f^{-1}y, f^{-1}y^{\prime }) + 1 = w$ ; hence,

$$\begin{align*}d_Y(y,y^{\prime}) \leq \rho(d_X(x,x^{\prime})) \leq \rho(w).\end{align*}$$

Therefore, $t \mapsto \rho (t) + t$ serves as an upper control for $Y^{\prime }_\infty \to Y$ .

If $d_Y(y,y^{\prime }) < \infty $ , then $\partial _\infty (y,y^{\prime }) \leq \Theta (d_Y(y,y^{\prime }))$ since there exists an internal edge between $y,y^{\prime }$ . Moreover, $\partial _\infty (y,y^{\prime }) = \infty $ only if $d_Y(y,y^{\prime }) = \infty $ by construction. Since $\Theta $ is proper, we obtain a lower control.

Proof By Lemma 4.3, the factorization $X \to Y ^\Theta _\infty \to Y$ of f yields an epi–mono factorization of $\bar f$ in $\overline {\boldsymbol {\mathsf {CLip}} }$ . The extremality of $\bar f$ implies that $Y^\Theta _\infty \to Y$ represents an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ and is hence a quasi-isometry. Therefore, this map admits a (strictly increasing) affine lower control $\theta $ . By assumption, there exists $\sigma \geq 0$ such that $t < \theta (\Theta (t) - 1)$ for all $t \geq \sigma $ .

Let $y,y^{\prime } \in Y$ satisfy $\partial ^\Theta _\infty (y,y^{\prime }) < \infty $ . Consider a weighted edge-path P in $\mathrm {Rips}_\infty ^\Theta (Y;f)$ passing through vertices $y = y_0, \ldots , y_n = y^{\prime }$ of length $L \in [\partial ^\Theta _\infty (y,y^{\prime }), \partial ^\Theta _\infty (y,y^{\prime }) + \frac {1}{3})$ . Note that each $d_Y(y_i,y_{i+1})$ is finite. If there exists some $0 \leq i \leq n$ such that $d_Y(y_i,y_{i+1}) \geq \sigma $ , then

$$\begin{align*}\theta\partial_\infty^\Theta(y_i,y_{i+1}) \leq d_Y(y_i,y_{i+1}) < \theta\left(\Theta {d_Y(y_i,y_{i+1})} - 1\right),\end{align*}$$

and hence $\partial _\infty ^\Theta (y_i,y_{i+1}) < \Theta {d_Y(y_i,y_{i+1})} - 1$ . Therefore, P does not run over the internal edge between $y_i$ and $y_{i+1}$ , for otherwise, we could replace that edge with a detour in $\mathrm {Rips}_\infty ^\Theta (Y;f)$ which reduces the length of P by at least $\frac {1}{2}$ , contradicting the choice of P. In other words, each edge appearing along P must be an edge in the subgraph $\mathrm {Rips}_\sigma ^\Theta (Y;f)$ . Thus, for all $y,y^{\prime }\in Y$ , the distance $\partial _\infty (y,y^{\prime })$ is attained by the infimal length of all weighted edge-paths from y to $y^{\prime }$ in $\mathrm {Rips}_\sigma ^\Theta (Y;f)$ . Consequently, $Y^\Theta _\sigma \to Y^\Theta _\infty $ is an isometry.

Proof (1) $\implies $ (2). Consider the factorization $X \xrightarrow {q_\sigma } Q_\sigma (f) \xrightarrow {f_\sigma } Y$ of f for $\sigma \geq 0$ . The weight function $\Theta (t) = 2^t$ satisfies the hypotheses of Lemmas 4.1–4.4, and hence $\bar f_\sigma $ is monic in $\overline {\boldsymbol {\mathsf {CLip}} }$ for $\sigma $ large; note that we require the extremality of $\bar f$ in order to apply Lemma 4.4. Since $\bar f$ is an extremal epi, $\bar f_\sigma $ is an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ , and hence $\bar f$ is a regular epi by Lemma 3.3. (2) $\implies $ (3). Let $d^{\prime }$ be a metric on Y such that the map $m \colon \thinspace (Y,d^{\prime }) \to (Y,d_Y)$ given by the underlying identity is a coarse equivalence and $f^{\prime } \colon \thinspace X \to (Y,d^{\prime })$ is coarsely Lipschitz. By Lemma 3.3, $K_\sigma (f) \rightrightarrows X \xrightarrow {\bar f} (Y,d)$ is a coequalizer diagram in $\overline {\boldsymbol {\mathsf {CLip}} }$ for $\sigma $ large. Note that $\bar f^{\prime } = \bar m \bar f$ coequalizes the parallel pair $K_\sigma (f) \rightrightarrows X$ . Therefore, by the universal property, there exists a unique morphism $\bar g \colon \thinspace (Y,d_Y) \to (Y,d^{\prime })$ in $\overline {\boldsymbol {\mathsf {CLip}} }$ such that $\bar f^{\prime } = \bar g \bar f$ . Since $\bar f$ is an epi in $\overline {\boldsymbol {\mathsf {Coarse}} }$ , we deduce that $\bar g = \bar m$ . Therefore, m is coarsely Lipschitz, and hence $d^{\prime } \prec d_Y$ .

(3) $\implies $ (4). Trivial.

(4) $\implies $ (1). Assume that $\bar f$ factorizes as $X \xrightarrow {\bar e} Z \xrightarrow {\bar m} Y$ of $\bar f$ in $\overline {\boldsymbol {\mathsf {CLip}} }$ , with $\bar m$ a mono. Since $\bar f$ is an epi, so is $\bar m$ . Thus, $m \colon \thinspace Z \to Y$ is a coarse equivalence. By Lemma 2.10, m factorizes as $Z \xrightarrow {g} (Y,d^{\prime }) \xrightarrow {m^{\prime }} (Y,d)$ for some metric $d^{\prime }$ with g a quasi-isometry and $m^{\prime }$ a coarse equivalence coinciding with the underlying identity. Note that $\bar m^{\prime } = \bar g^{-1} \bar m$ is a morphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ . Now, $f^{\prime } := ge$ is coarsely Lipschitz and coincides with f on underlying sets. By the maximality assumption, $(m^{\prime })^{-1}$ is coarsely Lipschitz, and hence $\bar m^{\prime }$ is an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ . Therefore, $\bar m = \bar g \bar m^{\prime }$ is an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ and so $\bar f$ is an extremal epi.

Proof (1) $\implies $ (2). Let $\Theta $ be any weight function satisfying the hypothesis of Lemma 4.2. For $\sigma \geq 0$ , consider the factorization

$$\begin{align*}X \xrightarrow{q_\sigma} Q_\sigma(f) \xrightarrow{g_\sigma} (Y, \partial^\Theta_\sigma) \xrightarrow{h_\sigma} (Y,d_Y)\end{align*}$$

of f, where $g_\sigma $ and $h_\sigma $ , respectively, coincide with f and the identity as functions. By construction, $\bar q_\sigma \colon \thinspace X \to Q_\sigma (f)$ is a regular epi in $\overline {\boldsymbol {\mathsf {CLip}} }$ for all $\sigma \geq 0$ . Appealing to Lemmas 4.1 and 4.2, and coarse surjectivity of f, we deduce that $g_\sigma $ is a quasi-isometry, and hence $\bar g_\sigma $ is an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ . Therefore, $\bar g_\sigma \bar q_\sigma $ is a regular epi in $\overline {\boldsymbol {\mathsf {CLip}} }$ . By assumption, $\bar f$ is a regular epi in $\overline {\boldsymbol {\mathsf {Coarse}} }$ and so, by Proposition 3.4, we may choose $\sigma $ large so that $f_\sigma = h_\sigma g_\sigma \colon \thinspace Q_\sigma (f) \to Y$ is a coarse equivalence. In particular, $\partial _\sigma ^\Theta $ and $d_Y$ are coarsely equivalent. Suppose that $d^{\prime }$ is a metric on Y such that the map $k \colon \thinspace (Y,d_Y) \to (Y,d^{\prime })$ given by the underlying identity is coarse equivalence, and that $f^{\prime }:= kf \colon \thinspace X \to (Y,d^{\prime })$ is coarsely Lipschitz. Since $\bar g_\sigma \bar q_\sigma $ is an extremal epi in $\overline {\boldsymbol {\mathsf {CLip}} }$ , it follows from Proposition 4.5 that $d^{\prime } \prec \partial ^\Theta _\sigma $ .

(2) $\iff $ (3). This follows from Proposition 4.5.

(2) $\implies $ (4). Let $d^{\prime }$ be a metric on Y satisfying the given $\prec $ -greatest condition. Then, by Proposition 4.5, $X \to (Y,d^{\prime }) \to (Y,d_Y)$ yields an extremal epi–mono factorization of $\bar f$ in $\overline {\boldsymbol {\mathsf {CLip}} }$ .

(4) $\implies $ (5). Assume that $\bar f$ factors as $X \xrightarrow {\bar e} Z \xrightarrow {\bar m} Y$ with $\bar e$ an extremal epi and $\bar m$ a regular mono in $\overline {\boldsymbol {\mathsf {CLip}} }$ . By the proof of (4) $\implies $ (1) in Proposition 4.5, $\bar m$ factors as $Z \xrightarrow {\bar g} (Y,d^{\prime }) \xrightarrow {\bar m^{\prime }} (Y,d)$ with $\bar g$ an isomorphism in $\overline {\boldsymbol {\mathsf {CLip}} }$ and m a coarse equivalence induced by the underlying identity. Therefore, $\bar g \bar e$ is an extremal epi in $\overline {\boldsymbol {\mathsf {CLip}} }$ represented by the induced map $(X,d_X) \to (Y,d^{\prime })$ .

(5) $\implies $ (1). By Proposition 4.5, the map $(X,d_X) \to (Y,d^{\prime })$ represents a regular epi in $\overline {\boldsymbol {\mathsf {CLip}} }$ . Appealing to Corollary 3.5, this map also represents a regular epi in $\overline {\boldsymbol {\mathsf {Coarse}} }$ . Since the underlying identity $(Y,d^{\prime }) \to (Y,d)$ is a coarse equivalence, it follows that $\bar f$ is a regular epi in $\overline {\boldsymbol {\mathsf {Coarse}} }$ .

Proof By Lemma 3.7, $\bar f$ admits a kernel pair in $\overline {\boldsymbol {\mathsf {CLip}} }$ if and only if $K_*(f)$ stabilizes for any representative f. The result follows using Lemma 2.6.

Proof First, we show that $\bar f$ is a regular epimorphism in $\overline {\boldsymbol {\mathsf {C}} }$ . Observe that $(a,b) \in K_0(f)$ if and only $aZ = bZ$ . Therefore, $Q_0(f)$ is the Cayley graph of E with respect to the generating set $(Z \cup \hat S_G)\big\backslash \left \{ {1} \right \} $ . The map $Q_0(f) \to G$ induced by f collapses each coset $aZ$ to a point $f(a)$ ; moreover, distinct cosets $aZ$ and $bZ$ map to adjacent vertices if and only if some vertex of $aZ$ is adjacent to a vertex in $bZ$ . Since each coset has diameter at most 1 in $Q_0(f)$ , this map is a quasi-isometry (see [Reference Disarlo, Pan, Randecker and TangDPRT21, Lemma 4.2]). Therefore, $\bar f$ is a regular epi in $\overline {\boldsymbol {\mathsf {C}} }$ by Lemma 3.3.

Next, we show that the coarse kernel filtration $K_*(f)$ stabilizes. Suppose that $(a,b) \in K_\sigma (f)$ for some $\sigma \geq 0$ . Then, $a,b\in E$ satisfy $d_{G}(1, f(a^{-1}b)) = d_{G}(f(a), f(b)) \leq \sigma $ . Thus, $f(a^{-1}b) \in G$ can be expressed as a word in at most $\sigma $ generators. By replacing each generator in this word with its lift in $\hat S_G$ , we obtain an element $h \in a^{-1}bZ$ such that $d_E(1,h) \leq \sigma $ . Note that $ahZ = bZ$ , and hence $(ah, b) \in K_0(f)$ . Since $d_E(a, ah) = d_E(1,h)\leq \sigma $ , we deduce $(a,b) \in N_\sigma (K_0(f))$ . Therefore, $K_*(f)$ stabilizes, and hence $\bar f$ is an effective epi in $\overline {\boldsymbol {\mathsf {C}} }$ by Propositions 3.9 and 4.11.

Proof Let $s^{\prime } \in \bar s$ be any representative. Then, there exists $\kappa \geq 0$ such that $fs^{\prime } \approx _\kappa 1_G$ . For any $g \in G$ , we have that $d_G(fs^{\prime }g, g) \leq \kappa $ . Thus, we may right-multiply $s^{\prime }g$ by a word of most $\kappa $ generators from $\hat S_G$ to obtain an element $s(g) \in f^{-1}(g)$ . Therefore, there exists a section $s \colon \thinspace G \to E$ of f satisfying $s \approx _\kappa s^{\prime }$ . Since $s^{\prime }$ is controlled, so is s. The result follows since any controlled map from the vertex set of a graph (with the combinatorial metric) is Lipschitz.