yielding

$$ \bar{B}(x_i, \operatorname{diam}(\phi_{q,i}(\Omega_q)) + r) \subset \operatorname{Int}(\phi_{q+1,i}(\Omega_{q+1})) $$

for all $i$ large enough by the $C^\infty$ convergence $\phi_{q,i}^*g_i \to g|_{\Omega_q}$. It follows that the composite $h'_{q,i} = \phi_{q+1,i}^{-1} \circ h_i \circ \phi_{q,i} : \Omega_q \to M$ is well defined for each $q$ and all $i$ large enough because $x_i \in \phi_{q,i}(\Omega_q)$ and $h_i(x_i) \in \bar{B}(x_i, r)$. On the one hand, from the $C^\infty$ convergence $\phi_{q,i}^*g_i \to g|_{\Omega_q}$ and $h_i(x_i) \in \bar{B}(x_i, r)$, we get the $C^\infty$ convergence $h'_{q,i}g \to g|_{\Omega_q}$ and $\limsup_i d(x, h'_{q,i}(x)) \le r$; in particular, for each $q$, the maps $h'_{q,i}$ are equi-quasi-isometries of order $\infty$. Therefore, by Lemma 8.2, some subsequence of $h'_{q,i}$ is $C^\infty$ convergent to some $C^\infty$ map $h'_q : \Omega_q \to M$, which is an isometric embedding satisfying $h'_q(x) \in \bar{B}(x, r)$.

For all $p \ge q$, the restrictions $h'_p|_{\Omega_q}$ form a sequence of isometric embeddings satisfying $h'_p(x) \in \bar{B}(x, r)$. Then, by Corollary 8.3, there is some sequence of positive integers $p(q, k)$ for each $q$ so that the subsequence $h'_{p(q,k)}|_{\Omega_q}$ of $h'_p|_{\Omega_q}$ is $C^\infty$ convergent as $k \to \infty$ to an isometric embedding $h''_q : \Omega_q \to M$ satisfying $h''_q(x) \in \bar{B}(x, r)$. We can assume that $p(q+1, k)$ is a subsequence of $p(q, k)$ for each $q$, yielding $h''_{q+1}|_{\Omega_q} = h''_q$. So the maps $h''_q$ can be combined to define an isometry $h : M \to M$ satisfying $h(x) \in \bar{B}(x, r)$.

Now fix any $q$ and let $z'_{p,i} = \phi_{p,i}^{-1}(z_i)$ for each $p \ge q$ and all $i$ large enough. From $z_i \in B(x_i, r)$ and the $C^\infty$ convergence $\phi_{p,i}^*g_i \to g|_{\Omega_p}$, it follows that $z'_{p,i}$ approaches the compact set $\bar{B}(x, r)$ as $i \to \infty$. Then, for each $p \ge q$, there is a sequence $z_{p,i}$ in $B(x, r)$ so that $d(z_{p,i}, z'_{p,i}) \to 0$. Hence, from the $C^\infty$ convergence $\phi_{p,i}^*g_i \to g|_{\Omega_p}$ and Claim 16, it follows that

$$ \begin{aligned} \sup\{d(z, h(z)) \mid z \in B(x, r)\} &= \sup\{d(z, h''_q(z)) \mid z \in B(x, r)\} \\ &\ge \sup\{\liminf_p d(z, h'_p(z)) \mid z \in B(x, r)\} \\ &\ge \sup\{\liminf_p \liminf_i d(z, h'_{p,i}(z)) \mid z \in B(x, r)\} \\ &\ge \liminf_p \liminf_i d(z_{p,i}, h'_{p,i}(z_{p,i})) \\ &= \liminf_p \liminf_i d(z'_{p,i}, h'_{p,i}(z'_{p,i})) \\ &\ge \liminf_i d_i(z_i, h_i(z_i)) \\ &\ge \delta. \end{aligned} $$

So $h \ne \mathrm{id}_M$, which is a contradiction because $h(x) \in \bar{B}(x, r)$. $\square$

<figure><img src="image_19.png" /></figure>

**Lemma 9.29.** For $n \ge 2$ and each point $[M, x] \in \mathcal{M}_{*,\text{la}}^\infty(n)$, there is some $r > 0$ such that, for each $\varepsilon \in (0, r)$, there is some neighbourhood $\mathcal{N}$ of $[M, x]$ in $\mathcal{M}_{*,\text{la}}^\infty(n)$ so that, if an equivalence class $\iota(L)$ of $\mathcal{M}_{*,\text{la}}^\infty(n)$ meets $\mathcal{N}$ at points $[L, y]$ and $[L, z]$, then either $d_L(y, z) < \varepsilon$ or $d_L(y, z) > r$, where $d_L$ denotes the distance function of $L$.

*Proof.* Since $M$ is locally aperiodic, there is some $r > 0$ such that

$$ (9.5) \qquad \{h \in \mathrm{Iso}(M) \mid d(x, h(x)) \le r\} = \{\mathrm{id}_M\}. $$
Suppose that the statement is false for this $r$. Then, given any $\varepsilon \in (0, r)$, there are sequences $[L_i, y_i]$ and $[L_i, z_i]$ in $M_{*,\text{la}}^\infty(n)$ converging to $[M, x]$ in $M_{*,\text{la}}^\infty(n)$ such that $\varepsilon \le d_i(y_i, z_i) \le r$ for all $i$.

Take a sequence of compact domains $\Omega_q$ of $M$ such that $x \in \Omega_q$ and $d(x, \partial\Omega_q) \to \infty$ as $q \to \infty$. For each $q$, there are of $C^\infty$ embeddings $\phi_{q,i} : \Omega_q \to M_i$ and $\psi_{q,i} : \Omega_q \to M_i$ for all $i$ large enough so that $\phi_{q,i}(x) = y_i$, $\psi_{q,i}(x) = z_i$, and $\phi_{q,i}^*g_i$, $\psi_{q,i}^*g_i \to g|_{\Omega_q}$ as $i \to \infty$ with respect to the $C^\infty$ topology. We can also assume that, for each $q$,

$$ \bar{B}(x, \operatorname{diam}(\Omega_q) + r) \subset \operatorname{Int}(\Omega_{q+1}), $$

yielding

$$ \bar{B}(y_i, \operatorname{diam}(\phi_{q,i}(\Omega_q)) + r) \subset \operatorname{Int}(\psi_{q+1,i}(\Omega_{q+1})) $$

for $i$ large enough by the $C^\infty$ convergence $\phi_{q,i}^*g_i$, $\psi_{q,i}^*g_i \to g|_{\Omega_q}$. It follows that the composite $h_{q,i} = \psi_{q+1,i}^{-1} \circ \phi_q : \Omega_q \to M$ is well defined for each $q$ and all $i$ large enough since $d_i(y_i, z_i) \le r$. From the $C^\infty$ convergence $\phi_{q,i}^*g_i$, $\psi_{q,i}^*g_i \to g|_{\Omega_q}$, we also get the $C^\infty$ convergence $h_{q,i}^*g_i \to g|_{\Omega_q}$, and moreover

$$ \liminf_i d(x, h_{q,i}(x)) \ge \varepsilon, \quad \limsup_i d(x, h_{q,i}(x)) \le r $$

because $\phi_{q,i}(x) = y_i$, $\psi_{q,i}(x) = z_i$ and $\varepsilon \le d_i(y_i, z_i) \le r$. Then, with arguments of the proof of Lemma 9.28, an isometry $h : M \to M$ can be constructed so that $\varepsilon \le d(x, h(x)) \le r$, which contradicts (9.5). $\square$

**Lemma 9.30.** Let $n \in \mathbb{N}$ and $r > 0$. For any convergent sequence $[M_i, x_i] \to [M, x]$ in $M_*^\infty(n)$ and each $y \in B(x, r)$, there are points $y_i \in B(x_i, r)$ such that $[M_i, y_i] \to [M, y]$ in $M_*^\infty(n)$.

*Proof.* Take a sequence of compact domains $\Omega_q$ of $M$ such that $x, y \in \Omega_q$ and $d(x, \partial\Omega_q) \to \infty$ as $q \to \infty$. For each $q$, there is some index $i_q$ such that, for each $i \ge i_q$ there is a $C^\infty$ embedding $\phi_{q,i} : \Omega_q \to M_i$ satisfying $\phi_{q,i}(x) = xi$ and $\phi_{q,i}^*g_i \to g|_{\Omega_q}$ as $i \to \infty$ with respect to the $C^\infty$ topology. Let $y_{q,i} = \phi_{q,i}(y)$ for all $i \ge i_q$. Then, for each $q$ and every $m \in \mathbb{Z}_+$, there is some index $i_{q,m} \ge i_q$ such that $d_i(x_i, y_{q,i}) < r$ and $\|\phi_{q,i}^*g_i - g\|_{C^m, \Omega_q} < 1/m$ for all $i \ge i_{q,m}$. Moreover we can assume that $i_{q,q} < i_{q+1,q+1}$ for all $q$. Now let $y_i$ be any point of $B(x_i, r)$ for $i < i_{1,1}$, and let $y_i = y_{q,i}$ for $i_{q,q} \le i < i_{q+1,q+1}$. It is easy to check that $[M_i, y_i] \to [M, y]$ in $M_*^\infty(n)$. $\square$

**Lemma 9.31.** For $n \in \mathbb{N}$, let $[M, x] \in M_*^\infty(n)$, and let $\mathcal{N}$ be a neighbourhood of $[M, x]$ in $M_*^\infty(n)$. Then there is some $\delta > 0$ and some neighbourhood $\mathcal{L}$ of $[M, x]$ in $M_*^\infty(n)$ such that, for all $[L, y] \in \mathcal{L}$ and all $z \in B(y, \delta)$, we have $[L, y] \in \mathcal{N}$.

*Proof.* There is some $m \in \mathbb{Z}_+$, some $\varepsilon > 0$ and a compact domain $\Omega$ of $M$ containing $x$ such that, for all $[L, z] \in M_*^\infty(n)$, if there is some $C^\infty$ embedding $\phi : \Omega \to L$ so that $\phi(x) = z$ and $\|\phi^*g_L - g_M\|_{C^m, \Omega} < \varepsilon$, then $[L, z] \in \mathcal{N}$. Take any compact domain $\Omega'$ of $M$ whose interior contains $\Omega$. There is some $\varepsilon_0 > 0$ and some neighborhood $\mathcal{H}$ of the $\mathrm{id}_M$ in the group of diffeomorphisms $M \to M$ with the $C^m$ topology such that, for all $h \in \mathcal{H}$, any metric tensor
$g'$ on $\Omega'$ satisfying $\|g' - g_M\|_{C^m, \Omega'} < \varepsilon_0$, we have $h(\Omega) \subset \Omega'$, $\|h^*g' - g_M\|_{C^m, \Omega} < \varepsilon$. Moreover $\varepsilon_0$ can be chosen so small that there is some $\delta > 0$ such that, for any metric tensor $g'$ satisfying the above condition, the $g'$-ball $B'(x, \delta)$ is contained in $\Omega$ and, for each $y \in B'(x, \delta)$, there is some $h \in \mathcal{H}$ so that $h(y) = x$. Then the result follows easily with $\mathcal{L}$ equal to the set of points $[L, y] \in \mathcal{M}^\infty_*(n)$ such that there is some $C^\infty$ embedding $\psi : \Omega' \to L$ so that $\psi(x) = y$ and $\|\psi^*g_L - g_M\|_{C^m, \Omega'} < \varepsilon_0$. $\square$

To construct foliated charts of $\mathcal{M}^\infty_{*,\text{la}}(n)$, we will use the exponential map of the Riemannian manifolds defining the equivalence classes of $\mathcal{M}^\infty_{*,\text{la}}(n)$. For that purpose, the “tangent bundle” and the “principal bundle of tangent frames” of $\mathcal{M}^\infty_{*,\text{la}}(n)$ can be defined as follows before showing that $\mathcal{M}^\infty_{*,\text{la}}(n)$ is a $C^\infty$ foliated space.

For any $C^\infty$ manifold $M$, let $\pi_M : TM \to M$ denote its tangent bundle, and $\pi_M : PM \to M$ its GL$(n)$-principal bundle of tangent frames. If $M$ is a Riemannian manifold, let $\pi_M : QM \to M$ be its O$(n)$-principal bundle of orthonormal tangent frames. The notation $\pi$ may be used instead of $\pi_M$ for simplicity. For each $n \in \mathbb{N}$, consider the class of pairs $(M, \xi)$, where $M$ is a complete connected Riemannian manifold of dimension $n$ and $\xi \in TM$. Define an equivalence relation on this class by declaring that $(M, \xi)$ is equivalent to $(N, \zeta)$ if there is an isometry $\phi : M \to N$ such that $\phi_*(\xi) = \zeta$. The class of a pair $(M, \xi)$ will be denoted by $[M, \xi]$, and the corresponding set of equivalence classes will be denoted by $\mathcal{T}_*(n)$. The $C^\infty$ topology on this set is the topology defined by the following convergence condition: a sequence $[M_i, \xi_i]$ converges to some point $[M, \xi]$ if, for any compact domain $\Omega$ containing $\pi(\xi)$, there is a $C^\infty$ embedding $\phi_i : \Omega \to M_i$ for $i$ large enough so that $\phi_{i*}(\xi) = \xi_i$ and $\phi_i^*g_i \to g|_{\Omega}$ with respect to the $C^\infty$ topology. The set $\mathcal{T}_*(n)$ with this topology will be denoted by $\mathcal{T}^\infty_*(n)$. If tangent frames are used instead of tangent vectors in the above definition, we get a space denoted by $\mathcal{P}^\infty_*(n)$; if only orthonormal frames are used, then we get a subspace $\mathcal{Q}^\infty_*(n) \subset \mathcal{P}^\infty_*(n)$. Let $\pi : \mathcal{T}^\infty_*(n) \to \mathcal{M}^\infty_*(n)$ and $\pi : \mathcal{P}^\infty_*(n) \to \mathcal{M}^\infty_*(n)$ be the maps defined by $\pi([M, \xi]) = [M, \pi_M(\xi)]$ and $\pi([M, f]) = [M, \pi_M(f)]$ for $[M, \xi] \in \mathcal{T}^\infty_*(n)$ and $[M, f] \in \mathcal{P}^\infty_*(n)$. It is easy to check that these maps are continuous and open. For each $[M, x] \in \mathcal{M}^\infty_*(n)$, let $\mathcal{T}^\infty_{[M,x]} = \pi^{-1}([M, x]) \subset \mathcal{T}^\infty_*(n)$.

**Lemma 9.32.** The space $\mathcal{T}^\infty_*(n)$ is a Riemannian vector bundle of rank $n$ over $\mathcal{M}^\infty_*(n)$ with the projection $\pi$ so that the canonical identity $\mathcal{T}^\infty_{[M,x]} \equiv T_x M$, given by $[M, \xi] \equiv \xi$, is an isometry for each $[M, x] \in \mathcal{M}^\infty_*(n)$. Moreover $\mathcal{P}^\infty_*(n)$, with the projection $\pi$, is canonically homeomorphic to the GL$(n)$-principal bundle of references of $\mathcal{T}^\infty_*(n)$, so that $\mathcal{Q}^\infty_*(n)$ corresponds to the O$(n)$-principal bundle of orthonormal references of $\mathcal{T}^\infty_*(n)$

*Proof.* There is a canonical continuous free action of GL$(n)$ on $\mathcal{P}^\infty_*(n)$ defined by $[M, f]a = [M, fa]$ for $[M, f] \in \mathcal{P}^\infty_*(n)$ and $a \in \text{GL}(n)$. The orbits of this action are the fibers of $\pi : \mathcal{P}^\infty_*(n) \to \mathcal{M}^\infty_*(n)$. Then, by ***, there is a local transversal of the orbits across each point of $\mathcal{P}^\infty_*(n)$. Let $\mathcal{S}$ be any of such local transversals, and set $\Theta = \pi(\mathcal{S})$, which is open in $\mathcal{M}^\infty_*(n)$. Then the restriction $\pi : \mathcal{S} \to \Theta$ is a homeomorphism, whose inverse defines a local section $\sigma : \Theta \to \mathcal{P}^\infty_*(n)$ of $\pi : \mathcal{P}^\infty_*(n) \to \mathcal{M}^\infty_*(n)$. By using the canonical identification of each frame of $T_x M$ with a linear map $T_x M \to \mathbb{R}^n$, we get a bijection $\theta : \pi^{-1}(\Theta) \to \mathbb{R}^n \times \Theta$ defined
by $\theta([M, \xi]) = (\sigma([M, x])(\xi), [M, \xi])$ if $\pi_M(\xi) = x$, whose inverse is given by $\theta^{-1}(v, [M, x]) = [M, \sigma(x)^{-1}(v)];$ thus $\theta$ is a homeomorphism because both $\theta$ and $\theta^{-1}$ are obviously continuous. If $\sigma' : \Theta' \to \mathcal{P}_*^\infty(n)$ is another of such local sections of : $\mathcal{P}_*^\infty(n) \to \mathcal{M}_*^\infty(n)$ defining a map $\theta' : \pi^{-1}(\Theta') \to \mathbb{R}^n \times \Theta'$ as above, and $[M, x] \in \Theta \cap \Theta'$, then the composite

$$ \mathbb{R}^n \equiv \mathbb{R}^n \times \{[M, x]\} \xrightarrow{\theta^{-1}} \mathcal{T}_{[M,x]}^\infty \xrightarrow{\theta'} \mathbb{R}^n \times \{[M, x]\} \equiv \mathbb{R}^n $$

is the linear map $\sigma'([M, x]) \circ \sigma([M, x])^{-1}$. It follows that $\mathcal{T}_*^\infty(n)$ with the projection $\pi$ is a vector bundle of rank $n$ over $\mathcal{M}_*^\infty(n)$. Now the mapping $[M, f] \mapsto ([M, v_1], \dots, [M, v_n])$, for $f = (v_1, \dots, v_n)$, defines a canonical homeomorphism between $\mathcal{P}_*^\infty(n)$ and the GL(n)-principal bundle of frames of $\mathcal{T}_*^\infty(n)$.

Consider the metrics on the fibers of $\pi: \mathcal{T}_*^\infty(n) \to \mathcal{M}_*^\infty(n)$ defined in the statement. As above, we get the existence of some local section $\sigma$ of $\pi: \mathcal{Q}_*^\infty(n) \to \mathcal{M}_*^\infty(n)$ defined on some neighborhood $\Theta$ of any point in $\mathcal{M}_*^\infty(n)$. In this case, the corresponding homeomorphism $\theta: \pi^{-1}(\Theta) \to \mathbb{R}^n \times \Theta$ restricts to an isometry $\mathcal{T}_{[M,x]}^\infty \to \mathbb{R}^n \times \{[M, x]\} \equiv \mathbb{R}^n$ for each $[M, x] \in \Theta$. So $\mathcal{T}_*^\infty(n)$ becomes a Riemannian vector bundle with this metric, and $\mathcal{Q}_*^\infty(n)$ is corresponds to the O(n)-principal bundle of orthonormal frames of $\mathcal{T}_*^\infty(n)$ by the above canonical homeomorphism. $\square$

According to Lemma 9.32, the spaces $\mathcal{T}_*^\infty(n)$, $\mathcal{P}_*^\infty(n)$ and $\mathcal{Q}_*^\infty(n)$ can be called the *tangent bungle*, the *principal bundle of tangent frames* and the *principal bundle of orthonormal tangent frames* of $\mathcal{M}_*^\infty(n)$, respectively.

For any complete Riemannian manifold $M$, let $\exp_M: TM \to M$ denote its exponential map, whose restriction to each tangent space $T_xM$ is denoted by $\exp_{M,x}$; the notation $\exp_x$ may be also used for simplicity, as well as the notation $\exp_i$ when a family of complete Riemannian manifolds $M_i$ is considered with $i$ running in some index set. By the compatibility of exponential maps and isometries, a map $\exp: \mathcal{T}_*^\infty(n) \to \mathcal{M}_*^\infty(n)$ is well defined by setting $\exp([M, \xi]) = [M, \exp_M(\xi)]$. This map is obviously continuous and can be called the *exponential map* of $\mathcal{M}_*^\infty(n)$. For each $[M, x] \in \mathcal{M}_*^\infty(n)$, the restriction $\exp: \mathcal{T}_{[M,x]}^\infty \to \mathcal{M}_*^\infty(n)$ will be denoted by $\exp_{[M,x]}$.

**Lemma 9.33.** Consider a convergent sequences $[M_i, f_i] \to [M, f]$ and $[M_i, f'_i] \to [M, f']$ in $\mathcal{P}_*^\infty(n)$ for some $n \in \mathbb{Z}_+$. Let $x = \pi_M(f)$, $x' = \pi_M(f')$, $x_i = \pi_{M_i}(f_i)$ and $x'_i = \pi_{M_i}(f'_i)$. Suppose that there is some $r > 0$ such that

$$ (9.6) \qquad \{h \in \mathrm{Iso}(M) \mid h(x) \in \bar{B}(x, 2r)\} = \{\mathrm{id}_M\}, $$

and $d(x, x'), d_i(x_i, x'_i) \le r$.

Then there is some compact domain $\Omega$ in $M$ whose interior contains $x$ and $x'$, and there are $C^\infty$ embeddings $\phi_i: \Omega \to M_i$ so that $\phi_{i*}(f) = f_i$, $\lim_i \phi_{i*}^{-1}(f'_i) = f'$ in PM, and $\lim_i \phi_i^*g_i = g|_{\Omega}$ with respect to the $C^\infty$ topology.

*Proof.* Let $\Omega_q$ be a sequence of compact domains in $M$ such that $x, x' \in \Omega_q$ and

$$ \mathrm{Pen}(\Omega_q, \mathrm{diam}(\Omega_q)) \subset \mathrm{Int}(\Omega_{q+1}). $$
By the convergence $[M_i, f_i] \to [M, f]$ and $[M_i, f'_i] \to [M, f']$ in $\mathbb{P}_*^\infty(n)$, for each $q$, there are $C^\infty$ embeddings $\phi_{q,i}, \psi_{q,i} : \Omega_q \to M_i$ so that $\phi_{q,i*}(f) = f_i$, $\phi_{q,i*}(f') = f'_i$, $\lim_i \phi_{q,i}^* g_i = g|_{\Omega_q}$ and $\lim_i \phi_{q,i}^* g_i = g|_{\Omega_q}$ with respect to the $C^\infty$ topology. Since $\phi_{q,i}(\Omega_q) \cap \psi_{q,i}(\Omega_q) \neq \emptyset$, we get

$$ \psi_{q,i}(\Omega_q) \subset \mathrm{Pen}(\phi_{q,i}(\Omega_q), \mathrm{diam}(\phi_{q,i}(\Omega_q))) \subset \mathrm{Int}(\phi_{q,i}(\Omega_{q+1})) $$

for $i$ large enough, depending on $q$, and thus the composite $h_{q,i} = \phi_{q+1,i}^{-1} \circ \psi_{q,i}$ is a well defined $C^\infty$ embedding $\Omega_q \to M$. Observe that $\lim_i h_{q,i}^* g = g|_{\Omega_q}$ with respect to the $C^\infty$ topology. Moreover

$$ \begin{aligned} \limsup_i d(x, h_{q,i}(x)) &= \limsup_i d(x, \phi_{q+1,i}^{-1} \circ \psi_{q,i}(x)) \\ &= \limsup_i d_i(x_i, \psi_{q,i}(x)) \\ &= \limsup_i d_i(x_i, x'_i) + \limsup_i d_i(x'_i, \psi_{q,i}(x)) \\ &= r + d(x', x) \\ &< 2r. \end{aligned} $$

If the statement is not true, then the above maps $\phi_{q,i}$ can be chosen such that there is some neighborhood $U$ of $f'$ in $PM$ containing no accumulation point of the sequence $\phi_{q+1,i*}^{-1}(f'_i) = \phi_{q+1,i*}^{-1} \circ \psi_{q,i*}(f') = h_{q,i*}(f')$ for each $q$. With the arguments of the proof of Lemma 9.28, it follows that there is some $h \in \mathrm{Iso}(M)$ such that $d(x, h(x)) \le 2r$ and $h_*(f') \notin U$, which contradicts (9.6). $\square$

**Proof of Theorem 9.15.** Let $M$ be a complete connected Riemannian manifold of dimension $n \ge 2$, and let $x \in M$. It is easy to construct a sequence of aperiodic small perturbations $g_i$ of the metric tensor $g$ so that $\lim_i g_i = g$ with respect to the $C^\infty$ topology. Then the corresponding Riemannian manifolds $M_i$ satisfy $\lim_i [M_i, x] = [M, x]$ in $\mathcal{M}_*^\infty(n)$. So $\mathcal{M}_*^\infty(n)$ is dense in $\mathcal{M}_*^\infty(n)$, and thus $\mathcal{M}_*^\infty(n)$ is dense too. On the other hand, the fact that $\mathcal{M}_*^\infty(n)$ is open in $\mathcal{M}_*^\infty(n)$ follows directly from Lemma 9.28; in particular, the space $\mathcal{M}_*^\infty(n)$ is polish by Lemma 9.14 and [?, ***].

For any point $[M, x] \in \mathcal{M}_*^\infty(n)$, choose some $r, \varepsilon > 0$ and some neighbourhood $\mathcal{N}$ of $[M, x]$ in $\mathcal{M}_*^\infty(n)$ satisfying the statement of Lemma 9.29 with $\varepsilon \le r/5$. We can assume that $\varepsilon$ and $\mathcal{N}$ are so small that, for any point $[L, y] \in \mathcal{N}$, the ball $B(y, \varepsilon)$ of $L$ satisfies the conditions of Proposition 9.26. Take any continuous function $\lambda : \mathcal{M}_*^\infty(n) \to [0, 1]$ supported in $\mathcal{N}$ and with $\lambda([M, x]) = 1$, whose existence is a simple consequence of the metrizability of $\mathcal{M}_*^\infty(n)$. For $[L, y] \in \mathcal{N}$, let $\lambda_{L,y} : L \to [0, 1]$ be the function defined by

$$ \lambda_{L,y}(z) = \begin{cases} \lambda([L, z]) & \text{if } d_L(y, z) \le \varepsilon \\ 0 & \text{if } d_L(y, z) \ge \varepsilon; \end{cases} $$

such a function is well defined and continuous by Lemma 9.29. Let $\omega_L$ denote the volume element of $L$, and set

$$ \bar{\lambda}_{L,y} = \frac{\lambda_{L,y}}{\int_L \lambda_{L,y} \omega_L}. $$
Then $\mu_{L,y} = \bar{\lambda}_{L,y} \omega_L$ is a probability measure on $L$, and the identity map $(L, \mu_{L,y}) \to L$ is a distribution of mass on $L$ satisfying the conditions of Proposition 9.26 with the ball $B(y, \varepsilon)$; thus its center of mass $\mathcal{C}_{L,y}$ is defined in $B(y, \varepsilon)$. Let $\mathbf{c}: \mathcal{N} \to \mathcal{M}_{*}^{\infty}(n)$ be the map given by $\mathbf{c}([L, y]) = [L, \mathcal{C}_{L,y}]$.

**Claim 17.** For equivalent points $[L, y]$, $[L, y'] \in \mathcal{N}$, we have $\mathbf{c}([L, y]) = \mathbf{c}([L, y'])$ if $d_L(y, y') \le \varepsilon$.

Take any point $z \in L$. If $[L, z] \notin \mathcal{N}$ or $d_L(y, z)$, $d_L(y', z) > \varepsilon$, then $\lambda_{L,y}(z) = \lambda_{L,y'}(z) = 0$. If $[L, z] \in \mathcal{N}$ and $d_L(y, z) \le \varepsilon$, then $d_L(y', z) \le 2\varepsilon$, yielding $d_L(y', z) \le \varepsilon$ by Lemma 9.29 since $5\varepsilon \le r$, and thus $\lambda_{L,y}(z) = \lambda_{L,y'}(z) = \lambda([L, z])$. If $[L, z] \in \mathcal{N}$ and $d_L(y', z) \le \varepsilon$, we similarly get $\lambda_{L,y}(z) = \lambda_{L,y'}(z)$. Therefore $\lambda_{L,y} = \lambda_{L,y'}$, yielding $\mathcal{C}_{L,y} = \mathcal{C}_{L,y'}$, and thus $\mathbf{c}([L, y]) = \mathbf{c}([L, y'])$, which finishes the proof of Claim 17.

**Claim 18.** The map $\mathbf{c}$ is continuous.

To prove this assertion, take any convergent sequence $[L_i, y_i] \to [L, y]$ in $\mathcal{N}$. Let $\Omega$ be a compact domain in $L$ whose interior contains $\bar{B}(y, \varepsilon)$. Then there is a $C^\infty$ embedding $\phi_i: \Omega \to L_i$ for each $i$ large enough so that $\lim_i \phi_i^* g_i = g|_{\Omega}$ with respect to the $C^\infty$ topology. It follows that $\lim_i \phi_i^* \mu_{L_i, y_i} = \mu_{L, y}|_{\Omega}$ with respect to the $C^0$ topology by the continuity of $\lambda$, and thus this convergence holds in the space of probability measures on $\Omega$ too ***. Since $\phi_i^{-1}(\mathcal{C}_{L_i, y_i})$ is the center of mass of the mass distribution on $\Omega$ defined by the probability measure $\phi_i^* \mu_{L_i, y_i}$, it follows from Corollary 9.27 that $\lim_i \phi_i^{-1}(\mathcal{C}_{L_i, y_i}) = \mathcal{C}_{L, y}$ in $L$. Therefore $\lim_i \mathbf{c}([L_i, y_i]) = \mathbf{c}([L, y])$ in $\mathcal{M}_{*}^{\infty}(n)$ because $\Omega$ is arbitrary, which completes the proof of Claim 18.

The set $\mathcal{Z} = \mathbf{c}(\mathcal{N})$ will play the rôle of local transversal but, since $\mathcal{Z}$ may not be contained in $\mathcal{N}$, we need a larger neighbourhood $\mathcal{N}'$ of $[M, x]$, which is defined as follows. Let $\mathcal{N}'$ be the set of points $[L, z] \in \mathcal{M}_{*}^{\infty}(n)$ such that $z \in B(c, \varepsilon)$ for some $c \in L$ with $[L, c] \in \mathcal{Z}$; that is, $\mathcal{N}'$ is the union of the sets $\iota_L(B(c, \varepsilon))$ with $[L, c]$ running in $\mathcal{Z}$. Notice that $\mathcal{N} \subset \mathcal{N}'$ because $d_M(y, \mathcal{C}_{L,y}) < \varepsilon$ for all $[L, y] \in \mathcal{N}$.

Let $\mathbf{c}' : \mathcal{N}' \to \mathcal{Z}$ be defined by the condition $\mathbf{c}'([L, z]) = [L, c]$ if $[L, c] \in \mathcal{Z}$ and $d_L(c, z) < \varepsilon$. To prove that $\mathbf{c}'$ is well defined, take another point $c' \in L$ satisfying $[L, c'] \in \mathcal{Z}$ and $d_L(c', z) < \varepsilon$, and let us check that $[L, c] = [L, c']$. Choose points $y, y' \in L$ such that $[L, y], [L, y'] \in \mathcal{N}$, $\mathbf{c}([L, y]) = [L, c]$ and $\mathbf{c}([L, y']) = [L, c']$. Then

$$d_L(y, y') \le d_L(y, c) + d_L(c, z) + d_L(z, c') + d_L(c', y') < 4\varepsilon,$$

yielding $d_L(y, y') \le \varepsilon$ by Lemma 9.29 since $5\varepsilon \le r$, which implies $[L, c] = [L, c']$ by Claim 17. Furthermore $\mathbf{c}'$ is an extension of $\mathbf{c}$ because $d_L(y, \mathcal{C}_{L,y}) < \varepsilon$ for all $[L, y] \in \mathcal{N}$. Note also that $\mathbf{c}'([L, c]) = [L, c]$ for all $[L, c] \in \mathcal{Z}$.

**Claim 19.** For equivalent points $[L, z]$, $[L, z'] \in \mathcal{N}'$, we have $\mathbf{c}'([L, z]) = \mathbf{c}'([L, z'])$ if $d_L(z, z') \le 2\varepsilon$.
Let $\mathbf{c}'([L, z]) = [L, c]$ and $\mathbf{c}'([L, z']) = [L, c']$. Choose points $[L, y], [L, y'] \in \mathcal{N}$ with $\mathbf{c}([L, y]) = [L, c]$ and $\mathbf{c}([L, y']) = [L, c']$. Then

$$d_L(y, y') \le d_L(y, c) + d_L(c, z) + d_L(z, z') + d_L(z', c') + d_L(c', y') < 5\epsilon.$$

From Lemma 9.29 and since $5\epsilon \le r$, it follows that $[L, c] = [L, c']$ as desired.

**Claim 20.** The map $\mathbf{c}'$ is continuous.

To prove this assertion, take any convergent sequence $[L_i, z_i] \to [L, z]$ in $\mathcal{N}'$. Let $\mathbf{c}'([L_i, z_i]) = [L_i, c_i]$ and $\mathbf{c}'([L, z]) = [L, c]$, and choose points $[L_i, y_i], [L, y] \in \mathcal{N}$ so that $\mathbf{c}([L_i, y_i]) = [L_i, c_i]$ and $\mathbf{c}([L, y]) = [L, c]$. Observe that $d_i(y_i, z_i), d(y, z) < 2\epsilon$. Then, by Lemma 9.30, there are points $y'_i \in B(z_i, 2\epsilon)$ such that $\lim_{i} [L_i, y'_i] = [L, y]$ in $M_*^\infty(n)$ as $i \to \infty$. Thus $[L_i, y'_i] \in \mathcal{N}$ for $i$ large enough, and moreover

$$d_i(y_i, y'_i) \le d_i(y_i, z_i) + d_i(z_i, y'_i) < 4\epsilon,$$

yielding $d_i(y_i, y'_i) \le \epsilon$ by Lemma 9.29 since $5\epsilon \le r$. By Claim 17, it follows that $\mathbf{c}([L_i, y'_i]) = \mathbf{c}([L_i, y_i]) = [L_i, c_i]$ for $i$ large enough, yielding $\lim_{i} [L_i, c_i] = [L, c]$ in $M_*^\infty(n)$ by Claim 18, which finishes the proof of Claim 20.

For each $[L, z] \in \mathcal{Z}$, let $B(0_z, \epsilon)$ be the ball of radius $\epsilon$ centered at the zero vector $0_z$ of $T_z L$. We can assume that $\epsilon$ and $\mathcal{N}$ are so small that the following properties hold for all $[L, y] \in \mathcal{N}$ and all $z \in B(y, \epsilon)$:

(a1) The map $\exp : B(0_z, \epsilon) \to B(z, \epsilon)$ is a diffeomorphism; and

$$(a2) \quad \{h \in \mathrm{Iso}(L) \mid h(z) \in \bar{B}(z, 4\epsilon)\} = \{\mathrm{id}_L\}.$$

Observe that the second property can be assumed by Lemma 9.28. Notice also that, in particular, both of the above assumptions hold for all $[L, z] \in \mathcal{Z}$. Let $\tilde{\mathcal{N}}'$ denote the set of points $[L, \xi] \in \mathcal{T}_*^\infty(n)$ such that $\pi([L, \xi]) \in \mathcal{Z}$ and $|\xi| < \epsilon$.

**Claim 21.** The map $\exp : \tilde{\mathcal{N}}' \to \mathcal{N}'$ is a homeomorphism.

This map is obviously surjective. For $i = 1, 2$, take points $[L_i, \xi_i] \in \tilde{\mathcal{N}}'$; thus $\xi_i \in T_{c_i} L_i$ for some points $[L_i, c_i] \in \mathcal{Z}$, and we have $\exp([L_i, \xi_i]) = [L_i, z_i]$ for $z_i = \exp_i(\xi_i)$. Suppose that $[L_1, z_1] = [L_2, z_2]$, which means that there is an isometry $\phi : L_1 \to L_2$ with $\phi(z_1) = z_2$. Then

$$(9.7) \qquad \exp_2 \circ \phi_*(\xi_1) = \phi \circ \exp_1(\xi_1) = \phi(z_1) = z_2 = \exp_2(\xi_2),$$

$$(9.8) \qquad d_2(\phi(c_1), c_2) \le d_2(\phi(c_1), z_2) + d_2(z_2, c_2) = d_1(c_1, z_1) + d_2(z_2, c_2) < 2\epsilon.$$

We get

$$[L_1, c_1] = \mathbf{c}'([L_1, c_1]) = \mathbf{c}'([L_2, \phi(c_1)]) = [L_2, c_2]$$

by Claim 19 and (9.8). So there is an isometry $\psi : L_1 \to L_2$ such that $\psi(c_1) = c_2$. Then the isometry $h = \psi^{-1} \circ \phi : L_1 \to L_1$ satisfies

$$d_1(c_1, h(c_1)) = d_2(c_2, \phi(c_1)) < 2\epsilon$$
by (9.8), yielding $h = \mathrm{id}_{L_1}$ by the assumption (a2). Hence $\phi(c_1) = \psi(c_1) = c_2$, yielding $\phi_*(\xi_1) = \xi_2$ by (9.7) and the assumption (a1) since $\xi_i \in T_{c_i} L_i$. Therefore $\exp : \hat{\mathcal{N}}' \to \mathcal{N}'$ is bijective. Moreover the continuity of $\exp^{-1} : \mathcal{N}' \to \hat{\mathcal{N}}'$ is a straightforward exercise, and Claim 21 follows.

By Lemma 9.32, there is some neighborhood $\Theta$ of $[M, x]$ in $\mathcal{M}_*^\infty(n)$ and some homeomorphism $\theta : \pi^{-1}(\Theta) \to \mathbb{R}^n \times \Theta$ so that the restriction $\theta : \pi^{-1}([L, y]) \to \mathbb{R}^n \times \{[L, y]\} \equiv \mathbb{R}^n$ is a linear isometry for all $[L, y] \in \Theta$. More precisely, according to the proof of Lemma 9.32, we can suppose that there is a local section $\sigma : \Theta \to \mathcal{Q}_*^\infty(n)$ of $\pi : \mathcal{Q}_*^\infty(n) \to \mathcal{M}_*^\infty(n)$ so that $\theta([L, \xi]) = (\sigma([L, y])(\xi), [L, \xi])$ if $\pi_L(\xi) = y$, by using the canonical identification of each frame of $T_y L$ with a linear map $T_y L \to \mathbb{R}^n$. We can assume that $Z \subset \Theta$ by Lemma 9.31. Hence the composite

$$ \mathcal{N}' \xrightarrow{\exp^{-1}} \hat{\mathcal{N}}' \xrightarrow{\theta} B_\varepsilon^n \times Z $$

is a homeomorphism $\Phi : \mathcal{N}' \to B_\varepsilon^n \times Z$, where $B_\varepsilon^n$ denotes the open ball of radius $\varepsilon$ centered at the origin in $\mathbb{R}^n$. This shows that $\mathcal{M}_{*,\mathrm{la}}^\infty(n)$, with its canonical equivalence relation, is a foliated space of dimension $n$, showing property (ii).

Now, for $k = 1, 2$, let $\Phi_k : \mathcal{N}_k' \to B_{\varepsilon_k}^n \times Z_k$ be two such homeomorphisms, which were constructed as above by using maps $c_k' : \mathcal{N}_k' \to Z_k$, restrictions $\exp : \hat{\mathcal{N}}_k' \to \mathcal{N}_k'$ of the exponential map, and local sections $\sigma_k : \Theta_k \to \mathcal{Q}_*^\infty(n)$. The following assertion shows that the this kind of charts define a $C^\infty$ structure on the foliated space $\mathcal{M}_{*,\mathrm{la}}^\infty(n)$.

**Claim 22.** The change of coordinates $\Phi_2 \circ \Phi_1^{-1} : \Phi_1(\mathcal{N}_1' \cap \mathcal{N}_2') \to \Phi_1(\mathcal{N}_1' \cap \mathcal{N}_2')$ is $C^\infty$ in the sense of ***.

This map has the expression

$$ \Phi_2 \circ \Phi_1^{-1}(v, [L, c]) = (\Psi(v, [L, c]), \Gamma([L, c])) $$

where $\Gamma : c_1'(\mathcal{N}_1' \cap \mathcal{N}_2') \to c_2'(\mathcal{N}_1' \cap \mathcal{N}_2')$ is the corresponding holonomy transformation, and $\Psi : \Phi_1(\mathcal{N}_1' \cap \mathcal{N}_2') \to \mathbb{R}^n$ is defined by

$$ \Psi(v, [L, c]) = \sigma_2([L, c']) \circ \exp_{[L,c']}^{-1} \circ \exp_{[L,c]} \circ \sigma_1([L, c])^{-1}(v), $$

where $[L, c'] = \Gamma([L, c])$. Let $[L, f] = \sigma_1([L, c])$ and $[L, f'] = \sigma_2([L, c'])$. We can take $c'$ so that $d(c, c') < \varepsilon_1 + \varepsilon_2$, and then

$$ \Psi(v, [L, c]) = f' \circ \exp_{c'}^{-1} \circ \exp_c \circ f^{-1}(v). $$

To prove that $\Psi$ is $C^\infty$ in the sense of ***, fix any $(v, [L, c]) \in \Phi_1(\mathcal{N}_1' \cap \mathcal{N}_2')$, and take $c', f$ and $f'$ as above. Let $V$ and $\mathcal{O}$ be open neighbourhoods of $v$ and $[L, c]$ in $\mathbb{R}^n$ and $Z_1$, respectively, such that $\bar{V} \times \mathcal{O} \subset \Phi_1(\mathcal{N}_1' \cap \mathcal{N}_2')$. Take any convergent sequence $[L_i, c_i] \to [L, c]$ in $\mathcal{O}$, and define $c'_i, f'_i$ and $f'_i$ as before for each $i$. Notice that $\Psi(v, [L, c])$ and $\Psi(v, [L_i, c_i])$ are defined for all $v \in V$, and let $\psi, \psi_i : V \to \mathbb{R}^n$ be the $C^\infty$ maps given by $\psi(v) = \Psi(v, [L, c])$ and $\psi_i(v) = \Psi(v, [L_i, c_i])$. We have to prove that $\lim_i \psi_i = \psi$ with respect to the weak $C^\infty$ topology.
Let $\Omega$ be any compact domain in $L$ whose interior contains $\bar{B}(c, \varepsilon_1 + 2\varepsilon_2)$, and thus $\bar{B}(c', \varepsilon_2) \subset \text{Int}(\Omega)$ too. Since the sections $\sigma_1$ and $\sigma_2$ are continuous, there are $C^\infty$ embeddings $\phi_i : \Omega \to L_i$ for $i$ large enough so that $\phi_{i*}(f) = f_i$ and $\lim_i \phi_i^* g_i = g|_{\Omega}$, and therefore $\phi_i(c) = c_i$. Hence $c'_i \in \phi_i(\text{Int}(\Omega))$ for $i$ large enough, and moreover $\lim_i \phi_{i*}^{-1}(f'_i) = f'$ by (a2) and Lemma 9.33. Observe that the composite $\hat{\psi} = \exp_{c'}^{-1} \circ \exp_c$ is defined on $W = f^{-1}(V) \subset B(0_c, \varepsilon_1) \subset T_c L$. It follows that the composite $\hat{\psi}_i = \phi_{i*}^{-1} \circ \exp_{c_i'}^{-1} \circ \exp_{c_i} \circ \phi_{i*}$ is also defined on $W$ for $i$ large enough, and moreover $\lim_i \hat{\psi}_i = \hat{\psi}$ in the space of $C^\infty$ maps $W \to T_c' L$ with the weak $C^\infty$ topology. So

$$ \lim_i \phi_{i*}^{-1}(f'_i) \circ \hat{\psi}_i \circ f^{-1} = f' \circ \hat{\psi} \circ f^{-1} = \psi $$

in the space of $C^\infty$ maps $V \to \mathbb{R}^n$ with the weak $C^\infty$ topology. Then Claim 22 follows because

$$ \phi_{i*}^{-1}(f'_i) \circ \hat{\psi}_i \circ f^{-1} = \phi_{i*}^{-1}(f'_i) \circ \hat{\psi}_i \circ (\phi_{i*}^{-1}(f_i))^{-1} = f'_i \circ \phi_{i*} \circ \hat{\psi}_i \circ \phi_{i*}^{-1} \circ f_i^{-1} = \psi_i. $$

Now we can consider the tangent bundle $TM_{*,la}^\infty(n)$; that is, the vector bundle of vectors tangent to the leaves of $M_{*,la}^\infty(n)$. Let $\mathcal{T}_{*,la}^\infty(n)$ denote the restriction of the Riemannian vector bundle $\mathcal{T}_*(n)$ to $M_{*,la}^\infty(n)$. For each leaf $\iota(M)$ of $M_{*,la}^\infty(n)$, the canonical homeomorphism $\bar{\iota} : \text{Iso}(M)\backslash M \to \iota(M)$ is a $C^\infty$ diffeomorphism, and $\iota_{*x} : T_x M \to T_{[M,x]} M_{*,la}^\infty(n)$ is an isomorphism for each $x \in M$. According to Lemma 9.32, we get a canonical bijection $TM_{*,la}^\infty(n) \to \mathcal{T}_{*,la}^\infty(n)$ defined by $\iota_{*x}(\xi) \mapsto [M, \xi]$. It is an easy exercise to prove that this bijection is an isomorphism of vector bundles. So the Riemannian structure on $\mathcal{T}_{*,la}^\infty(n)$ defined in Lemma 9.32 corresponds to a Riemannian structure on $TM_{*,la}^\infty(n)$, which can be easily proved to be $C^\infty$ by using the above kind of flow boxes of $M_{*,la}^\infty(n)$. It is elementary that each isomorphism $\iota_{*x} : T_x M \to T_{[M,x]} M_{*,la}^\infty(n)$ is an isometry, completing the proof of property (iii).

For any foliated space $X$, a path $\alpha : I = [0, 1] \to X$ will be called foliated when $\alpha(I)$ is contained in some leaf. The space of foliated paths in $X$ with the compact-open topology will be denoted by $C_{\text{fol}}(I, X)$. Property (iv) of the statement means that the following assertions holds.

**Claim 23.** The following properties hold for any point $[M, x] \in M_{*,la}^\infty(n)$, any path $\alpha : I = [0, 1] \to M$ with $\alpha(0) = x$, and any neighborhood $\mathcal{U}$ of $\iota \circ \alpha$ in $C_{\text{fol}}(I, M_{*,la}^\infty(n))$:

(a) If $\alpha(1) = x$ then, for each point $[N, y] \in M_{*,la}^\infty(n)$ close enough to $[M, x]$, there is a path $\beta \in \mathcal{U}$ with $\beta(0) = \beta(1) = [N, y]$.

(b) If $\alpha(1) \neq x$ then there is some path $\beta \in \mathcal{U}$ with $\beta(0) \neq \beta(1)$.

To prove Claim 23, consider a compact domain $\Omega$ in $M$ whose interior contains $\alpha(I)$, a point $[N, y] \in M_{*,la}^\infty(n)$, and a $C^\infty$ embedding $\phi : \Omega \to N$ with $\phi(x) = y$. Let $\beta = \iota \circ \phi \circ \alpha \in C_{\text{fol}}(I, M_{*,la}^\infty(n))$; that is, $\beta(t) = [N, \phi \circ \alpha(t)]$ for each $t \in I$. Observe that $\beta \in \mathcal{U}$ if $\Omega$ is large
enough and $\phi^*g_N$ is close enough to $g_M|_{\Omega}$ with respect to the $C^\infty$ topology, which in turn
can be assumed just when $[N, y]$ is close enough to $[M, x]$. If $\alpha(1) = x$, then

$$
\beta(0) = [N, \phi(x)] = [N, y] = [N, \phi \circ \alpha(1)] = \beta(1),
$$

showing property (a).

Suppose now that $\alpha(1) \neq x$. Let $N$ equal to the manifold $M$ with another metric tensor $g_N$. Define $\beta$ as above with $y$ equal to the point in $N$ that corresponds to $x$, and $\phi$ equal to the restriction of the identity map $M \to N$ to some compact compact domain $\Omega$ of $M$ whose interior contains $\alpha(I)$. If $\Omega$ is large enough and $g_N$ is a small enough local perturbation of $g_N$, then $\beta \in \mathcal{U}$ as before. Moreover we can choose $g_N$ so that $N$ is aperiodic because $n \ge 2$. Hence $\iota: N \to \mathcal{M}_{*,\text{la}}^{\infty}(n)$ is injective, yielding

$$
\beta(0) = \iota \circ \phi(x) \neq \iota \circ \phi \circ \alpha(1) = \beta(1),
$$

which proves property (b).

<figure><img src="image_20.png" /></figure>
# Chapter 10

# Distortions and bidistortions

The constructions of the Gromov space $\mathcal{M}_*$ and the space $\mathcal{M}_*^{\infty}(n)$ will be generalized with the theory of this chapter.

## 10.1 Preliminaries and notation

To begin with, consider the theory of *universes* developed by Grothendieck's school [?], and fix some *universe* $\mathfrak{U}$. A set is said to be *small* when it belongs to $\mathfrak{U}$.

The basic concepts and notation of category theory will be used in this chapter. Some definitions and notations are recalled and fixed in this section for further use (see e.g. [?]). Recall that a category $\mathcal{A}$ can be defined as a structure consisting of the following ingredients:

* A pair of sets, $\mathcal{A}_0$ and $\mathcal{A}_1$. The elements of $\mathcal{A}_0$ are called *objects* and the elements of $\mathcal{A}_1$ *morphisms* or *arrows*.

* An injective map $\mathcal{A}_0 \to \mathcal{A}_1$. For each $M \in \mathcal{A}_0$, the corresponding morphism is called the *identity* or *unit* at $M$, and denoted by $\mathrm{id}_M$ or $1_M$, or simply id or 1.

* Two maps $\mathcal{A}_1 \to \mathcal{A}_0$, called *source* and *target* maps (or *domain* and *codomain*). Each $f \in \mathcal{A}_1$ with source $M$ and target $N$ is usually represented by an arrow $M \to N$; the notation $f : M \to N$ or $M \xrightarrow{f} N$ can be used. Let $\mathcal{A}_2$ be the subset of $\mathcal{A}_1 \times \mathcal{A}_1$ consisting of the pairs $(g, f)$ which are *consecutive* in the sense that they form a diagram $L \xrightarrow{f} M \xrightarrow{g} N$.

* A map $\mathcal{A}_2 \to \mathcal{A}_1$, called *operation* or *product*, which assigns a morphism $gf : L \to N$ to each pair of consecutive morphisms $L \xrightarrow{f} M \xrightarrow{g} N$.

Moreover the following axioms are required:

* For each $M \in \mathcal{A}_0$, $\mathrm{id}_M$ is an identity element in the sense that $\mathrm{id}_M f = f$ and $g \mathrm{id}_M = g$ for all morphisms $f$ with target $M$ and $g$ with source $M$.
• The operation is *associative* in the sense that $(hg)f = h(gf)$ for all consecutive morphisms $L \xrightarrow{f} M \xrightarrow{g} N$.

Sometimes, it is convenient to identify each object with the corresponding unit; thus $\mathcal{A}_0$ is considered as a subset of $\mathcal{A}_1$, and $\mathcal{A}$ is identified to the set $\mathcal{A}_1$ with a partially defined operation satisfying some conditions. Both points of view will be used without further comment. For instance, this second interpretation will be used in notions involving the underlying set of each category, like inclusions, maps, products, etc. On the other hand, the first interpretation, using objects, is appropriate to describe certain examples and constructions. The categories with one object are just the *monoids*.

The notation $\mathcal{A}_0, \mathcal{A}_1$ and $\mathcal{A}_2$ can be continued for all $n \in \mathbb{N}$ by defining $\mathcal{A}_n$ as the set of sequences $(f_n, \dots, f_1)$ of $n$ morphisms which are consecutive; i.e., they form a diagram

$$M_0 \xrightarrow{f_1} M_1 \xrightarrow{f_2} \dots \xrightarrow{f_n} M_n.$$

Furthermore let $\mathcal{A}_\infty = \bigcup_{n=0}^\infty \mathcal{A}_n$. Two morphisms of $\mathcal{A}$ with the same source and target are called *parallel*. For $M, N \in \mathcal{A}_0$, let $\mathcal{A}(M, N)$ denote the family of morphisms $M \to N$ in $\mathcal{A}$. A morphism $f \in \mathcal{A}(M, N)$ is called an *isomorphism* when there is some $g \in \mathcal{A}(M, N)$ such that $gf = \mathrm{id}_M$ and $fg = \mathrm{id}_N$; this $g$ is unique; it is called the *inverse* of $f$ and denoted by $f^{-1}$. Let $\mathrm{Iso}_\mathcal{A}(M, N)$ denote the set of isomorphisms $M \to N$; the notation $\mathrm{Iso}_\mathcal{A}(M)$ is used when $M = N$. An object $M \in \mathcal{A}_0$ is said to be *aperiodic* if $\mathrm{Iso}_\mathcal{A}(M) = \{\mathrm{id}_M\}$.

Recall that a *subcategory* of a category $\mathcal{A}$ is a subset $\mathcal{C} \subset \mathcal{A}$ such that the source and target maps of $\mathcal{A}$, as well as its operation, can be restricted to $\mathcal{C}$ in an obvious way defining a category. A subcategory $\mathcal{C} \subset \mathcal{A}$ is said to be *full* if $\mathcal{C}(M, N) = \mathcal{A}(M, N)$ for all $M, N \in \mathcal{C}_0$. The mapping $\mathcal{C} \mapsto \mathcal{C}_0$ is a bijection between the family of full subcategories of $\mathcal{A}$ and the family of subsets of $\mathcal{A}_0$. For instance, the full subcategory of $\mathcal{A}$ given by the aperiodic objects will be denoted by $\mathcal{A}_a$.

Unless the contrary is indicated, we will only consider categories $\mathcal{A}$ with $\mathcal{A}_0 \subset \mathcal{U}$ and $\mathcal{A}(M, N) \in \mathcal{U}$ for all $M, N \in \mathcal{A}_0$; they will be called $\mathcal{U}$-categories. A category $\mathcal{A}$ is called *small* if its underlying set is small.

Recall that a category consisting only of isomorphisms is called a *groupoid*. Let $\mathcal{G}$ be a groupoid with surce and target maps $s, t: \mathcal{G} \to \mathcal{G}_0$. A *left action* of $\mathcal{G}$ on a set $X$ is given by the following data and condition:

• A map $r: X \to \mathcal{G}_0$. Thus we can consider the fibered product

$$\Omega = \{(g, x) \in \mathcal{G} \times X \mid t(g) = r(x)\}.$$

• A map $\Omega \to X$, $(g, x) \mapsto g \cdot x$.

• If $t(g) = r(x)$ for $g \in \mathcal{G}$ and $x \in X$, then $r(gx) = s(g)$ and $h \cdot (g \cdot x) = (hg) \cdot x$ for all $h \in \mathcal{G}$ with $t(h) = s(g)$.

<footnote n="1">¹Here, groupoids are not required to be small, which is a usual condition.</footnote>
Such a left action defines a partition of $X$ into the subsets $\mathcal{G} \cdot x = \{g \cdot x \mid (g, x) \in \Omega\}, x \in X$, which are called *orbits*. The corresponding quotient set is denoted by $\mathcal{G} \setminus X$.

The isomorphisms of any category $\mathcal{A}$ form a groupoid denoted by $\text{Iso}(\mathcal{A})$. There is a canonical left action of $\text{Iso}(\mathcal{A})$ on $\mathcal{A}_0$ so that $\phi \cdot N$ is defined if $N$ is the target of $\phi$, and then $\phi \cdot N$ is the source of $\phi$. The corresponding quotient set $\text{Iso}(\mathcal{A}) \setminus \mathcal{A}_0$ consists of the isomorphism classes of objects of $\mathcal{A}$.

With more generality, for an category $\mathcal{A}$, an equivalence relation “$\sim$” on $\mathcal{A}_0$ is defined by setting $M \sim N$ when $\mathcal{A}(M, N) \neq \emptyset \neq \mathcal{A}(N, M)$. The corresponding quotient set is denoted by $\mathcal{A} \setminus \mathcal{A}_0$. The class of each $M \in \mathcal{A}_0$ is denoted by $[M]$. These classes may not be small even when $\mathcal{A}$ is a $\mathfrak{U}$-category, and thus $\mathcal{A} \setminus \mathcal{A}_0$ may not be contained in $\mathfrak{U}$. But, if $\mathcal{A}$ is a $\mathfrak{U}$-category, there is a bijection between $\mathcal{A} \setminus \mathcal{A}_0$ and some subset of $\mathfrak{U}$ by the axiom of choice, and thus we can assume that $\mathcal{A} \setminus \mathcal{A}_0 \subset \mathfrak{U}$ after reparametrization.

Recall that a map between categories, $F : \mathcal{A} \to \mathcal{B}$, is called a *functor* when $F(\mathcal{A}_0) \subset \mathcal{B}_0$, $(F \times F)(\mathcal{A}_2) \subset \mathcal{B}_2$ and $F(g)F(f) = F(gf)$ for all $(g, f) \in \mathcal{A}_2$. Sometimes, the notation $F_M = F(M)$ will be used for each $M \in \mathcal{A}_0$. A functor $F : \mathcal{A} \to \mathcal{B}$ restricts to a map $F_0 : \mathcal{A}_0 \to \mathcal{B}_0$, and induces maps $F_n : \mathcal{A}_n \to \mathcal{B}_n$ for each $n \in \mathbb{N}$ in a canonical way. The maps $F_n$ can be combined to define a map $F_\infty : \mathcal{A}_\infty \to \mathcal{B}_\infty$. It also induces a map $\bar{F} : \mathcal{A} \setminus \mathcal{A}_0 \to \mathcal{B} \setminus \mathcal{B}_0$ defined by $\bar{F}([M]) = [F_M]$, and restricts a functor $\text{Iso}(F) : \text{Iso}(\mathcal{A}) \to \text{Iso}(\mathcal{B})$, which in turn induces the map $\text{Iso}(\bar{F}) : \text{Iso}(\mathcal{A}) \setminus \mathcal{A}_0 \to \text{Iso}(\mathcal{B}) \setminus \mathcal{B}_0$. With the operation of composition, the functors between $\mathfrak{U}$-categories form a category whose objects may not be in $\mathfrak{U}$; they may be in a larger universe. A functor is said to be *small* if its source category is small. An example of functor is given by the inclusion map of any subcategory.

It is said that a functor $F : \mathcal{A} \to \mathcal{B}$ is *commutative* when $F(\mathcal{A}(M, N))$ has at most one element for all $M, N \in \mathcal{A}_0$; this corresponds to the concept of a *commutative diagram* in the case of the functor defined by a diagram [?].

A *sub-functor* of a functor $F : \mathcal{A} \to \mathcal{B}$ is a functor $G : \mathcal{C} \to \mathcal{B}$ which is the composite of a functor $g : \mathcal{C} \to \mathcal{A}$ with $F$; this $g$ will be called a *reparametrization functor* defining $G$.

Recall that the *opposite* of a category $\mathcal{A}$ is the category $\mathcal{A}^{\text{op}}$ with $\mathcal{A}_0^{\text{op}} = \mathcal{A}_0$, $\mathcal{A}_1^{\text{op}} = \mathcal{A}_1$ and the same operation as $\mathcal{A}$, but so that the source (respectively, target) projection of $\mathcal{A}^{\text{op}}$ is the target (respectively, source) projection of $\mathcal{A}$. A functor $\mathcal{A} \to \mathcal{B}^{\text{op}}$ is called a *contravariant functor* $\mathcal{A} \to \mathcal{B}$.

For a set $X$, a category $\mathcal{B}$ and a map $\gamma : X \to \mathcal{B}_0$, the *pull-back* of $\mathcal{B}$ by $\gamma$ is the category $\gamma^*\mathcal{B}$ with $(\gamma^*\mathcal{B})_{0} = X$,

$$ (\gamma^*\mathcal{B})(x, y) = \{(x, y, \phi) \mid \phi \in \mathcal{B}(\gamma(x), \gamma(y))\} \equiv \mathcal{B}(\gamma(x), \gamma(y)) $$

for $x, y \in X$, and with the operation canonically induced by the operation of $\mathcal{B}$. There is a functor $\tilde{\gamma} : \gamma^*\mathcal{B} \to \mathcal{B}$ defined by setting $\tilde{\gamma}_0 = \gamma$ and $\tilde{\gamma}(x, y, \phi) = \phi$ for all $x, y \in X$ and $\phi \in \mathcal{B}(\gamma(x), \gamma(y))$. Observe that $\tilde{\gamma}$ is injective if so is $\gamma$.

For any functor $F : \mathcal{A} \to \mathcal{B}$, the pull-back $F_0^*\mathcal{B}$ will be denoted by $\tilde{\mathcal{B}}$ and called the *lift* of $\mathcal{B}$ by $F$. Observe that $F$ can be given as a composition

$$ (10.1) \qquad \mathcal{A} \longrightarrow \tilde{\mathcal{B}} \stackrel{\tilde{F}_0}{\longrightarrow} \mathcal{B}, $$
where the first functor is the identity on objects and induced by $F$ on morphisms: $\phi \mapsto (M, N, F(\phi))$ if $\phi \in \mathcal{A}(M, N)$ for $M, N \in \mathcal{A}_0$. Notice that $\tilde{F}_0$ is given by $F$ on objects and is essentially the identity map on morphisms: $(M, N, f) \mapsto f$ for $M, N \in \mathcal{A}_0$ and $f \in \mathcal{B}(F_M, F_N)$. A commutative diagram of functors

<figure><img src="image_21.png" /></figure>

induces a functor $\tilde{H} : \tilde{\mathcal{B}} \to \tilde{\mathcal{B}}'$, called the *lift* of $H$, given by $G$ on objects and induced by $H$ on morphisms: $(M, N, f) \mapsto (G_M, G_N, H(f))$ for $M, N \in \mathcal{A}_0$ and $f \in \mathcal{B}(F_M, F_N)$. The assignment of $\tilde{G}$ to such a diagram is functorial in the obvious sense.

Recall that the *product* of two categories $\mathcal{A}$ and $\mathcal{A}'$ is the category $\mathcal{A} \times \mathcal{A}'$ determined as follows: $(\mathcal{A} \times \mathcal{A}')_0 = \mathcal{A}_0 \times \mathcal{A}'_0$, $(\mathcal{A} \times \mathcal{A}')_1 = \mathcal{A}_1 \times \mathcal{A}'_1$, the source (respectively, target) map $(\mathcal{A} \times \mathcal{A}')_1 \to (\mathcal{A} \times \mathcal{A}')_0$ is the product of the source (respectively, target) maps of $\mathcal{A}$ and $\mathcal{A}'$, and its operation is given by $(g, g')(f, f') = (gf, g'f')$ whenever this product is defined. The factor projections of $\mathcal{A} \times \mathcal{A}'$ to $\mathcal{A}$ and $\mathcal{A}'$ are functors. If $F : \mathcal{A} \to \mathcal{B}$ and $F' : \mathcal{A}' \to \mathcal{B}'$ are functors, then the product $F \times F' : \mathcal{A} \times \mathcal{A}' \to \mathcal{B} \times \mathcal{B}'$ is a functor.

With more generality, for a diagram of functors

$$ (10.2) \qquad \mathcal{A} \xrightarrow{P} \mathcal{C} \xleftarrow{P'} \mathcal{A}' $$

the corresponding *fibered product* (over $\mathcal{C}$) or *pull-back* is the diagram

$$ \mathcal{A} \xleftarrow{\pi} \mathcal{A} \times_{\mathcal{C}} \mathcal{A}' \xrightarrow{\pi'} \mathcal{A}' $$

where $\mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$ is the subcategory of $\mathcal{A} \times \mathcal{A}'$ with

$$ \begin{aligned} (\mathcal{A} \times_{\mathcal{C}} \mathcal{A}')_0 &= \{(M, M') \in \mathcal{A}_0 \times \mathcal{A}'_0 \mid P_M = P'_{M'}\}, \\ (\mathcal{A} \times_{\mathcal{C}} \mathcal{A}')_1 &= \{(f, f') \in \mathcal{A}_1 \times \mathcal{A}'_1 \mid P(f) = P'(f')\}, \end{aligned} $$

and where $\pi$ and $\pi'$ are the canonical functors defined by the restrictions of the factor projections of $\mathcal{A} \times \mathcal{A}'$. This is the fibered product or pull-back in the category of functors between $\mathfrak{U}$-categories [?, p. 71]; in particular, the diagram

$$ (10.3) \qquad \begin{array}{ccc} \mathcal{A} \times_{\mathcal{C}} \mathcal{A}' & \xrightarrow{\pi} & \mathcal{A} \\ \pi' \downarrow & & \downarrow P \\ \mathcal{A}' & \xrightarrow{P'} & \mathcal{C} \end{array} $$

is commutative. Observe that

$$ (10.4) \qquad \mathrm{Iso}(\mathcal{A} \times_{\mathcal{C}} \mathcal{A}') = \mathrm{Iso}(\mathcal{A}) \times_{\mathrm{Iso}(\mathcal{C})} \mathrm{Iso}(\mathcal{A}') . $$
If $\mathcal{A}' = \mathcal{C}$ and $P' = 1$ is the identity map, then the canonical functor $\mathcal{A} \times_{\mathcal{C}} \mathcal{C} \to \mathcal{A}$ is an isomorphism, which can be considered as an identification $\mathcal{A} \times_{\mathcal{C}} \mathcal{C} \equiv \mathcal{A}$.

For a commutative diagram of functors

$$ (10.5) \qquad \begin{array}{ccc} \mathcal{A} & \xrightarrow{P} & \mathcal{C} \leftarrow\limits^{P'} \mathcal{A}' \\ \bigdownarrow_F & & \bigdownarrow_H \\ \mathcal{B} & \xrightarrow{Q} & \mathcal{D} \leftarrow\limits^{Q'} \mathcal{B}' \end{array} $$

the functor $F \times F'$ restricts to a functor $F \times_G F' : \mathcal{A} \times_{\mathcal{C}} \mathcal{A}' \to \mathcal{B} \times_{\mathcal{D}} \mathcal{B}'$ called the *fibered product* of $F$ and $F'$ over $H$; it may be denoted by $F \times_{\mathcal{C}} F'$ if $\mathcal{C} = \mathcal{D}$ and $H$ is the identity functor. If $\mathcal{A} = \mathcal{B}$, $\mathcal{A}' = \mathcal{B}'$, $F = \mathrm{id}_{\mathcal{A}}$, $F' = \mathrm{id}_{\mathcal{A}'}$ and $H$ is injective, then

$$ (10.6) \qquad \mathcal{A} \times_{\mathcal{C}} \mathcal{A}' = \mathcal{A} \times_{\mathcal{D}} \mathcal{A}' $$

and $\mathrm{id}_{\mathcal{A}} \times_H \mathrm{id}_{\mathcal{A}'} : \mathcal{A} \times_{\mathcal{C}} \mathcal{A}' \to \mathcal{A} \times_{\mathcal{D}} \mathcal{A}'$ is the identity functor.

For the sake of simplicity, let $\mathcal{A}'' = \mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$, $\mathcal{B}'' = \mathcal{B} \times_{\mathcal{D}} \mathcal{B}'$ and $F'' = F \times_G F'$. Consider the lifts $\tilde{\mathcal{B}}, \tilde{\mathcal{B}}', \tilde{\mathcal{B}}''$ and $\tilde{\mathcal{D}}$ defined by $F$, $F'$, $F''$ and $G$. The diagram

$$ \tilde{\mathcal{B}} \xrightarrow{\tilde{\mathcal{Q}}} \tilde{\mathcal{D}} \leftarrow\limits^{\tilde{\mathcal{Q}}'} \tilde{\mathcal{B}}' $$

defines the fibered product $\tilde{\mathcal{B}} \times_{\tilde{\mathcal{D}}} \tilde{\mathcal{B}}'$. Observe that

$$ (\tilde{\mathcal{B}}'')_0 = \mathcal{A}_0'' = \{(M, M') \in \mathcal{A}_0 \times \mathcal{A}'_0 \mid P_M = P'_{M'}\} = (\tilde{\mathcal{B}} \times_{\tilde{\mathcal{D}}} \tilde{\mathcal{B}}')_0 $$

because $\tilde{\mathcal{B}}_0 = \mathcal{A}_0$, $(\tilde{\mathcal{B}}')_0 = \mathcal{A}'_0$, $\tilde{\mathcal{Q}}_0 = P_0$ and $(\tilde{\mathcal{Q}}')_0 = P'_0$. Moreover, for objects $(M, M')$ and $(N, N')$ of $\tilde{\mathcal{B}}''$, we have

$$ \begin{align*} \tilde{\mathcal{B}}''(M, M'; N, N') &\equiv \mathcal{B}''(F_M, F'_{M'}; F_N, F_{N'}) \\ &= \{(f, f') \in \mathcal{B}(F_M, F_N) \times \mathcal{B}'(F'_{M'}, F'_{N'}) \mid Q(f) = Q'(f')\} \\ &= \{(f, f') \in \tilde{\mathcal{B}}(M, N) \times \tilde{\mathcal{B}}'(M', N') \mid \tilde{Q}(f) = \tilde{Q}'(f')\} \\ &= (\tilde{\mathcal{B}} \times_{\tilde{\mathcal{D}}} \tilde{\mathcal{B}}')(M, M'; N, N'). \end{align*} $$

Hence

$$ (10.7) \qquad \tilde{\mathcal{B}}'' \equiv \tilde{\mathcal{B}} \times_{\tilde{\mathcal{D}}} \tilde{\mathcal{B}}'. $$

An *involutive category* is a category $\mathcal{B}$ endowed with a contravariant functor $I : \mathcal{B} \to \mathcal{B}$, called an *involution*, such that $I_0 = \mathrm{id}_{\mathcal{B}_0}$ and $I^2 = \mathrm{id}_{\mathcal{B}}$. For $f \in \mathcal{B}_1$, the notation $\tilde{f} = I(f)$ will be used without further reference to $I$, as well as the notation $\bar{S} = \{\tilde{f} \mid f \in S\}$ for any $S \subset \mathcal{B}_1$. A subcategory of $\mathcal{B}$ invariant by the involution is called an *involutive subcategory*. A functor is said to be *involutive* if it is defined between involutive categories and is compatible with the involutions in the obvious sense.
**Example 10.1.** Any groupoid $\mathcal{G}$ has the involution given by $\bar{f} = f^{-1}$ for all $f \in \mathcal{G}_1$; in particular, $\text{Iso}(\mathcal{A})$ is involutive for every category $\mathcal{A}$.

**Example 10.2 (Involutive category $\mathcal{A}^{\text{in}}$ induced by any category $\mathcal{A}$).** Any category $\mathcal{A}$ induces another category $\mathcal{A}^{\text{in}}$ defined by the following conditions:

• $\mathcal{A}_0^{\text{in}} = \mathcal{A}_0$;

• $\mathcal{A}^{\text{in}}(M, N) = \mathcal{A}(M, N) \times \mathcal{A}(N, M)$ for $M, N \in \mathcal{A}_0$; and

• $(f', g')(f, g) = (f'f, gg')$ for consecutive morphisms $(f, g)$ and $(f', g')$ of $\mathcal{A}^{\text{in}}$.

In particular,

$$ \text{Iso}_{\mathcal{A}^{\text{in}}}(M, N) = \text{Iso}_{\mathcal{A}}(M, N) \times \text{Iso}_{\mathcal{A}}(N, M). $$

Notice that $\mathcal{A}^{\text{in}} \equiv \Delta^*(\mathcal{A} \times \mathcal{A}^{\text{op}})$, where $\Delta : \mathcal{A}_0 \to \mathcal{A}_0 \times \mathcal{A}_0^{\text{op}} = \mathcal{A}_0 \times \mathcal{A}_0$ is the diagonal map. Since $\Delta$ is injective, $\tilde{\Delta}$ realizes $\mathcal{A}^{\text{in}}$ as subcategory of $\mathcal{A} \times \mathcal{A}^{\text{op}}$.

Endow $\mathcal{A}^{\text{in}}$ with the involution defined by $(\overline{f}, g) = (g, f)$ for $(f, g) \in \mathcal{A}^{\text{in}}$. Any functor $F : \mathcal{A} \to \mathcal{B}$ induces an involutive functor $F^{\text{in}} : \mathcal{A}^{\text{in}} \to \mathcal{B}^{\text{in}}$ given by $F_0^{\text{in}} = F_0$ and $F^{\text{in}}(f, g) = (F(f), F(g))$ for $(f, g) \in \mathcal{A}_1^{\text{in}}$.

Any functor $F : \mathcal{A} \to \mathcal{B}$ induces an involutive functor $F^{\text{iso/in}} : \text{Iso}(\mathcal{A}) \to \mathcal{B}^{\text{in}}$; it is defined by $F_0^{\text{iso/in}} = F_0$ and $F^{\text{iso/in}}(f) = (F(f), F(f^{-1}))$ for $f \in \text{Iso}(\mathcal{A})_1$. This can be generalized as follows.

If $\mathcal{A}$ is itself involutive, then it can be canonically realized as an involutive subcategory of $\mathcal{A}^{\text{in}}$ via the injective functor $\mathcal{A} \to \mathcal{A}^{\text{in}}$, $f \mapsto (f, \bar{f})$. Thus, in this case, for any functor $F : \mathcal{A} \to \mathcal{B}$, the restriction of $F^{\text{in}}$ to $\mathcal{A} \subset \mathcal{A}^{\text{in}}$ is an involutive functor $\mathcal{A} \to \mathcal{B}^{\text{in}}$.

**Example 10.3 (Fibered product of involutions).** In (10.2), suppose that $P$ and $P'$ are involutive. Then $\mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$ has an induced involution given by $(\overline{f}, f') = (\overline{f}, \overline{f}')$, which will be called the *fibered product* of the involutions of $\mathcal{A}$ and $\mathcal{A}'$. Moreover the canonical functors of $\mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$ to $\mathcal{A}$ and $\mathcal{A}'$ are involutive. In (10.5), if $F, F', P, P', Q$ and $Q'$ are involutive, then $F \times_G F'$ is also involutive with the induced involutions of $\mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$ and $\mathcal{B} \times_D \mathcal{B}'$.

In the case of the fibered product $\mathcal{A}^{\text{in}} \times_{\mathcal{C}^{\text{in}}} \mathcal{A}'^{\text{in}}$ defined by the diagram

$$ \mathcal{A}^{\text{in}} \xrightarrow{P^{\text{in}}} \mathcal{C}^{\text{in}} \leftarrow_{P'^{\text{in}}} \mathcal{A}'^{\text{in}} $$

induced by (10.2), there is a canonical identity

$$ (10.8) \qquad (\mathcal{A} \times_{\mathcal{C}} \mathcal{A}')^{\text{in}} \equiv \mathcal{A}^{\text{in}} \times_{\mathcal{C}^{\text{in}}} \mathcal{A}'^{\text{in}}, \qquad ((f, f'), (g, g')) \equiv ((f, g), (f', g')) $$

so that the canonical involution of $(\mathcal{A} \times_{\mathcal{C}} \mathcal{A}')^{\text{in}}$ can be identified to the fibered product of the canonical involutions of $\mathcal{A}^{\text{in}}$ and $\mathcal{A}'^{\text{in}}$.

**Example 10.4.** For any category $\mathcal{A}$, let $\mathcal{P}_{\mathcal{A}}$ be the category such that $\mathcal{P}_{\mathcal{A},0} = \mathcal{A}_0$, $\mathcal{P}_{\mathcal{A}}(L, M)$ is the set of parts of $\mathcal{A}(L, M)$ for all $L, M \in \mathcal{A}_0$, and the product of two consecutive morphisms $L \xrightarrow{S} M \xrightarrow{T} N$ of $\mathcal{P}_{\mathcal{A}}$ is

$$ TS = \{gf \mid f \in S, g \in T\}. $$
When $\mathcal{A}$ is a monoid, then $\mathcal{P}_\mathcal{A}$ is the monoid of parts of $\mathcal{A}$. If $\mathcal{A}$ is involutive, then $\mathcal{P}_\mathcal{A}$ has an induced involution given by $\bar{S} = \{\bar{f} \mid f \in S\}$.

A functor $F : \mathcal{A} \to \mathcal{B}$ induces a functor $\mathcal{P}_F : \mathcal{P}_\mathcal{A} \to \mathcal{P}_\mathcal{B}$ with $\mathcal{P}_{F,0} = F_0$ and $\mathcal{P}_F(S) = F(S)$ for $S \in \mathcal{P}_{\mathcal{A},1}$; the notation $F : \mathcal{P}_\mathcal{A} \to \mathcal{P}_\mathcal{B}$ may be used too for this functor. If $F$ is involutive, then $\mathcal{P}_F$ is also involutive. Moreover the commutative diagram (10.3) induces a commutative diagram

$$ (10.9) \qquad \begin{tikzcd} \mathcal{P}_{\mathcal{A} \times_c \mathcal{A}'} & \xrightarrow{\pi} \mathcal{P}_{\mathcal{A}} \\ \mathcal{P}_{\mathcal{A}'} & \xleftarrow{\pi'} \mathcal{P}_{\mathcal{B}} \\ \mathcal{P}_{\mathcal{A}'} & \xrightarrow{P'} \mathcal{P}_{\mathcal{C}} \end{tikzcd} $$

Recall that any set $X$ endowed with a reflexive and transitive relation $\mathcal{R}$ can be considered as a category $\mathcal{A}$ with $\mathcal{A}_0 = X$ and $\mathcal{A}_1 = \mathcal{R} \subset X \times X$, and whose source and target maps are the restrictions of the factor projections $X \times X \to X$. The operation of $\mathcal{A}$ is determined by the above conditions, and is given by $(x, y)(y, z) = (x, z)$. For instance, we can consider the category $\mathcal{A}$ when $\mathcal{R}$ in an equivalence relation, an order relation or a direction. Observe that $\mathcal{A}$ is a groupoid just when $\mathcal{R}$ is an equivalence relation. Let $Y$ be another set endowed with a reflexive and transitive relation $\mathcal{Q}$, and let $\mathcal{B}$ be the corresponding category. A map $f : X \to Y$ is called relation preserving when $(f \times f)(\mathcal{R}) \subset \mathcal{Q}$. In this case, $f \times f$ defines a functor $f_* : \mathcal{A} \to \mathcal{B}$. Indeed any functor $F : \mathcal{A} \to \mathcal{B}$ is given in this way: the map $F_0 : X \to Y$ is relation preserving and induces $F$.

Let $\mathcal{J}$ be the category defined by a directed set. Since any functor between directed sets is determined by a direction preserving map, the term cofinal makes sense for such functor. A functor is said to be directed when its source category is a directed set. A directed sub-functor of a directed functor is said to be cofinal when it is defined by a cofinal reparametrization functor.

Any directed functor is commutative. The following kind of “weak direction”, without yielding commutativity, was also introduced [?, page 207]. A category $\mathcal{J}$ is said to be weakly filtered if, for all $i, j \in \mathcal{J}_0$, there are morphisms $i \to k$ and $j \to k$ of $\mathcal{J}$ for some $k \in \mathcal{J}_0$; if moreover, for any pair of parallel morphisms $f, g : i \to j$ in $\mathcal{J}$, there is some morphism $h : j \to k$ in $\mathcal{J}$ such that $hf = hg$, then $\mathcal{J}$ is called a filtered category. A functor is said to be (weakly) filtered if its source category is (weakly) filtered. Any weakly filtered category $\mathcal{J}$ defines a direction “$\leq$” on $\mathcal{J}_0$ by setting $i \leq j$ if $\mathcal{J}(i, j) \neq \emptyset$. Thus it makes sense to study the existence of the limit as $i \to \infty$ in $\mathcal{J}_0$ of a map of $\mathcal{J}_0$ to a topological space. For a functor $F$ between weakly filtered categories, its restriction $F_0$ preserves the corresponding directions. In this case, $F$ is said to be cofinal if $F_0$ is cofinal. In particular, a weakly filtered subcategory of a weakly filtered category is called cofinal when the corresponding inclusion functor is cofinal. Moreover a weakly filtered sub-functor of a weakly filtered functor is called cofinal when it can be defined by a cofinal reparametrization functor. It is easy to prove that the product of (weakly) filtered categories is a (weakly) filtered category, and the corresponding factor projections are cofinal (weakly) filtered functors. It follows that any pair of (weakly) filtered functors, $\mathcal{J} \to \mathcal{A}$ and $\mathcal{J} \to \mathcal{B}$, have corresponding cofinal (weakly)
filtered sub-functors with the same source category: consider the factor projections of $\mathcal{J} \times \mathcal{J}$ to $\mathcal{J}$ and $\mathcal{J}$ as reparametrization functors.

Recall that a *natural transformation* between two parallel functors, $F, F' : \mathcal{J} \to \mathcal{A}$, is a map $t : \mathcal{J}_0 \to \mathcal{A}_1$, $i \mapsto t_i$, such that $t_i \in \mathcal{A}(F_i, F'_i)$ for all $i \in \mathcal{J}_0$, and the diagram

$$ (10.10) \qquad \begin{tikzcd} F_i \arrow[r, "t_i"] & F'_i \\ F(f) \arrow[r, "t_j"] \arrow[d] & F'(f) \\ F_j \arrow[r, "t_j"] & F'_j \end{tikzcd} $$

is commutative for any morphism $f : i \to j$ of $\mathcal{J}$. The notation $t : F \to F'$ will be used for such a natural transformation. A natural transformation is said to be *small* (respectively, *weakly filtered*) when it is defined between small (respectively, weakly filtered) functors. With the operation induced by the operation of $\mathcal{A}$, the natural transformations between functors $\mathcal{J} \to \mathcal{A}$ form a category denoted by $\mathcal{A}^{\mathcal{J}}$, whose objects are small if $\mathcal{J}$ is small. When $\mathcal{A}$ is an involutive category, for each natural transformation $t : F \to F'$, there is a natural transformation $\bar{t} : F' \to F$ defined by $\bar{t}_i = \bar{t}_{i'}$; this defines an involution of $\mathcal{A}^{\mathcal{J}}$.

Let $F, F' : \mathcal{J} \to \mathcal{A}$ be functors and $t : F \to F'$ a natural transformation. For any functor $\mathbf{a} : \mathcal{J} \to \mathcal{J}$, consider the corresponding sub-functors $A = F \circ \mathbf{a}$ of $F$ and $A' = F' \circ \mathbf{a}$ of $F'$. We get a natural transformation $u = t\mathbf{a} : A \to A'$ defined by $u_j = t_{\mathbf{a}_j}$, which is called the *sub-natural transformation* of $t$ defined by the reparametrization functor $\mathbf{a}$. When $t$ and $u$ are weakly filtered, it is said that the sub-natural transformation $u$ is *cofinal* if it can be defined by a cofinal reparametrization functor. On the other hand, any functor $G : \mathcal{A} \to \mathcal{B}$ defines a natural transformation $v = Gt : GF \to GF'$ given by $v_i = G(t_i)$.

The following concept will be also used. A *source-target-preserving map* (or simply an *stp-map*) is a map between categories, $F : \mathcal{J} \to \mathcal{A}$, satisfying $F(\mathcal{J}_0) \subset \mathcal{A}_0$ and $F(\mathcal{J}_2) \subset \mathcal{A}_2$. Here, the compatibility of functors with the operations is dropped down, but keeping their compatibility with units, and with the source and target maps. Thus the image of a diagram with arrows by an stp-map is a diagram of the same type, but only functors preserve commutativity of diagrams. Like in the case of functors, the restriction $\mathcal{J}_0 \to \mathcal{A}_0$ of such an stp-map $F : \mathcal{J} \to \mathcal{A}$ is denoted by $F_0$, and $F$ induces maps $F_n : \mathcal{J}_n \to \mathcal{A}_n$ for all $n \in \mathbb{N} \cup \{\infty\}$. A map between categories, $A : \mathcal{J} \to \mathcal{A}$, is called a *sub-stp-map* of $F$ if $A = F\mathbf{a}$ for some stp-map $\mathbf{a} : \mathcal{J} \to \mathcal{J}$, which is called an *stp-reparametrization* defining $A$. If $\mathcal{J}$ is small (respectively, weakly filtered), then the stp-map $F$ is called *small* (respectively, weakly filtered). If $\mathcal{J}$ and $\mathcal{A}$ are weakly filtered, then $F_0$ preserves the corresponding directions, and $F$ is called *cofinal* if $F_0$ is cofinal. If $F$ is weakly filtered, a sub-stp-map of $F$ is called *cofinal* if it can be defined by a cofinal stp-reparametrization. Any pair of stp-maps, $\mathcal{J} \to \mathcal{A}$ and $\mathcal{J} \to \mathcal{B}$, have corresponding cofinal weakly filtered sub-stp-maps with the same source category defined by taking the factor projections of $\mathcal{J} \times \mathcal{J}$ to $\mathcal{J}$ and $\mathcal{J}$ as stp-reparametrizations. Like in the case of functors, we can define *commutative stp-maps*, or *involutive stp-maps* between involutive categories; notice that commutative stp-maps are functors. The compositions of consecutive stp-maps are stp-maps; thus stp-maps form a category whose objects may not be small. *Contravariant stp-maps* can be also defined like in the case of functors.
For two parallel stp-maps, $F, F' : \mathcal{J} \to \mathcal{A}$, a transformation $t : F \to F'$ is a map $t : \mathcal{J}_0 \to \mathcal{A}$ such that $t_i \in \mathcal{A}(F_i, F'_i)$ for all $i \in \mathcal{J}_0$ (the commutativity of (10.10) is not required). A transformation is said to be *small* (respectively, *weakly filtered*) when it is defined between small (respectively, weakly filtered) functors. Like in the case of natural transformations, the operation of $\mathcal{A}$ induces an operation on the family of transformations between stp-maps $\mathcal{J} \to \mathcal{A}$, defining a category which will be denoted by $\mathcal{A}_{\text{stp}}^{\mathcal{J}}$, and any involution of $\mathcal{A}$ also induces an involution of $\mathcal{A}_{\text{stp}}^{\mathcal{J}}$. Observe that $\mathcal{A}^{\mathcal{J}}$ is a subcategory of $\mathcal{A}_{\text{stp}}^{\mathcal{J}}$.

Let $F, F' : \mathcal{J} \to \mathcal{A}$ be parallel stp-maps and $t : F \to F'$ a transformation. Like in the case of natural transformations, any stp-map $\mathbf{a} : \mathcal{J} \to \mathcal{J}$ defines a transformation $u = \mathbf{ta} : F\mathbf{a} \to F'\mathbf{a}'$, which is called the *sub-transformation* of $t$ defined by the stp-reparametrization $\mathbf{a}$. When $t$ and $u$ are weakly filtered, the sub-transformation $u$ is called *cofinal* if it can be defined by a cofinal weakly filtered stp-reparemetrization. Any stp-map $G : \mathcal{A} \to \mathcal{B}$ also defines a transformation $Gt : GF \to GF'$ like in the case of natural transformations.

## 10.2 Underlying functors

The following is an abstraction of what is called a *forgetful* or *underlying functor* [?, page 14]. A *forgetful* or *underlying functor* (or simply a *u-functor*) is a functor $|\cdot| : \mathcal{A} \to \mathcal{B}$ between *U*-categories satisfying the following conditions:

(u1) The fibers of $|\cdot|$ in $\mathcal{A}_0$ are small.

(u2) For all $M, N \in \mathcal{A}_0$, the restriction $|\cdot| : \mathcal{A}(M, N) \to \mathcal{B}(|M|, |N|)$ is injective.

The image of objects and morphisms by $|\cdot|$ are called *underlying objects* and *underlying morphisms*. A morphism of $\mathcal{A}$ may be identified to its underlying morphism by (u2). If $\mathcal{B}$ is some familiar category, the usual terminology and notation of $\mathcal{B}$ can be used for $\mathcal{A}$ referring to the underlying objects and morphisms. The image of a subcategory $\mathcal{A}' \subset \mathcal{A}$ by $|\cdot|$ will be denoted by $|\mathcal{A}'|$. A u-functor may be indicated by saying that $\mathcal{A}$ is a $\mathcal{B}$-category, a category over $\mathcal{B}$, a category subordinated to $\mathcal{B}$, or with the quotient notation $\mathcal{A}/\mathcal{B}$; then the symbol $|\cdot|$ (or $|\cdot|_{\mathcal{B}}$) is used to denote underlying objects and morphisms without previous reference to it. The notation $\mathcal{A}/\mathcal{B}/\mathcal{C}/\dots$ is used for a sequence of consecutive u-functors; in this case, the term *level of subordination* may be used with the obvious sense. The composite of u-functors is a u-functor. For any subcategory $\mathcal{A}' \subset \mathcal{A}$, the restriction $|\cdot| : \mathcal{A}' \to \mathcal{B}$ is a u-functor; in this case, $\mathcal{A}'$ is called a $\mathcal{B}$-subcategory or a *subcategory over* $\mathcal{B}$.

A u-functor $\mathcal{A}/\mathcal{B}$ is said to be *full* if its restriction $\mathcal{A}(M, N) \to \mathcal{B}(|M|, |N|)$ is onto (and thus bijective) for all $M, N \in \mathcal{A}_0$. It is said that $\mathcal{A}/\mathcal{B}$ is *complete* if, for all $X \in \mathcal{B}_0$, $M \in \mathcal{A}_0$ and $f \in \text{Iso}_{\mathcal{B}}(|M|, X)$, there is some $N \in \mathcal{A}_0$ and some $\phi \in \text{Iso}_{\mathcal{A}}(M, N)$ such that $|N| = X$ and $|\phi| = f$. It is said that $\mathcal{A}/\mathcal{B}$ is *faithful* when an isomorphism of $\mathcal{A}$ is an identity if its underlying isomorphism is an identity of $\mathcal{B}$. Finally, $\mathcal{A}/\mathcal{B}$ is said to be *strongly faithful* when a morphism of $\mathcal{A}$ is an identity if its underlying morphism is an identity of $\mathcal{B}$. Observe that, for u-functors $\mathcal{A}/\mathcal{B}/\mathcal{C}$, if the composite $\mathcal{A}/\mathcal{C}$ is (strongly) faithful, then so is $\mathcal{A}/\mathcal{B}$.
**Proposition 10.5.** If Iso($\mathcal{A}$) is a faithful and complete $\mathcal{B}$-subcategory of $\mathcal{A}$, then $\mathcal{A}$ is a $\mathcal{B}$-subcategory of some faithful and complete $\mathcal{B}$-category $\mathcal{A}'$ such that any object of $\mathcal{A}'$ is isomorphic to some object of $\mathcal{A}$.

*Proof.* Consider triples $(M, X, f)$, where $M \in \mathcal{A}_0$, $X \in \mathcal{B}_0$ and $f \in \text{Iso}_\mathcal{B}(|M|, X)$. Two such triples, $(M, X, f)$ and $(N, Y, g)$, are declared to be equivalent if $X = Y$ and there is some $\phi \in \text{Iso}_\mathcal{A}(M, N)$ such that $g|\phi|f^{-1} = \text{id}_X$. Let $[M, X, f]$ denote the equivalence class of each triple $(M, X, f)$. The set of these classes may not be contained to $\mathfrak{A}$, but it is bijective to some subset of $\mathfrak{A}$ by the axiom of choice; so, after reparametrization, we can assume that it is a subset of $\mathfrak{A}$. A morphism between two such classes, $[M, X, f] \to [N, Y, g]$, is defined to be a morphism $X \to Y$ in $\mathcal{B}$. These morphisms between classes of triples form a category $\mathcal{A}'$ with the operation induced by the operation of $\mathcal{B}$. Let $|\cdot|: \mathcal{A}' \to \mathcal{B}$ be the functor defined by $|[M, X, f]|' = X$ and $|h|' = h$ for all $[M, X, f] \in \mathcal{A}'_0$ and all $h \in \mathcal{A}'_1$. This functor clearly satisfies (u2).

For each $X \in \mathcal{B}_0$, let $S(X)$ and $S'(X)$ be the fibers of $|\cdot|$ and $|\cdot|'$ over $X$. If $S'(X) \neq \emptyset$, then we can fix some $X_0 \in |\mathcal{A}|_0$ and some $f_0 \in \text{Iso}_\mathcal{B}(X_0, X)$. For a triple of the form $(M, X, f)$, the product $f^{-1}f_0: X_0 \to |M|$ is an isomorphism of $\mathcal{B}$. Since $\text{Iso}(\mathcal{A})$ is a faithful and complete $\mathcal{B}$-subcategory of $\mathcal{A}$, there is a unique $K \in S(X_0)$ and a unique $\sigma \in \text{Iso}_\mathcal{A}(K, M)$ with $|\sigma| = f^{-1}f_0$. If $(M, X, f)$ is equivalent to another triple of the form $(N, X, g)$, then there is some $\phi \in \text{Iso}_\mathcal{A}(M, N)$ such that $g|\phi|f^{-1} = \text{id}_X$, yielding $|\phi\sigma| = |\phi|f^{-1}f_0 = g^{-1}f_0$. So the mapping $(M, X, f) \mapsto K$ induces a map $f_0^* : S'(X) \to S(X_0)$, which is easily checked to be a bijection by using again the assumption that $\text{Iso}(\mathcal{A})$ is a faithful and complete $\mathcal{B}$-subcategory of $\mathcal{A}$. Therefore, by reparametrization, we can assume that each $S'(X) \in \mathfrak{A}$; so $|\cdot|$ satisfies (u1) too, and thus it is a $\mathfrak{U}$-functor.

A functor $F: \mathcal{A} \to \mathcal{A}'$ is defined by $F_M = [M, |M|, \text{id}_M]$ and $F(\phi) = |\phi|$ for $M \in \mathcal{A}_0$ and $\phi \in \mathcal{A}_1$, which is injective and compatible with the $\mathfrak{U}$-functors $|\cdot|$ and $|\cdot|'$. Hence we can suppose that $\mathcal{A}$ is a $\mathcal{B}$-subcategory of $\mathcal{A}'$.

Finally, for every $[M, X, f] \in \mathcal{A}'_0$, the isomorphism $f$ of $\mathcal{B}$ is an isomorphism of $\mathcal{A}'$ between $[M, |M|, \text{id}_M] \equiv M$ and $[M, X, f]$. $\square$

A double functor of a $\mathfrak{U}$-functor $\mathcal{A}/\mathcal{B}$ to a $\mathfrak{U}$-functor $\mathcal{C}/\mathcal{D}$ is a pair of functors, $F: \mathcal{A} \to \mathcal{C}$ and $G: \mathcal{B} \to \mathcal{D}$, such that the diagram

$$ \begin{array}{ccc} \mathcal{A} & \xrightarrow{F} & \mathcal{C} \\ |\cdot|_{\mathcal{B}} \downarrow & & |\cdot|_{\mathcal{D}} \downarrow \\ \mathcal{B} & \xrightarrow{G} & \mathcal{D} \end{array} $$

is commutative. It may be also said that $F$ is a $G$-functor, a functor over $G$ or a functor subordinated to $G$, and the notation $F/G: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{D}$ is used. If $\mathcal{B} = \mathcal{D}$ and $G = \text{id}_\mathcal{B}$, then $F$ is called a $\mathcal{B}$-functor, a functor over $\mathcal{B}$ or a functor subordinated to $\mathcal{B}$, and the notation $F/\mathcal{B}: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{B}$ is preferred. If $F$ and $G$ are $\mathfrak{U}$-functors too, then $F/G$ is called a double $\mathfrak{U}$-functor; it may be also said that $\mathcal{A}/\mathcal{B}$ is a $\mathcal{C}/\mathcal{D}$-functor, a $\mathfrak{U}$-functor over $\mathcal{C}/\mathcal{D}$ or a $\mathfrak{U}$-functor subordinated to $\mathcal{C}/\mathcal{D}$, and the notation $(\mathcal{A}/\mathcal{B})/(\mathcal{C}/\mathcal{D})$ is used; in this case, the double
u-functor $(\mathcal{A}/\mathcal{C})/(\mathcal{B}/\mathcal{D})$ can be considered as well. If $\mathcal{C} = \mathcal{D}$ and $\mathcal{C}/\mathcal{D}$ is the identity functor, then $\mathcal{A}/\mathcal{B}$ is called a $\mathcal{C}$-u-functor, a $\mathcal{u}$-functor over $\mathcal{C}$ or a $\mathcal{u}$-functor subordinated to $\mathcal{C}$, and the notation $(\mathcal{A}/\mathcal{B})/\mathcal{C}$ is preferred. A sequence of consecutive double u-functors may be denoted by

$$ (\mathcal{A}/\mathcal{B})/(\mathcal{A}'/\mathcal{B}')/(\mathcal{A}''/\mathcal{B}'')/\cdots; $$

here, if e.g. $\mathcal{A}' = \mathcal{B}'$ and $\mathcal{A}'/\mathcal{B}'$ is the identity functor, then the notation

$$ (\mathcal{A}/\mathcal{B})/\mathcal{A}'/(\mathcal{A}''/\mathcal{B}'')/\cdots $$

is better. For double u-functors $(\mathcal{A}/\mathcal{B})/(\mathcal{E}/\mathcal{F})$ and $(\mathcal{C}/\mathcal{D})/(\mathcal{E}/\mathcal{F})$, and a double functor $F/G: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{D}$, if $F$ and $G$ are functors over $\mathcal{E}$ and $\mathcal{F}$, respectively, then $F/G$ is called an $\mathcal{E}/\mathcal{F}$-double functor, a double functor over $\mathcal{E}/\mathcal{F}$ or a double functor subordinated to $\mathcal{E}/\mathcal{F}$; if moreover $\mathcal{E} = \mathcal{F}$ and $\mathcal{E}/\mathcal{F}$ is the identity functor, then $\mathcal{E}$ is used instead of $\mathcal{E}/\mathcal{F}$ in the above terminology.

Double functors between u-functors, with the operation of double composite defined in the obvious way, form a category whose objects may not be in $\mathfrak{U}$.

A double functor $F/G: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{D}$ is called a semi-pull-back if, for all $M, N \in \mathcal{A}_0$, the diagram

$$ \begin{tikzcd} \mathcal{A}(M, N) \arrow[r, "F"] & \mathcal{C}(F_M, F_N) \arrow[d, "|_{\mathcal{B}}"] \arrow[d, "|_{\mathcal{D}}"] & \mathcal{B}(|M|_{\mathcal{B}}, |N|_{\mathcal{B}}) \arrow[r, "G"] & \mathcal{D}(G_{|M|_{\mathcal{B}}}, G_{|N|_{\mathcal{B}}}) \end{tikzcd} $$

is a pull-back; i.e., for $f \in \mathcal{B}(|M|_{\mathcal{B}}, |N|_{\mathcal{B}})$ and $f' \in \mathcal{C}(F_M, F_N)$, if $G(f) = |f'|_{\mathcal{D}}$, then there is a unique $\phi \in \mathcal{A}(M, N)$ such that $|\phi|_{\mathcal{B}} = f$ and $F(\phi) = f'$.

**Proposition 10.6.** Let $\mathcal{A}/\mathcal{B}$ and $\mathcal{C}/\mathcal{B}$ be u-functors, and $F: \mathcal{A} \to \mathcal{B}$ a $\mathcal{B}$-functor. Then $F_1$ is injective if so is $F_0$.

*Proof.* Since $F_0$ is injective, it is enough to prove that the restriction of $F_1$ to $\mathcal{A}(M, N)$ is injective for $M, N \in \mathcal{A}_0$. Let $\phi, \psi \in \mathcal{A}(M, N)$ such that $F(\phi) = F(\psi)$. Then

$$ |\phi| = |F(\phi)| = |F(\psi)| = |\psi| $$

because $F$ is $\mathcal{B}$-functor, yielding $\phi = \psi$ by (u2). $\square$

**Example 10.7** (Inclusion functors and lifts). The inclusion functor of any subcategory, $\mathcal{A} \hookrightarrow \mathcal{B}$, is a strongly faithful u-functor, which is full if and only if $\mathcal{A}$ is a full subcategory of $\mathcal{B}$. Vice versa, for any u-functor $|\cdot|: \mathcal{A} \to \mathcal{B}$, the first functor of the corresponding factorization (10.1) is injective, and thus realizes $\mathcal{A}$ as a subcategory of the lift $\tilde{\mathcal{B}}$ of $\mathcal{B}$ by $|\cdot|$. We have $\tilde{\mathcal{B}}_0 = \mathcal{A}_0$, $\tilde{\mathcal{B}}(M, N) = \mathcal{B}(|M|, |N|)$ for all $M, N \in \mathcal{A}_0$, and the injection of $\mathcal{A}$ into $\tilde{\mathcal{B}}$ essentially identifies each morphism of $\mathcal{A}$ with its underlying morphism of $\mathcal{B}$. Of course, we may lose information by considering $\mathcal{A}/\tilde{\mathcal{B}}$ instead of $\mathcal{A}/\mathcal{B}$. For a double u-functor $F/G: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{D}$, the lift $\tilde{G}: \tilde{\mathcal{B}} \to \tilde{\mathcal{D}}$ of $G$ is a u-functor.
**Example 10.8.** A particular case of Example 10.7 is the involutive u-functor given by the canonical injection any involutive category $\mathcal{A}$ into $\mathcal{A}^{\text{in}}$.

**Example 10.9 (Restrictions of u-functors).** Let $|\cdot| : \mathcal{A} \to \mathcal{B}$ be a u-functor, and $\mathcal{C} \subset \mathcal{A}$ and $\mathcal{D} \subset \mathcal{B}$ subcategories such that $|\mathcal{C}| \subset \mathcal{D}$. Then the restriction $|\cdot| : \mathcal{C} \to \mathcal{D}$ is a u-functor, obtaining a double u-functor $(\mathcal{C}/\mathcal{D})/(\mathcal{A}/\mathcal{B})$, which is a semi-pull-back if $\mathcal{C}$ is full in $\mathcal{A}$. The corresponding u-functor between lifts, $\tilde{\mathcal{D}}/\tilde{\mathcal{B}}$, is an inclusion map of a subcategory.

**Example 10.10.** Consecutive u-functors $\mathcal{A}/\mathcal{B}/\mathcal{C}$ define a double u-functor $(\mathcal{A}/\mathcal{B})/(\mathcal{A}/\mathcal{C})$. Again, the corresponding u-functor $\tilde{\mathcal{B}}/\tilde{\mathcal{C}}$ is an inclusion map of a subcategory.

**Example 10.11 (Fibered product of u-functors).** Given u-functors $\mathcal{A}/\mathcal{C}$ and $\mathcal{A}'/\mathcal{C}$, the canonical functors of $\mathcal{A} \times_{\mathcal{C}} \mathcal{A}'$ to $\mathcal{A}$, $\mathcal{A}'$ and $\mathcal{C}$ are u-functors. The product of two u-functors is easily seen to be a u-functor. So the fibered product of two u-functors is a u-functor because it is a restriction of their product.

**Example 10.12 (The involutive u-functor $\mathcal{A}^{\text{in}}/\mathcal{B}^{\text{in}}$ induced by a u-functor $\mathcal{A}/\mathcal{B}$).** For any u-functor $|\cdot| : \mathcal{A} \to \mathcal{B}$, the induced involutive functor $|\cdot|^{\text{in}} : \mathcal{A}^{\text{in}} \to \mathcal{B}^{\text{in}}$ is a u-functor. Moreover a double functor $F/G : \mathcal{A}/\mathcal{B} \to \mathcal{A}'/\mathcal{B}'$ induces the double functor $F^{\text{in}}/G^{\text{in}} : \mathcal{A}^{\text{in}}/\mathcal{B}^{\text{in}} \to \mathcal{A}'^{\text{in}}/\mathcal{B}'^{\text{in}}$.

**Example 10.13 (The involutive u-functor $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ induced by a u-functor $\mathcal{A}/\mathcal{B}$).** With the notation of Example 10.2, any u-functor $\mathcal{A}/\mathcal{B}$ induces an involutive u-functor $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$. In particular, we get an involutive u-functor $\text{Iso}(\mathcal{A})/\mathcal{A}^{\text{in}}$ for any category $\mathcal{A}$. Let $\tilde{\mathcal{B}}$ and $\tilde{\mathcal{B}}^{\text{in}}$ denote the lifts of $\mathcal{B}$ and $\mathcal{B}^{\text{in}}$ by $\mathcal{A}/\mathcal{B}$ and $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$, respectively. Then $\tilde{\mathcal{B}}^{\text{in}} = \tilde{\mathcal{B}}^{\text{in}}$. A double functor $F/G : \mathcal{A}/\mathcal{B} \to \mathcal{A}'/\mathcal{B}'$ induces the double functor $\text{Iso}(F)/G^{\text{in}} : \text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}} \to \text{Iso}(\mathcal{A}')/\mathcal{B}'^{\text{in}}$. As a particular case of (10.6), we get

$$ \text{Iso}(\mathcal{A}) \times_{\text{Iso}(\mathcal{C})} \text{Iso}(\mathcal{A}') = \text{Iso}(\mathcal{A}) \times_{\mathcal{C}^{\text{in}}} \text{Iso}(\mathcal{A}') . $$

**Example 10.14.** Let $\mathcal{A}$ be the groupoid defined by an equivalence relation $\mathcal{R}$ on some set $X$. Then the trivial groupoid homomorphism of $\mathcal{A}$ to the trivial group is a full and complete u-functor; moreover this u-functor is faithful if and only if each $\mathcal{R}$-equivalence class consists of one element. More generally, let $Y$ be another set with another equivalence relation, and $\mathcal{B}$ the corresponding groupoid. The groupoid homomorphism $f_* : \mathcal{A} \to \mathcal{B}$ induced by any relation preserving map $f : X \to Y$ is a full u-functor, which is faithful if only if $f$ has injective restrictions between the equivalence classes of $X$ and $Y$, and is complete if and only if those restrictions are onto.

**Example 10.15.** Let $\mathcal{A}$ be the groupoid defined by any right action of a group $G$ on a set $X$. Recall that $\mathcal{A}_0 = X$, $\mathcal{A}_1 = X \times G$, the source map $\mathcal{A}_1 \to \mathcal{A}_0$ is the first factor projection, the target map $\mathcal{A}_1 \to \mathcal{A}_0$ is the given action, and the operation of $\mathcal{A}$ is defined by $(x,g)(xg,h) = (x,gh)$. Then the groupoid homomorphism $|\cdot| : \mathcal{A} \to G$, defined by $|(x,g)| = g$, is a faithful and complete u-functor, which is full if and only if the action is trivial. On the other hand, let $\mathcal{B}$ be the groupoid defined by the equivalence relation on $X$ whose equivalence classes are the orbits of the $G$-action, and let $F : \mathcal{A} \to \mathcal{B}$ be the functor defined by $F(x,g) = (x,xg)$. Then $F$ is a u-functor if and only if the action is free.
**Example 10.16.** Let Set be the category of maps between small sets; Top the category of continuous maps between small topological spaces; Met the category of isometric injections between small metric spaces; Path the category of arc-wise isometries between small length spaces [?, page 9], [?, page 22]; Man the full subcategory of Top whose objects are manifolds; Man$^{∞}$ the category of $C^{∞}$ maps between small $C^{∞}$ manifolds; Riem the category of infinitesimal isometries between small Riemannian manifolds; and Simp the category of simplicial maps between small simplicial complexes. Furthermore let PropMet $\subset$ Met and PropPath $\subset$ Path be the full subcategory whose objects have proper metrics; ConRiem $\subset$ Riem the full subcategory whose objects are connected; ComConRiem $\subset$ Riem the full subcategory whose objects are connected and complete; and ConSimp $\subset$ Simp the full subcategory whose objects are connected. The usual underlying structures give rise to canonical faithful complete u-functors

$$ \begin{array}{ll} \text{Met} / \text{Top} / \text{Set}, & \text{Riem} / \text{Man}^{\infty} / \text{Man} / \text{Top}, \\ \text{Simp} / \text{Top}, & \text{ConRiem} / \text{Path}, \\ \text{ConSimp} / \text{PropPath}, & \text{ComConRiem} / \text{PropPath}. \end{array} $$

All of these u-functors, except Top/Set and Man$^{∞}$/Man, are strongly faithful too. Canoni-
cally, Met is also a category over the category of Lipschitz maps between metric spaces, and
observe that this u-functor is not complete. Even though Path$_0 \subset$ Met$_0$, there are many
more arc-wise isometries than isometries, and thus Path is not a Met-category in a canonical
way.

Given a u-functor $\mathcal{A}/\mathcal{B}$, let $\mathcal{S}(X)$ denote its fiber over each $X \in \mathcal{B}_0$, which is small by (u1).
For each morphism $f: X \to Y$ of $\mathcal{B}$, let $\mathcal{S}(f) \subset \mathcal{S}(X) \times \mathcal{S}(Y)$ be the relation consisting of
the pairs $(M, N)$ such that there is some $\phi \in \mathcal{A}(M, N)$ with $|\phi| = f$. This defines a functor
$\mathcal{S}: \mathcal{B} \to \text{Rel}$, called the *structure functor* of $\mathcal{A}/\mathcal{B}$, where $\text{Rel}$ is the category of relations
between small sets with the operation of composition of relations: For relations $R \subset X \times Y$
and $S \subset Y \times Z$, the *composite* $S \circ R \subset X \times Z$ is the set of pairs $(x, z)$ such that $(x, y) \in R$
and $(y, z) \in S$ for some $y \in Y$; thus Set is a subcategory of $\text{Rel}$. If $\mathcal{A}/\mathcal{B}$ is strongly faithful
and complete, then $\mathcal{S}$ restricts to a functor $\text{Iso}(\mathcal{B}) \to \text{Set}$.

**Example 10.17** (U-functor defined by a species of structures and a class of morphisms). This generalizes the u-functors of Example 10.16. Let $\Sigma$ be a species of structures over sets [?] and $\sigma$ a class of morphisms of these structures (as defined by the conditions of [?, page E IV.11]). Let $\mathcal{A}$ be the category of $\sigma$-morphisms between small sets endowed with structures of the species $\Sigma$, with the operation of composition; i.e., each object of $\mathcal{A}$ is a pair $(X, \alpha)$, where $X$ is a small set and $\alpha$ a structure of the species $\Sigma$ over $X$. Then $\mathcal{A}$ is a Set-category by forgetting the structure. Moreover $\mathcal{A}/\text{Set}$ is faithful and complete since, if a $\sigma$-morphism is bijective and its inverse is a $\sigma$-morphism, then it is an isomorphism of structures of the species $\Sigma$ [?, page E IV.12]. In this case, $\mathcal{S}(X)$ can be considered as the set of structures of the species $\Sigma$ over each set $X$. Hence faithful and complete u-functors may be considered as a way to get rid of the involved definition of a species of structures. Let $\mathcal{B}$ be the Set-category defined by a second species of structures $\Theta$ over sets and a class of morphisms $\theta$ of $\Theta$. If $\Theta$
is subordinated to $\Sigma$ [?, page E IV.8] and any $\sigma$-morphism is a $\theta$-morphism (with respect to the underlying structures), then $\mathcal{A}$ becomes a faithful and complete $\mathcal{B}$-category.

**Example 10.18.** For sets $X$ and $Y$, any map of a subset of $X$ to $Y$ is called a *partial map* of $X$ to $Y$; even the empty map $\emptyset: \emptyset \to Y$ is a partial map of $X$ to $Y$. If $f$ is a partial map of $X$ to $Y$, and $g$ a partial map of $Y$ to another set $Z$, then the composite of the maps

$$f^{-1}(\operatorname{dom} g) \xrightarrow{f} \operatorname{dom} g \xrightarrow{g} Z$$

is called the *partial composition* of $f$ and $g$, and denoted by $g \circ f$. Let $\operatorname{ParSet}$ be the category with $\operatorname{ParSet}_0 = \operatorname{Set}_0$,

$$\operatorname{ParSet}(X, Y) = \{(X, Y, f) \mid f \text{ is a partial map of } X \text{ to } Y\}$$

for $X, Y \in \operatorname{Set}_0$, and the operation defined by

$$(Y, Z, g)(X, Y, f) = (X, Z, g \circ f).$$

Each morphism $(X, Y, f)$ of $\operatorname{ParSet}$ is simply denoted by $f$, but without forgetting that it determines the objects $X$ and $Y$. Similarly, we get the $\operatorname{ParSet}$-categories: $\operatorname{ParTop}$, of continuous partial maps between small topological spaces, and $\operatorname{ParMet}$, of isometric partial maps between small metric spaces. Also, let $\operatorname{ParPropMet} \subset \operatorname{ParMet}$ be the full subcategory whose objects have proper metrics. Observe that $\operatorname{Set} \subset \operatorname{ParSet}$, $\operatorname{Top} \subset \operatorname{ParTop}$, etc., and thus any category over $\operatorname{Set}$, $\operatorname{Top}$, etc. becomes a category over $\operatorname{ParSet}$, $\operatorname{ParTop}$, etc. Moreover there are canonical faithful and complete u-functors

$$\operatorname{ParMet} / \operatorname{ParTop} / \operatorname{ParSet}.$$

Let $\mathcal{A}$ be a $\operatorname{ParTop}$-category. Its full subcategory with compact objects will be denoted by $\mathcal{A}_c$. An object $M$ of $\mathcal{A}$ is called *locally aperiodic* if each $x \in M$ has a neighborhood $U_x$ such that

$$\{f \in \operatorname{Iso}_{\mathcal{A}}(M) \mid f(x) \in U_x\} = \{\operatorname{id}_M\}.$$

The full subcategory of $\mathcal{A}$ given by the locally aperiodic objects will be denoted by $\mathcal{A}_{\text{la}}$.

**Proposition 10.19.** If the objects of a $\operatorname{ParTop}$-category $\mathcal{A}$ are separable and metrizable, then there is a bijection between $\operatorname{Iso}(\mathcal{A})\backslash\mathcal{A}_0$ and some small set.

*Proof.* The hypothesis is equivalent to the existence of an embedding of each $M \in \mathcal{A}_0$ into the Hilbert cube $[0, 1]^N$ (by the Urysohn metrization theorem [?, p. 166]). Then the result follows by (u1) since the set of homeomorphism classes of subspaces of $[0, 1]^N$ is small. $\square$

If the conditions of Proposition 10.19 are satisfied, we can assume that $\operatorname{Iso}(\mathcal{A})\backslash\mathcal{A}_0$ is small after reparametrization.

Let $\mathcal{A}$ be a $\operatorname{Set}$-category. An *embedding* of $\mathcal{A}$ is an injective morphism $f: N \to M$ of $\mathcal{A}$ such that, for each $L \in \mathcal{A}_0$, any map $g: L \to N$ is a morphism of $\mathcal{A}$ if $f \circ g$ is a morphism of $\mathcal{A}$. An *identification* of $\mathcal{A}$ is a surjective morphism $f: M \to N$ of $\mathcal{A}$ such that, for each
$L \in \mathcal{A}_0$, any map $g : N \to L$ is a morphism of $\mathcal{A}$ if $g \circ f$ is a morphism of $\mathcal{A}$. The embeddings form a subcategory, and the identifications too.

A sub-object<fnref n="2" /> of an object $M \in \mathcal{A}_0$ is another object $N$ of $\mathcal{A}$ such that $|N| \subset |M|$ and the inclusion map $N \hookrightarrow M$ is an embedding of $\mathcal{A}$, called the *inclusion morphism*. Observe that, given a morphism $\phi : M \to N$ of $\mathcal{A}$, and sub-objects $A \subset M$ and $B \subset N$ such that $\phi(A) \subset B$, the restriction $\phi : A \to B$ is a morphism of $\mathcal{A}$ too. A *quotient* of an object $M$ of $\mathcal{A}$ is another object $N$ of $\mathcal{A}$ such that $|N| = |M|/\mathcal{R}$ for some equivalence relation $\mathcal{R}$ on $M$, and moreover the quotient map $M \to N$ is an identification of $\mathcal{A}$, called a *quotient morphism*; the notation $M/\mathcal{R}$ will be used for $N$, as well as $G\backslash M$ or $M/G$ when $\mathcal{R}$ is defined by the left or right action of a group $G$. A sub-object of a sub-object of $M$ is a sub-object of $M$, and a quotient of a quotient of $M$ is a quotient of $M$. If $\mathcal{A}$/Set is complete, then any embedding of $\mathcal{A}$ is the composition of an isomorphism and the inclusion morphism of some sub-object, and any identification of $\mathcal{A}$ is the composition of a quotient morphism and an isomorphism. Given any $M \in \mathcal{A}_0$, its sub-objects and quotients are determined by their underlying sets, which follows from the universal characteristic properties of embeddings and identifications. If a subset $X \subset M$ underlies a sub-object $N \subset M$, it is said that $M$ can be *restricted* to $X$; in this case, $N$ is called the *restriction* of $M$ to $X$ and may be denoted by $M|_X$, or simply by $X$.

**Example 10.20.** In Set, the embeddings and identifications are the injective and surjective maps, and the sub-objects and quotients are the subsets and quotient sets. In Top, these definitions of embeddings and identifications coincide with their usual meanings, and the sub-objects and quotients are the subspaces and quotient spaces. In Met, any morphism is an embedding, every identification is an isomorphism, the sub-objects are the metric subspaces, and each metric space is the only quotient of itself. In Man$^\infty$, the embeddings are the injective $C^\infty$ immersions, and the identifications are the surjective $C^\infty$ submersions; thus the sub-objects are the immersed $C^\infty$ submanifolds. In Riem, the embeddings are the injective infinitesimally isometric immersions, and the identifications are the surjective Riemannian submersions; thus a subset of an object of Riem is a sub-object when it is an immersed Riemannian submanifold. In the case of a Set-category defined by a species of structures over sets and a class of morphisms, the sub-objects are the subsets endowed with the induced structures if they are of the same species [?, page E IV.16].

**Example 10.21** (ParSet-categories defined by a good class of sub-objects). This generalizes the u-functors of Example 10.18. Let $\mathcal{A}$ be a Set-category. A *class of sub-objects* in $\mathcal{A}$ is a map $\Xi$ that assigns to each $M \in \mathcal{A}_0$ a family $\Xi_M$ of sub-objects of $M$. A class $\Xi$ of sub-objects in $\mathcal{A}$ is called *good* if, for all $M, N \in \mathcal{A}_0$, $A \in \Xi_M$, $B \in \Xi_N$ and any morphism $\phi : A \to N$ of $\mathcal{A}$, we have $\phi^{-1}(B) \in \Xi_M$; in particular, $\Xi_N$ is closed under finite intersections for all $N \in \mathcal{A}_0$. Equivalently, $\Xi$ is good if and only if the following conditions are satisfied:

* If $M \in \mathcal{A}_0$ and $A \in \Xi_M$, then $\Xi_A \subset \Xi_M$;

<footnote n="2">²This term is used only for the scope of this work because it is used with a different meaning in category theory.</footnote>
• if $\phi: M \to N$ is in $\mathcal{A}_1$ and $B \in \Xi_N$, then $\phi^{-1}(B) \in \Xi_M$.

Fix a good class $\Xi$ of sub-objects in $\mathcal{A}$. For $M, N \in \mathcal{A}_0$ and $A \in \Xi_M$, a morphism $A \to N$ of $\mathcal{A}$ will be called a $\Xi$-partial morphism of $M$ to $N$. If a $\Xi$-partial morphism is an embedding, it is called a $\Xi$-partial embedding. If the image of a $\Xi$-partial embedding of $M$ to $N$ is in $\Xi_N$, then it is called a $\Xi$-partial isomorphism; i.e., the $\Xi$-partial isomorphisms of $N$ to $N$ can be considered as the isomorphisms of $\mathcal{A}$ between objects in $\Xi_M$ to objects in $\Xi_N$. Like in Example 10.18, the $\Xi$-partial morphisms, $\Xi$-partial embeddings and $\Xi$-partial isomorphisms between objects of $\mathcal{A}$ form ParSet-categories $\mathrm{Par}(\mathcal{A}, \Xi)$, $\mathrm{ParEmb}(\mathcal{A}, \Xi)$ and $\mathrm{ParIso}(\mathcal{A}, \Xi)$, respectively, with the operation of partial composition. They satisfy

$$ \mathrm{Par}(\mathcal{A}, \Xi) \subset \mathrm{ParEmb}(\mathcal{A}, \Xi) \subset \mathrm{ParIso}(\mathcal{A}, \Xi). $$

If $\mathcal{A}$ and $\Xi$ are fixed and clear from the context, then they can be removed from this terminology and notation. Observe that $\mathrm{Par}(\mathcal{A}, \Xi)/\mathrm{ParSet}$ is faithful (respectively, complete) if and only if $\mathcal{A}/\mathrm{Set}$ is faithful (respectively, complete). Since $\mathcal{A}$ is a subcategory of $\mathrm{Par}(\mathcal{A}, \Xi)$, any $\mathcal{A}$-category becomes a $\mathrm{Par}(\mathcal{A}, \Xi)$-category.

$\mathrm{ParIso}(\mathcal{A}, \Xi)$ is an involutive category in the following way. Let $M, N \in \mathcal{A}_0$, and let $\phi \equiv (M, N, \phi) \in \mathrm{ParIso}(M, N)$. There is some $A \in \Xi_M$ such that $\phi$ is an embedding $A \to N$ of $\mathcal{A}$, and there is some $B \in \Xi_N$ such that $\phi(A) = B$. Then the involution $\phi^{-1} \equiv (N, M, \phi^{-1})$ of $\phi$ is given by the composite

$$ \mathrm{im} \, \phi \xrightarrow{\phi^{-1}} \mathrm{dom} \, \phi \hookrightarrow M. $$

If $\mathcal{A}/\mathrm{Set}$ is complete, then $\mathrm{im} \, \phi$ is a sub-object of $N$ for any $\Xi$-partial embedding $\phi$ of $M$ to $N$, but $\mathrm{im} \, \phi$ may not be in $\Xi_N$; $\phi$ is a partial isomorphism just when $\mathrm{im} \, \phi \in \Xi_N$.

Consider u-functors $\mathcal{A}/\mathcal{B}/\mathrm{Set}$, and let $\Xi$ and $\Theta$ be good classes of sub-objects in $\mathcal{A}$ and $\mathcal{B}$, respectively, such that $\Xi_M \subset \Theta_{|M|_B}$ for all $M \in \mathcal{A}_0$. Then $\mathcal{A}/\mathcal{B}$ canonically extends to a u-functor $\mathrm{Par}(\mathcal{A}, \Xi)/\mathrm{Par}(\mathcal{B}, \Theta)$, which restricts to a u-functors $\mathrm{ParEmb}(\mathcal{A}, \Xi)/\mathrm{ParEmb}(\mathcal{B}, \Theta)$ and $\mathrm{ParIso}(\mathcal{A}, \Xi)/\mathrm{ParIso}(\mathcal{B}, \Theta)$.

Some interesting ParSet-categories, like $\mathrm{ParPropMet}$, are not induced by any good class of sub-objects of a Set-category in the sense of Example 10.21.

**Example 10.22** (The pointed category of a ParSet-category). Let $\mathcal{A}$ be a ParSet-category. For any $M \in \mathcal{A}_0$ and any $x \in M$, the pair $(M, x)$ is called a *pointed object* of $\mathcal{A}$. For another pointed object $(N, y)$, a morphism $\phi \in \mathcal{A}(M, N)$ is called a *pointed morphism* of $(M, x)$ to $(N, y)$ if $x \in \mathrm{dom} \, \phi$ and $\phi(x) = y$. The pointed morphisms between pointed objects of $\mathcal{A}$ form a category with the induced operation, which is denoted by $A_*$ and called the *pointed category* of $\mathcal{A}$. A strongly faithful u-functor $A_*/\mathcal{A}$ is defined by forgetting the distinguished point. This u-functor is complete if and only if the canonical action of $\mathrm{Iso}_\mathcal{A}(M)$ on $M$ is transitive for all $M \in \mathcal{A}_0$.

If $\mathcal{A} = \mathrm{Par}(\mathcal{B}, \Xi)$ for a good class of sub-objects $\Xi$ of some Set-category $\mathcal{B}$, then $A_* = \mathrm{Par}(\mathcal{B}_*, \Xi_*)$, where $\Xi_*$ is the good class of sub-objects of $\mathcal{B}_*$ defined by

$$ \Xi_{*,M,x} = \{(A, x) \mid A \in \Xi_M, x \in A\} $$
for $(M, x) \in \mathcal{B}_{*,0}$.

A functor $F : \mathcal{A} \to \mathcal{B}$ over ParSet induces a functor $F_* : \mathcal{A}_* \to \mathcal{B}_*$ defined by $F_*(M, x) = (F_M, x)$ and $F_*(h) \equiv F(h)$ for all $M \in \mathcal{A}_0$ and $h \in \mathcal{A}_1$, yielding a double functor $F_*/F : \mathcal{A}_*/\mathcal{A} \to \mathcal{B}_*/\mathcal{B}$. Moreover $F_*$ is a u-functor if $F$ is a u-functor; in this case, $(\mathcal{A}_*/\mathcal{A})/(\mathcal{B}_*/\mathcal{B})$ is a semi-pull-back, and thus so is $(\mathcal{A}_*/\mathcal{B}_*)/(\mathcal{A}/\mathcal{B})$ is also a semi-pull-back. In particular, $(\mathcal{A}_*/\mathcal{A})/(\mathrm{ParSet}_*/\mathrm{ParSet})$ is a semi-pull-back, as well as $(\mathcal{A}_*/\mathcal{A})/(\mathrm{Set}_*/\mathrm{Set})$ if $\mathcal{A}$ is a Set-category. Any $\mathrm{ParSet}_*$-category $\mathcal{A}$ is isomorphic to a subcategory of the pointed category of a ParSet-category; indeed, $\mathcal{A}$ is a ParSet-category too, and there is a canonical injective functor $F : \mathcal{A} \to \mathcal{A}_*$ defined by $F_N = (N, x_N)$ for $N \in \mathcal{A}_0$, where $x_N$ is the distinguished point of $N$, and $F(\phi) \equiv \phi$ for $\phi \in \mathcal{A}_1$.

When $\mathcal{A}$ is a ParTop-category, if an object $M \in \mathcal{A}_0$ is locally aperiodic, then $(M, x)$ is aperiodic in $\mathcal{A}_*$ for any $x \in M$.

**Example 10.23.** Let $\mathcal{A}$ be a Top-category whose objects can be restricted to open subsets (see Example 10.21). Then the open subsets form a good class $\Xi$ of sub-objects, and we can consider $\mathrm{Par}(\mathcal{A}, \Xi)$, $\mathrm{ParEmb}(\mathcal{A}, \Xi)$ and $\mathrm{ParIso}(\mathcal{A}, \Xi)$ according to Example 10.21. These categories respectively consist of the partial morphisms of $\mathcal{A}$ with open domains, the partial embeddings of $\mathcal{A}$ with open domains and the partial embeddings of $\mathcal{A}$ with open domains and open images.

**Example 10.24 (The category of local morphisms).** With the notation of Example 10.23, for $M, N \in \mathcal{A}_0$, a local morphism (respectively, a local embedding) of $M$ to $N$ is a partial map $\phi$ of $M$ to $N$ with open domain and such that any $x \in \mathrm{dom}\,\phi$ has some open neighborhood $U$ in $\mathrm{dom}\,\phi$ so that the restriction $\phi : U \to N$ is a morphism (respectively, an embedding) of $\mathcal{A}$. All local morphisms (respectively, local embeddings) between objects of $\mathcal{A}$ form a category $\mathrm{Loc}(\mathcal{A})$ (respectively, $\mathrm{LocEmb}(\mathcal{A})$) with the operation of partial composite. Since $\mathrm{Par}(\mathcal{A}, \Xi) \subset \mathrm{Loc}(\mathcal{A})$, any $\mathrm{Par}(\mathcal{A}, \Xi)$-category becomes a $\mathrm{Loc}(\mathcal{A})$-category; in particular, this applies to any $\mathcal{A}$-category. If $\mathcal{B}$ is another Top-category whose objects can be restricted to open subsets, then any functor $F : \mathcal{A} \to \mathcal{B}$ over Top extends to a functor $\mathrm{Loc}(F) : \mathrm{Loc}(\mathcal{A}) \to \mathrm{Loc}(\mathcal{B})$, which is locally defined by $F$ in the obvious sense. If $F$ is a u-functor, then so is $\mathrm{Loc}(F)$.

**Example 10.25.** If $\mathcal{A}$ is Top, Path, Man, $\mathrm{Man}^\infty$ or Riem, then $\mathrm{Loc}(\mathcal{A}) = \mathrm{Par}(\mathcal{A}, \Xi)$ with the notation of Examples 10.23 and 10.24. Nevertheless $\mathrm{Loc}(\mathrm{Met}) \neq \mathrm{Par}(\mathcal{A}, \Xi)$. The notation $\mathrm{LocTop}$, $\mathrm{LocMet}$, $\mathrm{LocPath}$, $\mathrm{LocMan}$, $\mathrm{LocMan}^\infty$ and $\mathrm{LocRiem}$ will be used in these examples. Let $\mathrm{LocPropPath}$ be the full subcategory of $\mathrm{LocPath}$ whose objects have proper metrics, and let $\mathrm{LocConRiem}$ and $\mathrm{LocComConRiem}$ be the subcategories of $\mathrm{LocRiem}$ whose objects are connected Riemannian manifolds and complete connected Riemannian manifolds, respectively. For a $\mathrm{LocMan}$-category $\mathcal{A}$ and $n \in \mathbb{N}$, its full subcategory with $n$-dimensional objects will be denoted by $\mathcal{A}(n)$.

**Example 10.26 (The extended category).** Let $\mathcal{A}/\mathcal{B}$ be a u-functor over Set. Let $f : |M| \to P$ be an identification of $\mathcal{B}$ for some $M \in \mathcal{A}_0$ and $P \in \mathcal{B}_0$. Any $\phi \in \mathrm{Iso}_\mathcal{A}(M)$ satisfying $f = f \circ \phi$ is called a *deck transformation* of the triple $(M, P, f)$. The deck transformations
of $(M, P, f)$ form a subgroup of $\text{Iso}_A(M)$. It is said that $(M, P, f)$ is an *extended object* of $\mathcal{A}$ over $\mathcal{B}$ if the fibers of $f$ are the orbits of the canonical action of the group of deck transformations of $(M, P, f)$ on $M$. An extended object $(M, P, f)$ is called *regular* if $f \in |\mathcal{A}_1|$ (thus $P \in |\mathcal{A}_0|$); otherwise, $(M, P, f)$ is called *singular*. A *morphism* between extended objects, $(M, P, f) \to (N, Q, g)$, is a morphism $\psi : M \to N$ of $\mathcal{A}$ such that there is some morphism $h : P \to Q$ of $\mathcal{B}$ satisfying $h \circ f = g \circ \psi$; notice that $\psi$ determines $h$. The morphisms between extended objects, with the operation defined by the operation of $\mathcal{A}$, form a category $\mathcal{A}^e$ in $\mathcal{U}$, called the *extended category* of $\mathcal{A}$ over $\mathcal{B}$. There is a u-functor $\mathcal{A}^e/\mathcal{A}$ defined by $|(M, P, f)| = M$ and $|\psi| \equiv \psi$ for $(M, P, f) \in \mathcal{A}_0^e$ and $\psi \in \mathcal{A}_1^e$; thus $\mathcal{A}^e$ becomes also a $\mathcal{B}$-category. There is another canonical functor $\pi : \mathcal{A}^e \to \mathcal{B}$ defined by $\pi(M, P, f) = P$ and $\pi(\psi) = h$ for $\psi : (M, P, f) \to (N, Q, g)$ and $h : P \to Q$ satisfying $h \circ f = g \circ \psi$. Finally, there is a canonical injective functor $\iota : \mathcal{A} \to \mathcal{A}^e$ defined by $\iota(M) = (M, |M|, \text{id}_{|M|})$ and $\iota(\phi) \equiv \phi$ for $M \in \mathcal{A}_0$ and $\phi \in \mathcal{A}_1$; thus $\mathcal{A}$ can be considered as a subcategory of $\mathcal{A}^e$. Observe that $\iota$ is a section of the u-functor $\mathcal{A}^e/\mathcal{A}$, and therefore this u-functor is surjective. The deck transformations form a subgroupoid $\text{Deck}(\mathcal{A}^e) \subset \text{Iso}(\mathcal{A}^e)$. For the canonical left action of $\text{Deck}(\mathcal{A}^e)$ in $\mathcal{A}^e$, the element in the orbit space $\text{Deck}(\mathcal{A}^e)\backslash\mathcal{A}_0^e$ defined by each $(M, P, f) \in \mathcal{A}^e$ will be denoted by $[M, P, f]$. Obviously, the functor $\pi$ is invariant by the action of $\text{Deck}(\mathcal{A}^e)$, and thus induces a map $\bar{\pi} : \text{Deck}(\mathcal{A}^e)\backslash\mathcal{A}_0^e \to \mathcal{B}_0$. Since the quotient objects of each object of $\mathcal{B}$ are determined by their underlying quotient sets, there is a canonical injective functor of the extended category of $\mathcal{A}$ over $\mathcal{B}$ into the extended category of $\mathcal{A}$ over Set, and this injection is bijective if $\mathcal{B}$/Set is complete and $\mathcal{B}$ has enough quotients in an obvious sense. For instance, the extended categories of $\mathcal{A}$ over Top and over Set can be identified in this way.

**Example 10.27** (U-functors defined by compactifications). ***

## 10.3 Local categories

**Definition 10.28.** A local category is a faithful Top-category $\mathcal{A}$ satisfying the following properties:

(L0) Each object of $\mathcal{A}$ can be restricted to any open subset (see Example 10.21).

(L1) Let $X$ be a topological space, and $\{U_i\}$ an open covering of $X$. If each $U_i$ underlies some $M_i \in \mathcal{A}_0$ such that $M_i|_{U_i \cap U_j} = M_j|_{U_i \cap U_j}$ for all $i, j$, then $X$ underlies some $M \in \mathcal{A}_0$ satisfying $M|_{U_i} = M_i$ for all $i$.

(L2) Let $M, N \in \mathcal{A}_0$, $f : |M| \to |N|$ a continuous map, $\{U_i\}$ an open covering of $M$, and $\{V_i\}$ a collection of open subsets of $N$ such that $f(U_i) \subset V_i$. Then $f$ underlies some morphism in $\mathcal{A}(M, N)$ if and only if each restriction $f : U_i \to V_i$ underlies some morphism in $\mathcal{A}(M|_{U_i}, N|_{V_i})$.

It may be also said that $\mathcal{A}$/Top is local.

**Remarks.** (i) Note that (L2) implies the unicity of $M$ in (L1) since $\mathcal{A}$/Top is faithful.
(ii) Observe also that (L2) means that $\mathrm{Loc}(\mathcal{A}) = \mathrm{Par}(\mathcal{A}, \Xi)$, where $\Xi$ is the good class of open sub-objects in $\mathcal{A}$.

(iii) A morphism of $\mathrm{Loc}(\mathcal{A})$ is in $\mathcal{A}$ if and only it is globally defined on its source object.

(iv) *** local categories and sheaves ***

**Example 10.29.** Top, Path, Man, $\mathrm{Man}^\infty$ and Riem are local categories, whilst Simp and Met are not local.

**Example 10.30** (Local category defined by a species of local structures and a local class of morphisms). Let $\Sigma$ be a species of *local* structures over sets in the sense of C. Ehresmann [?, ?, ?, ?]. A class $\sigma$ of morphisms of $\Sigma$ is called a class of *local morphisms* when the obvious analogue of (L2) is satisfied. In this case, the category defined by $\Sigma$ and $\sigma$, according to Example 10.17, is local.

**Example 10.31** (Local category defined by a pseudomonoid). Generalizing the concept of pseudogroup, a *pseudomonoid* $\mathcal{P}$ on a space $Z$ is a family of continuous maps from open subsets of $Z$ to $Z$ which contains $\mathrm{id}_Z$ and is closed under the operations of partial composition, restriction to open subsets and combination. A pseudomonoid $\mathcal{P}$ on $Z$ is *generated* by a subset $S \subset \mathcal{P}$ when $\mathcal{P}$ can be obtained from $S$ by adding $\mathrm{id}_Z$ and using the above operations. Any pseudogroup generates a pseudomonoid whose elements are local homeomorphisms. *Vice versa*, any pseudomonoid $\mathcal{P}$ on $Z$ induces a pseudogroup $\Gamma$ consisting of the homeomorphisms $h$ between open subsets of $Z$ such that $h$ and $h^{-1}$ are defined by elements of $\mathcal{P}$.

Suppose that a space $Z$ is *locally small* in the sense that it can be covered by small open sets. As a particular case of Example 10.30, a local category can be defined as follows by any pseudomonoid $\mathcal{P}$ on $Z$. According to [?], the pseudogroup $\Gamma$ induced by $\mathcal{P}$ defines a species $\Sigma$ of local structures on sets in the following way. A $Z$-valued *chart* of a set $X$ is a partial injection of $X$ to $Z$ whose image is open. For two $Z$-valued charts on $X$, $\theta$ and $\zeta$, the partial composites $\zeta \circ \theta^{-1}$ and $\theta \circ \zeta^{-1}$ are called *changes of coordinates*. A collection of $Z$-valued charts whose domains cover $X$ is called a *$Z$-valued atlas*. A $Z$-valued atlas of $X$ is said to be *compatible* with $\Gamma$ when all of its changes of coordinates are in $\Gamma$. Then the maximal $Z$-valued atlases compatible with $\Gamma$ are local structures over sets of a species $\Sigma$. On the other hand, $\mathcal{P}$ defines a class $\sigma$ of local morphisms of $\Sigma$ in the following way. If $\alpha$ and $\beta$ are structures of the species $\Sigma$ on sets $X$ and $Y$, then the $\sigma$-morphisms $(X, \alpha) \to (Y, \beta)$ are the maps $f: X \to Y$ satisfying that $\zeta \circ f \circ \theta^{-1}$ is in $\mathcal{P}$ for all $Z$-valued charts $\theta \in \alpha$ and $\zeta \in \beta$. Then $\Sigma$ and $\sigma$ induce a local category in the sense of Example 10.30.

**Example 10.32.** For a local category $\mathcal{A}$, a full subcategory $\mathcal{B} \subset \mathcal{A}$ is a local category if and only if $\mathcal{B}_0$ is closed by the operation of restrictions to open subsets in $\mathcal{A}_0$ (given by (L0)).

For an arbitrary local category $\mathcal{A}$, let $\mathcal{C} \subset \mathrm{Loc}(\mathcal{A})$ be any full subcategory so that any $M \in \mathcal{A}_0$ is covered by the domains of open local embeddings to objects of $\mathcal{C}$ (see Example 10.24); e.g., we can take $\mathcal{C} = \mathrm{Loc}(\mathcal{A})$. Let $Z$ be the topological sum of the spaces $|C|_{\mathrm{Top}}$ with $C$ running in $\mathcal{C}_0$, which is a locally small space. The morphisms of $\mathcal{C}$ generate a pseudomonoid $\mathcal{P}$ on $Z$; indeed, the maps in $\mathcal{P}$ are combinations of morphisms of $\mathcal{C}$.
If $\mathcal{C}_0$ is small, then $Z$ is also small. In this case, $Z$ underlies some object of $\mathcal{A}$ by (L1), also denoted by $Z$; thus we can choose $\mathcal{C}$ with $\mathcal{C}_0 = \{Z\}$, yielding $\mathcal{P} \equiv \mathcal{C}_1$.

Let $\mathcal{B}$ be the local category defined by $\mathcal{P}$ according to Example 10.31.

**Proposition 10.33.** *A can be canonically realized as a full subcategory of B. Moreover A ≡ B if and only A/Top is complete.*

*Proof.* For each $M \in \mathcal{A}_0$, let $\alpha_M$ be the collection of all open embeddings of open subsets of $M$ to $Z$ which are combinations of morphisms of LocEmb($\mathcal{A}$) with source $M$ and targets the objects of $\mathcal{C}$. With the notation of Example 10.31, it is easy to see that $\alpha_M$ is a structure of the species $\Sigma$ on $|M|_{\text{Set}}$, and define $F_M = (|M|_{\text{Set}}, \alpha_M) \in \mathcal{B}_0$. For each morphism $\phi : M \to N$ of $\mathcal{A}$, the underlying map $|\phi|_{\text{Set}} : |M|_{\text{Set}} \to |N|_{\text{Set}}$ obviously defines a morphism $F(\phi) : F_M \to F_N$ of $\mathcal{B}$. This gives a canonical definition of a functor $F : \mathcal{A} \to \mathcal{B}$ over Top.

Suppose that $F_M = F_N$ for some $M, N \in \mathcal{A}_0$. Then $|M|_{\text{Set}} = |N|_{\text{Set}}$ and $\alpha_M = \alpha_N$, which is denoted by $\alpha$. It easily follows that $|M|_{\text{Top}} = |N|_{\text{Top}}$ and, for the domain $U$ of any element of $\alpha$, the identity map $\text{id}_U$ underlies a morphism $M|_U \to N|_U$ of $\mathcal{A}$, yielding $M|_U = N|_U$ because $\mathcal{A}/\text{Top}$ is faithful. Hence $M = N$ by (L2). This shows that $F_0$ is injective. Therefore $F_1$ is injective as well by Proposition 10.6, and thus $\mathcal{A}$ can be considered as a subcategory of $\mathcal{B}$ via $F$.

Take any $f \in \mathcal{B}(F_M, F_N)$ for some $M, N \in \mathcal{A}_0$. By (L0) and (L2), there are embeddings $\theta_i : M|_{U_i} \to C_i$ and $\zeta_i : N|_{V_i} \to D_i$ of $\mathcal{A}$, where $C_i, D_i \in \mathcal{C}_0, \{U_i\}$ is a covering of $M$, and each $V_i$ is an open subset $N$ containing $f(U_i)$. Since $\zeta_i \circ f \circ \theta_i^{-1} \in \mathcal{P}$ for each $i$, this composition underlies some morphism in $\mathcal{A}(C_i|_{\theta_i(U_i)}, D_i|_{\zeta_i(V_i)})$ by (L0) and (L2), which is also denoted by $\zeta_i \circ f \circ \theta_i^{-1}$. Hence each restriction $f : U_i \to V_i$ underlies the product of morphisms

$$M|_{U_i} \xrightarrow{\theta_i} C_i|_{\theta_i(U_i)} \xrightarrow{\zeta_i \circ f \circ \theta_i^{-1}} D_i|_{\zeta_i(V_i)} \xrightarrow{\zeta_i^{-1}} N|_{V_i}.$$

So $f$ underlies some morphism in $\mathcal{A}(M, N)$ by (L2). This shows that $F : \mathcal{A}(M, N) \to \mathcal{B}(F_M, F_N)$ is surjective; i.e., $\mathcal{A}$ is full in $\mathcal{B}$ via $F$.

If $F$ is an isomorphism, then $\mathcal{A}/\text{Top}$ is complete because $\mathcal{B}/\text{Top}$ is complete.

Now, assume that $\mathcal{A}/\text{Top}$ is complete. To prove that $F$ is an isomorphism, it is enough to show that $F_0$ is surjective. Let $X$ be any object of $\mathcal{B}_0$. Consider an atlas of $X$ consisting of charts of charts $\theta_i : U_i \to Z$ of $X$, and let $U_{i,j} = U_i \cap U_j$, $V_i = \theta_i(U_i)$ and $V_{i,j} = \theta_i(U_{i,j})$. We can suppose that $V_i \subset C_i$ for some $C_i \in \mathcal{C}_0$, and that each change of coordinates $\theta_j \circ \theta_i^{-1}$ underlies some morphism in $\mathcal{A}(C_i|_{V_{i,j}}, C_j|_{V_{j,i}})$, also denoted by $\theta_j \circ \theta_i^{-1}$. Since $\mathcal{A}/\text{Top}$ is complete and by (L0), there is some $M_i \in \mathcal{A}_0$ so that $|M_i|_{\text{Top}} = U_i$ and $\theta_i : U_i \to V_i$ underlies some isomorphism in $\mathcal{A}(M_i, C_i|_{V_i})$, denoted by $\theta_i$ too. The product of isomorphisms

$$M_i|_{U_{i,j}} \xrightarrow{\theta_i} C_i|_{V_{i,j}} \xrightarrow{\theta_j \circ \theta_i^{-1}} C_j|_{V_{j,i}} \xrightarrow{\theta_j^{-1}} M_j|_{U_{i,j}}$$

is over the identity map on each $U_{i,j}$, and thus $M_i|_{U_{i,j}} = M_j|_{U_{i,j}}$ because $\mathcal{A}/\text{Top}$ is faithful. Therefore there is some $M \in \mathcal{A}_0$ whose restriction to each $U_i$ is $M_i$. It easily follows that $F_M = X$, and thus $F_0$ is surjective. $\square$
**Remark.** According to Proposition 10.33, all local categories are essentially given by the combination of Examples 10.31 and 10.32.

A local category $\mathcal{A}$ is over Man (respectively, $\text{Man}(n)$) if and only if it is a full subcategory of the local category defined by a pseudomonoid on $\bigsqcup_n \mathbb{R}^n$ (respectively, $\mathbb{R}^n$).

## 10.4 Distortions

The following notation will be useful. For a set $X$, let “$\ll$” be the pre-order relation between sets of functions $X \to [0, \infty]$ defined by setting $C \ll D$ if, for all $\varepsilon > 0$, there is some $\delta_\varepsilon > 0$ such that, if $d(x) < \delta_\varepsilon$ for all $d \in D$ and $x \in X$, then $c(x) < \varepsilon$ for all $c \in C$ and $x \in X$. Write $C \approx D$ when $C \ll D \ll C$. For $C = \{c_1, \dots, c_m\}$ and $D = \{d_1, \dots, d_n\}$, the relation $C \ll D$ is equivalent to the existence of a symmetric continuous function $f : [0, \infty]^n \to [0, \infty]$ continuous at $(0, \dots, 0)$, and such that $f(0, \dots, 0) = 0$ and

$$c_k(x) \le f(d_1(x), \dots, d_n(x))$$

for all $x \in X$ and all $k \in \{1, \dots, m\}$; it will be said that $C \ll D$ holds with $f$. The notation

$$\{c_1(x), \dots, c_m(x)\} \ll \{d_1(x), \dots, d_n(x)\}$$

may be used too, or simply

$$c(x) \ll \{d_1(x), \dots, d_n(x)\}, \quad \{c_1(x), \dots, c_m(x)\} \ll d(x), \quad c(x) \ll d(x)$$

if $C = \{c\}$ or $D = \{d\}$. The same simplification of the notation will be used with “$\approx$”. Observe that

$$(10.11) \qquad C \approx \sup C,$$

$$(10.12) \qquad C \ll D, C' \ll D' \implies C \cup C' \ll D \cup D'$$

Of course, (10.12) extends to finite unions, but not to infinite unions. Since all norms on $\mathbb{R}^n$ are equivalent, it follows from (10.11) that

$$(10.13) \qquad \|(c_1(x), \dots, c_n(x))\| \approx \{c_1(x), \dots, c_n(x)\}$$

for any norm $\| \cdot \|$ on $\mathbb{R}^n$.

The restriction $\| \cdot \| : [0, \infty)^n \to [0, \infty)$ of any norm $\| \cdot \|$ on $\mathbb{R}^n$ extends to a map $[0, \infty]^n \to [0, \infty]$, also denoted by $\| \cdot \|$, whose value is $\infty$ on the complement of $[0, \infty)^n$.

**Definition 10.34.** A distortion of a u-functor $\mathcal{A}/\mathcal{B}$ is a map $\mathfrak{d} : \tilde{\mathcal{B}}_1 \to [0, \infty]$ subject to the following conditions:

$$(\text{d1}) \quad \mathfrak{d}^{-1}(0) = \mathcal{A}_1.$$

$$(\text{d2}) \quad \mathfrak{d}(f_1 \cdots f_n) \ll \mathfrak{d}(f_1) + \cdots + \mathfrak{d}(f_n) \text{ for } (f_1, \dots, f_n) \in \tilde{\mathcal{B}}_\infty.$$
The pair $(A/\mathcal{B}, \mathfrak{d})$ is called a *distortion functor*. It is said that $\mathfrak{d}$ is *involutive* when $A/\mathcal{B}$ is involutive and $\mathfrak{d}(\bar{f}) = \mathfrak{d}(f)$ for all $f \in \tilde{\mathcal{B}}_1$.

**Remarks.** (i) For fixed $n$, the relation of (d2) is equivalent to

$$ \mathfrak{d}(f_n \cdots f_1) \preceq \{\mathfrak{d}(f_1), \dots, \mathfrak{d}(f_n)\} $$

by (10.13), but this relation is stronger when $n$ is arbitrary.

(ii) Observe that the definition of distortion bears comparison with the definition of metric. The condition (d1) corresponds to the condition that the null set of a metric is the diagonal, and (d2) is some kind of “infinitesimal logarithmic triangle inequality” for consecutive morphisms of $\tilde{\mathcal{B}}$. In fact, it will be shown in Example 10.105 that metrics can be considered as distortions.

Let $(A/\mathcal{B}, \mathfrak{d})$ be a distortion functor. For a morphism $f$ of $\tilde{\mathcal{B}}$, think of $\mathfrak{d}(f)$ as a measure of its lack to be a morphism of $A$. Sometimes, for $M, N \in A_0$, the restriction of $\mathfrak{d}$ to $\tilde{\mathcal{B}}(M, N)$ will be denoted by $\mathfrak{d}_{M,N}$. The condition (d2) can be improved in the following way. Let $\hat{\mathfrak{d}} : \tilde{\mathcal{B}}_1 \to [0, \infty]$ be the function defined by

$$ (10.14) \qquad \hat{\mathfrak{d}}(f) = \inf\{\mathfrak{d}(f_1) + \dots + \mathfrak{d}(f_n) \mid (f_1, \dots, f_n) \in \tilde{\mathcal{B}}_\infty, f_n \cdots f_1 = f\}. $$

**Proposition 10.35.** The map $\hat{\mathfrak{d}}$ is a distortion of $A/\mathcal{B}$ and satisfies

$$ \hat{\mathfrak{d}}(gf) \le \hat{\mathfrak{d}}(f) + \hat{\mathfrak{d}}(g) $$

for all $(f, g) \in \tilde{\mathcal{B}}_2$.

*Proof.* It is obvious that $\hat{\mathfrak{d}}$ satisfies the stated inequality, which is stronger than (d2). On the other hand, (d1) for $\hat{\mathfrak{d}}$ is a consequence of (d1) and (d2) for $\mathfrak{d}$. $\square$

The inequality of Proposition 10.35 can be really called a “logarithmic triangle inequality”. It could be used instead of (d2) to define distortions, but (d2) allows easier constructions of examples. Moreover (d2) will be more appropriate for the definitions of Section 10.7.

Fix any norm $\| \cdot \|$ on $\mathbb{R}^2$ invariant by permutation of components and satisfying

$$ (10.15) \qquad |r_1| \le |s_1|, |r_2| \le |s_2| \implies \|(r_1, r_2)\| \le \|(s_1, s_2)\|. $$

For instance, the standard norm on $\mathbb{R}^2$ satisfies (10.15), but the norm on $\mathbb{R}^2$ defined by $\|(r_1, r_2)\| = \max\{|r_1 + r_2|, 2|r_1 - r_2|\}$ does not satisfy this condition. Notice that (10.15) is also satisfied by the induced map $\| \cdot \| : [0, \infty]^2 \to [0, \infty]$. Then let $d_\mathfrak{d} : A_0 \times A_0 \to [0, \infty]$ be defined by

$$ (10.16) \qquad d_{\mathfrak{d}}(M, N) = \|(\inf \hat{\mathfrak{d}}_{M,N}, \inf \hat{\mathfrak{d}}_{N,M})\|. $$

in particular,

$$ (10.17) \qquad \begin{gathered} \tilde{\mathcal{B}}(M, N) = \emptyset \implies d_{\mathfrak{d}}(M, N) = \infty, \\ A(M, N) \neq \emptyset \neq A(N, M) \implies d_{\mathfrak{d}}(M, N) = 0. \end{gathered} $$
**Proposition 10.36.** $d_\mathfrak{D}$ is a pseudo-metric with possible infinite values on $\mathcal{C}_0$, which is invariant by the equivalence relation on $\mathcal{C}_0$ defined by the existence of morphisms of $\mathcal{C}$.

*Proof.* This map is obviously symmetric, and satisfies the triangle inequality by (10.15) and the definition of $\hat{\mathfrak{d}}$. The last part of the statement follows from (10.17) and the triangle inequality. $\square$

According to Proposition 10.36, $d_\mathfrak{D}$ induces a pseudo-metric with possible infinite values on $A\backslash A_0$, which will be also denoted by $d_\mathfrak{D}$.

**Lemma 10.37.** $\mathfrak{D}(\psi f\phi) \preceq \mathfrak{D}(f)$ for $f \in \tilde{\mathcal{B}}_1$ and $\phi, \psi \in A_1$ such that the product $\psi f\phi$ is defined; in particular, $\mathfrak{D}(\psi f\phi) \approx \mathfrak{D}(f)$ when $\phi, \psi \in \text{Iso}(A)_1$.

*Proof.* This is a direct consequence of (d1) and (d2). $\square$

Let $\| \|$ be any norm on $\mathbb{R}^2$ invariant by permutation of components. Let $S$ be the structure functor of a u-functor $A/B$ with a distortion $\mathfrak{d}$. For each $X \in \mathcal{B}_0$ and all $M, N \in S(X)$, let $\mathcal{L}(M, N)$ be the set of finite sequences $(L_0, \dots, L_n)$ in $S(X)$, with arbitrary length $n$, such that $L_0 = M$ and $L_n = N$. Let

$$ \mathfrak{s}(L_0, \dots, L_n) = \sum_{i=1}^{n} \| (\mathfrak{d}_{L_{i-1}, L_i}(\text{id}_X), \mathfrak{d}_{L_i, L_{i-1}}(\text{id}_X)) \|, $$

for every $(L_0, \dots, L_n) \in \mathcal{L}(M, N)$, and set

$$ d_X(M, N) = \inf\{\mathfrak{s}(L_0, \dots, L_n) \mid (L_0, \dots, L_n) \in \mathcal{L}(M, N)\} $$

in $[0, \infty]$. This defines a pseudo-metric $d_X$ with possible infinite values in $S(X)$. If $A/B$ is faithful, then $d_X$ is a metric with possible infinite values. In this way, if $A/B$ is strongly faithful and complete, and $d_X$ is finite valued for all $X \in \mathcal{B}_0$, then the functor $S: \text{Iso}(B) \to \text{Set}$ lifts to a functor $S: \text{Iso}(B) \to \text{Met}$.

The notation $F/G: (A/B, \mathfrak{d}) \to (A'/B', \mathfrak{d}')$ is used for a double functor between distortion functors; the notation $(A'/B', \mathfrak{d}')/(A/B, \mathfrak{d})$ is also used when $F/G$ is a double u-functor. Recall that $\tilde{G}: \tilde{B} \to \tilde{B}'$ denotes the lift of $G$ (Example 10.7). The following concepts are introduced:

* $F/G: (A/B, \mathfrak{d}) \to (A'/B', \mathfrak{d}')$ is called *continuous* when, for all $\varepsilon > 0$, there is some $\delta > 0$ such that, $\mathfrak{d}'(\tilde{G}(f)) < \varepsilon$ if $\mathfrak{d}(f) < \delta$ for every $f \in \tilde{\mathcal{B}}_1$. If $F/G$ is a doudle u-functor, it may be said that $\mathfrak{d}$ is *continuous over $\mathfrak{d}'$*.

* $F/G: (A/B, \mathfrak{d}) \to (A'/B', \mathfrak{d}')$ is said to be *Lipschitz* if there is some $K > 0$ such that $\mathfrak{d}'(\tilde{G}(f)) \le K \mathfrak{d}(f)$ for every $f \in \tilde{\mathcal{B}}_1$. If $F/G$ is a doudle u-functor, it may be said that $\mathfrak{d}$ is *Lipschitz over $\mathfrak{d}'$*.

* Two distortions $\mathfrak{d}$ and $\mathfrak{d}'$ of a u-functor $A/B$ are said to be *equivalent* if they are continuous over each other (using $\text{id}_A / \text{id}_B$).

* Two distortions $\mathfrak{d}$ and $\mathfrak{d}'$ of a u-functor $A/B$ are said to be *Lipschitz equivalent* if they are Lipschitz over each other.
## 10.5 Examples of distortions

**Example 10.38 (The top distortion).** Any u-functor $\mathcal{A}/\mathcal{B}$ has a distortion $\mathfrak{d}$ defined by

$$ \mathfrak{d}(f) = \begin{cases} 0 & \text{if } f \in \mathcal{A}_1 \\ \infty & \text{otherwise.} \end{cases} $$

In this case, the condition (d2) holds with any function $f: [0, \infty] \to [0, \infty]$ continuous at 0, and fixing 0 and $\infty$. This will be called the *top distortion*.

**Example 10.39 (The top extension).** For u-functors $\mathcal{A}/\mathcal{B}/\mathcal{C}$, let $\tilde{\mathcal{B}}$ and $\tilde{\mathcal{C}}$ denote the lifts of $\mathcal{B}$ and $\mathcal{C}$ to $\mathcal{A}$. We can canonically consider $\tilde{\mathcal{B}}$ as a subcategory of $\tilde{\mathcal{C}}$ according to Example 10.10. Then any distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{B}$ induces a distortion $\tilde{\mathfrak{d}}$ of $\mathcal{A}/\mathcal{C}$ defined by

$$ \tilde{\mathfrak{d}}(f) = \begin{cases} \mathfrak{d}(f) & \text{if } f \in \tilde{\mathcal{B}}_1 \\ \infty & \text{otherwise.} \end{cases} $$

If $\mathfrak{d}$ satisfies (d2) with some function $f$, then $\tilde{\mathfrak{d}}$ satisfies (d2) with the function $\tilde{f}$ given by

$$ \tilde{f}(r) = \begin{cases} f(r) & \text{if } r < \infty \\ \infty & \text{if } r = \infty. \end{cases} $$

This $\tilde{\mathfrak{d}}$ is called the *top extension* of $\mathfrak{d}$ to $\mathcal{A}/\mathcal{C}$. If $\mathcal{B} = \mathcal{A}$, then $\mathfrak{d}$ is the zero function and $\tilde{\mathfrak{d}}$ is the top distortion of $\mathcal{A}/\mathcal{C}$ (Example 10.38).

**Example 10.40 (Pull-back of a distortion).** Consider a double functor $F/G: \mathcal{A}/\mathcal{B} \to \mathcal{C}/\mathcal{D}$, and let $\tilde{G}: \tilde{\mathcal{B}} \to \tilde{\mathcal{D}}$ be the lift of $G$ (Example 10.7). If $F/G$ is a semi-pull-back, then, for any distortion $\mathfrak{d}$ of $\mathcal{C}/\mathcal{D}$, the composite

$$ \tilde{\mathcal{B}}_1 \xrightarrow{\tilde{G}} \tilde{\mathcal{D}}_1 \xrightarrow{\mathfrak{d}} [0, \infty] $$

is a distortion $\tilde{\mathfrak{d}}$ of $\mathcal{A}/\mathcal{B}$ called the *pull-back* of $\mathfrak{d}$ by $F/G$. Observe that $\tilde{\mathfrak{d}}$ satisfies (d2) with the same function as $\mathfrak{d}$.

**Example 10.41 (Lift or restriction of a distortion).** In Example 10.40, if $F/G$ is a double u-functor, then $\tilde{\mathfrak{d}}$ may be also called the *lift* or the *restriction* of $\mathfrak{d}$ to $\mathcal{A}/\mathcal{B}$. According to Example 10.9, a particular case is given by a u-functor $\mathcal{A}/\mathcal{B}$ and subcategories $\mathcal{A}' \subset \mathcal{A}$ and $\mathcal{B}' \subset \mathcal{B}$ such that $|\mathcal{A}'| \subset \mathcal{B}'$. If $(\mathcal{A}'/\mathcal{B}')/(\mathcal{A}/\mathcal{B})$ is a semi-pull-back (e.g. when $\mathcal{A}'$ is full in $\mathcal{A}$), then the lift (or restriction) of any distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{B}$ to $\mathcal{A}'/\mathcal{B}'$ is its restriction to $(\tilde{\mathcal{B}}')_1$.

**Example 10.42.** Another particular instance of Example 10.40 is given by u-functors $\mathcal{A}/\mathcal{B}/\mathcal{C}$. They form a semi-pull-back $(\mathcal{A}/\mathcal{B})/(\mathcal{A}/\mathcal{C})$ (Example 10.10). Thus any distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{C}$ lifts (or restricts) to a distortion $\tilde{\mathfrak{d}}$ of $\mathcal{A}/\mathcal{B}$, which is the restriction of $\mathfrak{d}$ to $\tilde{\mathcal{B}}_1$.
Let $\mathfrak{d}$ be a distortion of a u-functor $\mathcal{A}/\mathcal{B}$ over Met/Set. With the notation of Examples 10.45 and 10.46, $\mathfrak{d}$ is said to be of Lipschitz type (respectively, large scale Lipschitz type) if it is Lipschitz over $\mathfrak{d}_{\text{Lip}}$ (respectively, $\mathfrak{d}_{\text{lsl}}$); this is independent of the chosen norm in Example 10.46. “Arc-wise versions” of Examples 10.45 and 10.46 can be considered as follows.

**Example 10.47** (The arc-wise Lipschitz distortion). For any map $f: M \to N$ between path metric spaces, let

$$ \mathfrak{d}_{\text{aLip}}(f) = \inf\{\log C \mid C \text{ is an arc-wise bi-Lipschitz distortion of } f\} $$

in $[0, \infty]$. This defines a distortion $\mathfrak{d}_{\text{aLip}}$ called the *arc-wise Lipschitz distortion* of Path/Set. Property (d2) is satisfied in this case with the identity function on $[0, \infty]$.

**Example 10.48** (The arc-wise large scale Lipschitz distortion). Fix any norm $\|\|$ on $\mathbb{R}^2$, and let $Q > 0$ be given by (10.19). For any map $f: M \to N$ between path metric spaces, let

$$ \mathfrak{d}_{\text{alsl}}(f) = \inf\{\|(\log \lambda, \mu)\| \mid (\lambda, \mu) \text{ is an arc-wise large scale bi-Lipschitz distortion of } f\} $$

in $[0, \infty]$. This defines a distortion $\mathfrak{d}_{\text{alsl}}$ of Path/Set, called the *arc-wise large scale Lipschitz distortion*. From ***, it easily follows that property (d2) is satisfied with the function $f$ defined by

$$ f(r) = \begin{cases} Q \|(r, r)\| & \text{if } r \le 1/Q \\ \infty & \text{if } r > 1/Q. \end{cases} $$

Another norm on $\mathbb{R}^2$ satisfying (10.15) gives rise to a Lipschitz equivalent distortion.

Let $\mathfrak{d}$ be a distortion of a u-functor $\mathcal{A}/\mathcal{B}$ over Path/Set. With the notation of Examples 10.47 and 10.48, $\mathfrak{d}$ is said to be of *arc-wise Lipschitz type* (respectively, *arc-wise large scale Lipschitz type*) if it is Lipschitz over $\mathfrak{d}_{\text{aLip}}$ (respectively, $\mathfrak{d}_{\text{alsl}}$).

**Example 10.49** (The Riemannian distortion). Fix any norm $\|\|$ on the vector space $\ell^\infty = \ell^\infty(\mathbb{N})$ satisfying

$$ (10.20) \qquad r_n \le s_n \quad \forall n \in \mathbb{N} \implies \|(r_n)\| \le \|(s_n)\| $$

for all $(r_n), (s_n) \in \ell^\infty$. For any $C^\infty$ map $f: M \to N$ between Riemannian manifolds, let

$$ \mathfrak{d}_{\text{Riem}}(f) = \inf\{\|(\min\{1, \log C_i\})\| \mid (C_i) \text{ is a Lipschitz distortion of order } \infty \text{ *** of } f\} $$

in $[0, \infty]$. This defines a distortion $\mathfrak{d}_{\text{Riem}}$ of Riem/Man$^\infty$, called the *Riemannian distortion* (of order $\infty$). Its restriction to ConRiem/Man$^\infty$, also denoted by $\mathfrak{d}_{\text{Riem}}$, is a distortion of arc-wise Lipschitz type. By ***, this distortion satisfies the condition (d2) with the function $f$ defined by

$$ f(r) = \begin{cases} \|(r, r, \dots)\| & \text{if } r < \infty \\ \infty & \text{if } r = \infty. \end{cases} $$
Let $\mathfrak{d}$ be a distortion of a u-functor $\mathcal{A}/\mathcal{B}$ over Riem $/ \text{Man}^{\infty}$. With the notation of Example 10.49, $\mathfrak{d}$ is said to be of *Riemannian type* if it is Lipschitz over $\mathfrak{d}_{\text{Riem}}$.

**Example 10.50** (The $\dagger$-extension of a distorsion). By using Examples 10.22 and 10.40, for u-functors $\mathcal{A}/\mathcal{B}/\text{ParSet}$, any distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{B}$ lifts to a distortion $\tilde{\mathfrak{d}}$ of $\mathcal{A}_*/\mathcal{B}_*$, called the $\dagger$-lift of $\mathfrak{d}$. In turn, according Example 10.39, $\tilde{\mathfrak{d}}$ can be trivially extended to a distortion $\mathfrak{d}^*$ of $\mathcal{A}_*/\mathcal{B}$ called the $\dagger$-extension of $\mathfrak{d}$. Observe that, for a morphism $f : (M, x) \to (N, y)$ of the lift $\tilde{\mathcal{B}}$ defined by $\mathcal{A}_*/\mathcal{B}$, we have

$$ \mathfrak{d}^*(f) = \begin{cases} \mathfrak{d}(f) & \text{if } x \in \text{dom } f \text{ and } f(x) = y \\ \infty & \text{otherwise.} \end{cases} $$

For instance, with the notation of Examples 10.45–10.49, we get distortions $\mathfrak{d}_{\text{Lip}}^*$ and $\mathfrak{d}_{\text{lsl}}^*$ of $\text{Met}_*/\text{Set}$, $\mathfrak{d}_{\text{aLip}}^*$ and $\mathfrak{d}_{\text{aLsl}}^*$ of $\text{Path}_*/\text{Set}$, and $\mathfrak{d}_{\text{Riem}}^*$ of $\text{ConRiem}_*/\text{Man}^\infty$.

**Example 10.51** (The $\dagger$-extension of a distorsion). Let $\mathfrak{d}$ be a distortion of large scale Lipschitz type (respectively, arc-wise large scale Lipschitz type) of a u-functor $\mathcal{A}/\mathcal{B}$ over $\text{Met}/\text{Set}$ (respectively, $\text{Path}/\text{Set}$). Fix any norm $\| \cdot \|$ on $\mathbb{R}^2$. Then, for $(M, x), (N, y) \in \mathcal{A}_{*,0}$ and $f \in \mathcal{B}(|M|, |N|)$, let

$$ \mathfrak{d}_{M,x;N,y}^{\dagger}(f) = \|(\mathfrak{d}(f), d_N(f(x), y))\| $$

This defines a distortion $\mathfrak{d}^{\dagger}$ of $\mathcal{A}_*/\mathcal{B}$ called the $\dagger$-extension of $\mathfrak{d}$, and its Lipschitz equivalence class is independent of the chosen norm on $\mathbb{R}^2$. If $\mathfrak{d}$ satisfies (d2) with a function $f$, then (d2) holds for $\mathfrak{d}^{\dagger}$ with the function $f^{\dagger}$ defined by

$$ f^{\dagger}(r) = \begin{cases} \|(f(Qr), 2(C+1)Qr)\| & \text{if } e^{CQr} \le C+1 \\ \infty & \text{otherwise,} \end{cases} $$

for some $C > 0$, where $Q$ is given by (10.19). Notice that $f^{\dagger}$ is continuous at 0 and fixes 0; thus we only have to check the inequality of (d2). To prove it, take some $C > 0$ such that $(e^{C\mathfrak{d}(f)}, C\mathfrak{d}(f))$ is a large scale Lipschitz distortion of any $f \in \tilde{\mathcal{B}}_1$; this constant $C$ is given by the Lipschitz condition of $\mathfrak{d}$ over $\mathfrak{d}_{\text{lsl}}$ (respectively, $\mathfrak{d}_{\text{aLsl}}$) and the constant given by (10.19) with the norm used in Example 10.46 (respectively, 10.48). Take $L, M, N \in \mathcal{A}_0$, $f \in \mathcal{B}(|L|, |M|)$ and $g \in \mathcal{B}(|M|, |N|)$, and choose points $x \in L$, $y \in M$ and $z \in N$. Then

$$ \begin{align} d_N(g \circ f(x), z) &\le d_N(g \circ f(x), g(y)) + d_N(g(y), z) \tag{10.21} \\ &\le e^{C\mathfrak{d}(f)} d_M(f(x), y) + C\mathfrak{d}(f) + d_N(g(y), z). \end{align} $$

Now, let $M_0, \dots, M_n \in \mathcal{A}_0$, and take $x_i \in M_i$ and $f_i \in \mathcal{B}(|M_{i-1}|_{\mathcal{B}}, |M_i|_{\mathcal{B}})$ for $i \in \{1, \dots, n\}$. Set $r = f^{\dagger}(f_1) + \dots + f^{\dagger}(f_n)$, and let us prove by induction on $n$ that

$$ (10.22) \qquad e^{CQr} \le C+1 \implies d_{M_n}(f_n \circ \dots \circ f_1(x_0), x_n) \le 2(C+1)Qr. $$
For $n = 2$, from (10.21) and the condition of (10.22), we get

$$ 
\begin{aligned}
d_{M_1}(f_2 \circ f_1(x_0), x_n) &\le e^{CQ\mathfrak{D}^\dagger(f_1)} Q\mathfrak{D}^\dagger(f_1) + CQ\mathfrak{D}^\dagger(f_1) + Q\mathfrak{D}^\dagger(f_2) \\
&\le e^{CQr} Qr + (C+1)Qr \\
&\le 2(C+1)Qr.
\end{aligned}
$$

For $n > 2$, by (10.21), the induction hypothesis and the condition of (10.22), it follows that

$$ 
\begin{aligned}
d_{M_n}(f_n \circ \cdots \circ f_1(x_0), x_n) &\le e^{C\mathfrak{D}(f_1)} d_{M_1}(f_1(x_0), x_1) + C\mathfrak{D}(f_1) \\
&\quad + d_{M_n}(f_n \circ \cdots \circ f_2(x_2), x_n) \\
&\le e^{CQr} Q\mathfrak{D}^\dagger(f_1) + CQ\mathfrak{D}^\dagger(f_1) \\
&\quad + 2(C+1)Q(\mathfrak{D}^\dagger(f_2) + \cdots + \mathfrak{D}^\dagger(f_n)) \\
&\le (2C+1)Q\mathfrak{D}^\dagger(f_1) + 2(C+1)Q(\mathfrak{D}^\dagger(f_2) + \cdots + \mathfrak{D}^\dagger(f_n)) \\
&\le 2(C+1)Qr.
\end{aligned}
$$

This completes the proof of (10.22), which easily yields (d2) with $f^\dagger$.

For instance, with the notation of Examples 10.45–10.49, we get distortions $\mathfrak{D}^\dagger_{\text{Lip}}$ and $\mathfrak{D}^\dagger_{\text{lsl}}$ of $\text{Met}_* / \text{Set}$, $\mathfrak{D}^\dagger_{\text{aLip}}$ and $\mathfrak{D}^\dagger_{\text{alsl}}$ of $\text{Path}_* / \text{Set}$, and $\mathfrak{D}^\dagger_{\text{Riem}}$ of $\text{ConRiem}_* / \text{Man}^\infty$.

With the notation of Examples 10.50 and 10.51, a distortion $\mathfrak{D}$ of a u-functor over $\text{Met}_* / \text{Set}$ is said to be of \*-Lipschitz type (respectively, \*-large scale Lipschitz type, $\dagger$-Lipschitz type or $\dagger$-large scale Lipschitz type) if it is Lipschitz over $\mathfrak{D}^*_{\text{Lip}}$ (respectively, $\mathfrak{D}^*_{\text{lsl}}$, $\mathfrak{D}^\dagger_{\text{Lip}}$ or $\mathfrak{D}^\dagger_{\text{lsl}}$); a distortion $\mathfrak{D}$ of a u-functor over $\text{Path}_* / \text{Set}$ is said to be of \*-arc-wise Lipschitz type (respectively, \*-arc-wise large scale Lipschitz type, $\dagger$-arc-wise Lipschitz type or $\dagger$-arc-wise large scale Lipschitz type) if it is Lipschitz over $\mathfrak{D}^*_{\text{aLip}}$ (respectively, $\mathfrak{D}^*_{\text{alsl}}$, $\mathfrak{D}^\dagger_{\text{aLip}}$ or $\mathfrak{D}^\dagger_{\text{alsl}}$); and a distortion $\mathfrak{D}$ of a u-functor over $\text{ConRiem}_* / \text{Man}^\infty$ is said to be of \*-Riemannian type (respectively, $\dagger$-Riemannian type) if it is Lipschitz over $\mathfrak{D}^*_{\text{Riem}}$ (respectively, $\mathfrak{D}^\dagger_{\text{Riem}}$). These conditions are independent of the choice of the norm in Example 10.51. The \*-extension (respectively, $\dagger$-extension) of any distortion of (arc-wise) large scale Lipschitz type is of \*- (arc-wise) large scale Lipschitz type (respectively, $\dagger$- (arc-wise) large scale Lipschitz type). Any distortion of \*- (arc-wise) large scale Lipschitz type is of $\dagger$- (arc-wise) large scale Lipschitz type. Also, any distortion of \*-Riemannian type is of $\dagger$-Riemannian type.

**Example 10.52** (The partial map extension of a distortion). Let $\mathcal{A}/\mathcal{B}$ be a u-functor over $\text{Met}_* / \text{Set}$ (respectively, $\text{Path}_* / \text{Set}$), and let $\Xi$ be good class of sub-objects in $\mathcal{B}$. Then the ParSet-category $\text{Par} = \text{Par}(\mathcal{B}, \Xi)$ is defined according to Example 10.21. Assume that, for every $M \in \mathcal{A}_0$ with distinguished point $x$, each $B \in \Xi_{|M|}$ containing $x$ underlies some sub-object of $M$; this sub-object is unique because it is determined by its underlying set. Let $\mathfrak{D}$ be a distortion of $\mathcal{A}/\mathcal{B}$ of $\dagger$-large scale Lipschitz type (respectively, $\dagger$-arc-wise large scale Lipschitz type). Let $M, N \in \mathcal{A}_0$, with respective distinguished points $x$ and $y$, and let $f \in \widetilde{\text{Par}}(M, N)$. If $x \in \text{dom } f$, then $\text{dom } f$ underlies a unique sub-object of $M$, also denoted
by dom $f$, and $f \in \mathcal{B}(|\text{dom } f|_{\mathcal{B}}, |N|_{\mathcal{B}})$; so we can consider $\mathfrak{d}(f) = \mathfrak{d}_{\text{dom } f, N}(f)$. Let

$$R(f) = \sup\{r \ge 0 \mid B_M(x, r) \subset \text{dom } f\}$$

when $x \in \text{dom } f$. Then fix any norm $\|\cdot\|$ on $\mathbb{R}^2$, and let

$$\mathfrak{d}_{M,N}^{\ddagger}(f) = \begin{cases} \|(\mathfrak{d}(f), 1/R(f))\| & \text{if } x \in \text{dom } f \text{ and } \mathfrak{d}(f) < \infty \\ \infty & \text{otherwise.} \end{cases}$$

This defines a distortion $\mathfrak{d}^{\ddagger}$ of $\mathcal{A}/\text{Par}$, which is called the *partial map extension* or $\ddagger$-extension of $\mathfrak{d}$. To check it, suppose that $\mathfrak{d}$ satisfies (d2) with a function $f$, and let us show that (d2) holds for $\mathfrak{d}^{\ddagger}$ with a function $f^{\ddagger} : [0, \infty] \to [0, \infty]$ defined by

$$f^{\ddagger}(r) = \begin{cases} \|(f(Qr), Qr)\| & \text{if } Q\sqrt{2C}r \le 1 \text{ and } e^{CQr} + 2C(Qr)^2 \le 1 \\ \infty & \text{otherwise,} \end{cases}$$

for some $C > 0$, where $Q > 0$ is defined by (10.19) with the given norm. Notice that $f^{\ddagger}$ is continuous at 0 and fixes 0, and thus we only have to prove the inequality of (d2). By the conditions on $\mathfrak{d}$, there is some $C > 0$ such that, for any $f$ as above, $d_N(f(x), y) \le C\mathfrak{d}(f)$, and $(e^{C\mathfrak{d}(f)}, C\mathfrak{d}(f))$ is a large scale Lipschitz distortion of $f$ when $\mathfrak{d}(f) < \infty$; this $C$ depends on the Lipschitz condition of $\mathfrak{d}$ over $\mathfrak{d}_{\text{sl}}^{\ddagger}$ (respectively, $\mathfrak{d}_{\text{alsl}}^{\ddagger}$), and on the constant (10.19) for the norms used in Exemples 10.46 (respectively, 10.48) and 10.51. Now, take $L, M, N \in \mathcal{A}_0$ with corresponding distinguished points $x, y$ and $z$, and consider morphisms $f \in \text{Par}(|L|_{\text{Par}}, |M|_{\text{Par}})$ and $g \in \text{Par}(|M|_{\text{Par}}, |N|_{\text{Par}})$ such that $x \in \text{dom } f$ and $y \in \text{dom } g$. For all $x' \in \text{dom } f$, we have

$$\begin{aligned} d_M(f(x'), y) &\le d_M(f(x'), f(x)) + d_M(f(x), y) \\ &\le e^{C\mathfrak{d}(f)} d_L(x', x) + 2C\mathfrak{d}(f), \end{aligned}$$

yielding

$$d_L(x', x) < e^{-C\mathfrak{d}(f)}(R(g) - 2C\mathfrak{d}(f)) \implies d_M(f(x'), y) < R(g).$$

It follows that, if $2C\mathfrak{d}(f) < R(g)$, then

$$\text{dom } f \cap B_L(x, e^{-C\mathfrak{d}(f)}(R(g) - 2C\mathfrak{d}(f))) \subset \text{dom}(g \circ f),$$

obtaining

$$R(g \circ f) \ge \min\{R(f), e^{-C\mathfrak{d}(f)}(R(g) - 2C\mathfrak{d}(f))\}.$$

Therefore

$$(10.23) \qquad 2C\mathfrak{d}(f) < R(g) \implies \frac{1}{R(g \circ f)} \le \max \left\{ \frac{1}{R(f)}, \frac{e^{C\mathfrak{d}(f)}}{R(g) - 2C\mathfrak{d}(f)} \right\}.$$

Take $M_0, \dots, M_n \in \mathcal{A}_0$, and let $f_i \in \text{Par}(|M_{i-1}|_{\text{Par}}, |M_i|_{\text{Par}})$ for each $i \in \{1, \dots, n\}$. For $r = \mathfrak{d}^{\ddagger}(f_1) + \dots + \mathfrak{d}^{\ddagger}(f_n)$, we have

$$(10.24) \qquad \mathfrak{d}(f_1) + \dots + \mathfrak{d}(f_n) \le Qr, \qquad \frac{1}{R(f_1)} + \dots + \frac{1}{R(f_n)} \le Qr.$$
Let us prove by induction on $n$ that

$$ (10.25) \qquad Q\sqrt{2C}r, e^{CQr} + 2C(Qr)^2 \le 1 \implies \frac{1}{R(f_n \circ \cdots \circ f_1)} \le Qr. $$

For $n = 2$, $f_1$ and $f_2$ satisfy the condition of (10.23) by the first condition of (10.24). So, from (10.23), (10.24) and the conditions of (10.25), we get

$$ \begin{aligned} \frac{1}{R(f_2 \circ f_1)} &\le \max \left\{ Qr, \frac{e^{CQr}}{\frac{1}{Qr} - 2CQr} \right\} \\ &\le \max \left\{ Qr, \frac{1 - 2C(Qr)^2}{\frac{1}{Qr} - 2CQr} \right\} \\ &= Qr. \end{aligned} $$

For $n > 2$, by induction and using (10.23) again, we get

$$ \begin{aligned} \frac{1}{R(f_n \circ \cdots \circ f_1)} &\le \max \left\{ \frac{1}{R(f_1)}, \frac{e^{C\mathfrak{D}(f_1)}}{R(f_n \circ \cdots \circ f_2) - 2C\mathfrak{D}(f_1)} \right\} \\ &\le \max \left\{ Qr, \frac{e^{CQr}}{\frac{1}{Qr} - 2CQr} \right\}, \end{aligned} $$

which is $\le Qr$ by the conditions of (10.25), as we saw. This finishes the proof of (10.25), from which (d2) follows easily with $f^\ddagger$.

**Example 10.53** ($\dagger$- and $\ddagger$-extensions of a distorsion). Let $\mathcal{A}/\mathcal{B}$ be a u-functor over $\mathrm{Met}/\mathrm{Set}$ (respectively, $\mathrm{Path}/\mathrm{Set}$), let $\mathfrak{D}$ be a distortion of $\mathcal{A}/\mathcal{B}$ of large scale Lipschitz type (respectively, arc-wise large scale Lipschitz type), and let $\Xi$ be a good class of sub-objects of $\mathcal{B}$ such that any $B \in \Xi_{|M|}$ underlies some sub-object of $M$ in $\mathcal{A}$ for all $M \in \mathcal{A}_0$, and therefore $B$ also underlies some sub-object of $(M, x)$ in $\mathcal{A}_*$ for all $x \in B$. Then, according to Examples 10.50–10.52, $\mathfrak{D}$ has extensions $\mathfrak{D}^{*\dagger}$ and $\mathfrak{D}^{\ddagger}$ to $\mathcal{A}_*/\mathrm{Par}(\mathcal{A}, \Xi)$, respectively called $*\dagger$-extension and $\ddagger$-extension. For instance, we get distortions $\mathfrak{D}_{\mathrm{Lip}}^{*\dagger}$, $\mathfrak{D}_{\mathrm{Isl}}^{*\dagger}$, $\mathfrak{D}_{\mathrm{Lip}}^{\ddagger}$ and $\mathfrak{D}_{\mathrm{Isl}}^{\ddagger}$ of $\mathrm{Met}_*/\mathrm{ParSet}$; $\mathfrak{D}_{\mathrm{aLip}}^{*\ddagger}$ and $\mathfrak{D}_{\mathrm{aLip}}^{\ddagger}$ of $\mathrm{Path}_*/\mathrm{ParSet}$; and $\mathfrak{D}_{\mathrm{Riem}}^{*\ddagger}$ and $\mathfrak{D}_{\mathrm{Riem}}^{\ddagger}$ of $\mathrm{ConRiem}_*/\mathrm{LocMan}^\infty$.

By using the distortions of Example 10.53, the terminology $*\dagger$-Lipschitz type, $*\ddagger$-large scale Lipschitz type, etc. can be introduced as above. The $*\dagger$-extension (respectively, $\ddagger$-extension) of any distortion of (arc-wise) large scale Lipschitz type is of $*\dagger$-(arc-wise) large scale Lipschitz type (respectively, $\ddagger$-(arc-wise) large scale Lipschitz type). Any distortion of $*\dagger$-(arc-wise) large scale Lipschitz type is of $\ddagger$-(arc-wise) large scale Lipschitz type, any distortion of $*\ddagger$-Riemannian type is of $\ddagger$-Riemannian type, and any distortion of $*\ddagger$-Riemannian type (respectively, $\ddagger$-Riemannian type) is of $*\ddagger$-arc-wise Lipschitz type (respectively, $\ddagger$-arc-wise Lipschitz type).

**Example 10.54** (Distortions of u-functors defined by compactifications). ***
## 10.6 Crossings

Let $\mathcal{D}$ be a category, and consider the category $\mathcal{P}_{\mathcal{D}}$ given in Example 10.4. The following concept will be used.

**Definition 10.55.** A crossing $\times$ of $\mathcal{D}$ is stp-map $\mathcal{P}_{\mathcal{D}} \to \mathcal{P}_{\mathcal{D}}$, denoted by $S \mapsto S^{\times}$, such that

(c1) $\emptyset^{\times} = \emptyset$;

(c2) $S \subset S^{\times}$; and

(c3) $S \subset T \Rightarrow S^{\times} \subset T^{\times}$.

A crossing $\times$ of $\mathcal{D}$ is called a *pairing* if $\{\sigma\}^{\times} = \{\sigma\}$ for all $\sigma \in \mathcal{D}_1$ and $S^{\times} = \bigcup_{\sigma, \tau \in S} \{\sigma, \tau\}^{\times}$ for all $S \in \mathcal{P}_{\mathcal{D},1}$. It is said that $\times$ is *functorial* if it is a functor, and it is said that $\times$ is *semi-functorial* if $(TS)^{\times} \subset T^{\times}S^{\times}$ for all $(S, T) \in \mathcal{P}_{\mathcal{D},2}$.

When $\mathcal{D}$ is involutive, we can consider an involutive crossing $\times$ of $\mathcal{D}$ with respect to the induced involution on $\mathcal{P}_{\mathcal{D}}$; i.e., $\overline{S^{\times}} = \overline{S^{\times}}$ for all $S \in \mathcal{P}_{\mathcal{D},1}$.

**Remarks.** (i) A crossing of $\mathcal{D}$ induces the identity map on $\mathcal{P}_{\mathcal{D},0} = \mathcal{D}_0$ by (c2).

(ii) *Semi-involutive crossings* could be defined by requiring either $\overline{S^{\times}} \subset \overline{S^{\times}}$ or $\overline{S^{\times}} \subset \overline{S^{\times}}$, but they would be involutive with any of these conditions; e.g., the second one yields

$$ \overline{S^{\times}} = \overline{\overline{S}^{\times}} \subset \overline{\overline{S}^{\times}} = \overline{S^{\times}}. $$

**Example 10.56 (Trivial crossing).** Define the *trivial crossing* on any category by $S^{\times} = S$.

**Example 10.57 (Canonical pairing of $\mathcal{B}^{\text{in}}$).** For any category $\mathcal{B}$, its involutive category $\mathcal{B}^{\text{in}}$ has a canonical functorial involutive pairing defined by

$$ S^{\times} = \{(f, g') | (f, g), (f', g') \in S \text{ for some } f', g' \in \mathcal{B}_1\}. $$

This pairing satisfies $S^{\times\times} = S^{\times}$.

**Example 10.58 (Fibered product of crossings).** For functors

$$ (10.26) \qquad \mathcal{D} \xrightarrow{F} \mathcal{F} \xleftarrow{F'} \mathcal{D}', $$

let $\mathcal{D}'' = \mathcal{D} \times_{\mathcal{F}} \mathcal{D}'$, and let $\pi : \mathcal{D}'' \to \mathcal{D}$ and $\pi' : \mathcal{D}'' \to \mathcal{D}'$ denote the canonical functors. Suppose that $\mathcal{D}$ and $\mathcal{D}'$ are endowed with respective crossings $\times$ and $\times'$. Then the *fibered product* of $\times$ and $\times'$ is the crossing $\times''$ on $\mathcal{D}''$ defined by

$$ S^{\times''} = (\pi(S)^{\times} \times \pi'(S)^{\times'}) \cap \mathcal{D}'_1 $$

for $S \in \mathcal{P}_{\mathcal{D}''1}$.
The crossing $\times$ of $\mathcal{D}$ induces another crossing of $\mathcal{D}_{\text{stp}}^j$, also denoted by $\times$, which is defined as follows. For $F$, $F'$ and $S$ as above, $S^\times$ is the set of transformations $t: F \to F'$ satisfying

$$ (10.30) \qquad \{F'(f) t_i, t_j F(f)\}^\times \subset \bigcup_{u,v \in S} \{F'(f) u_i, v_j F(f)\}^\times $$

for all $i, j \in \mathcal{I}_0$ and all $f \in \mathcal{I}(i, j)$. When $\mathcal{D}$ is involutive, the crossing $\times$ of $\mathcal{D}_{\text{stp}}^j$ may not be involutive even when the crossing $\times$ of $\mathcal{D}$ is involutive. In that case, an involutive crossing $\times'$ can be defined by setting $S^{\times'}$ equal to the set of transformations $t: F \to F'$ satisfying (10.30) and

$$ \{t_i \overline{F(f)}, \overline{F'(f)} t_j\}^\times \subset \bigcup_{u,v \in S} \{u_i \overline{F(f)}, \overline{F'(f)} v_j\}^\times $$

for all $i, j \in \mathcal{I}_0$ and all $f \in \mathcal{I}(i, j)$. The crossings $\times$ and $\times'$ of $\mathcal{D}_{\text{stp}}^j$ may not be pairings or functorial even when the crossing $\times$ of $\mathcal{D}$ is a pairing or functorial.

**Example 10.60 (Lift of a crossing).** For any u-functor $\mathcal{C}/\mathcal{D}$, any crossing of $\mathcal{D}$ induces a crossing of $\tilde{\mathcal{D}}$ by using the identity $\tilde{\mathcal{D}}(M, N) = \mathcal{D}(|M|, |N|)$ for all $M, N \in \mathcal{C}_0$. This crossing of $\tilde{\mathcal{D}}$ will be called the *lift* of the crossing of $\mathcal{D}$. If the crossing of $\mathcal{D}$ is a pairing, functorial or involutive, then the corresponding crossing of $\tilde{\mathcal{D}}$ is also a pairing, functorial or involutive, respectively.

**Example 10.61.** For any category $\mathcal{D}$, the category $\mathcal{P}_{\mathcal{D}}$ has a functorial crossing $\times$ defined as follows. For each $S \in \mathcal{P}_{\mathcal{D},1}$, $S^\times$ is the family of unions of elements of $S$. If $\mathcal{D}$ is involutive, then $\times$ is involutive with respect to the induced involution of $\mathcal{P}_{\mathcal{D}}$. Moreover $S^{\times\times} = S^\times$.

**Example 10.62 (Pairing induced by a crossing).** Any crossing $\times$ of a category $\mathcal{D}$ induces a pairing $\times_p$ of $\mathcal{D}$ defined by $\emptyset^{\times_p} = \emptyset$, $\{f\}^{\times_p} = \{f\}$ for any $f \in \mathcal{D}_1$, and

$$ S^{\times_p} = \bigcup_{f,g \in S} \{f,g\}^\times $$

for any $S \in \mathcal{P}_{\tilde{\mathcal{D}},1}$. If $\times$ is functorial (respectively, involutive), then $\times_p$ is also functorial (respectively, involutive).

## 10.7 Bidistortions

Let $\mathcal{C}/\mathcal{D}$ be an involutive u-functor. Suppose that $\mathcal{D}$ is endowed with a crossing $\times$, and consider its lift to $\tilde{\mathcal{D}}$ (Example 10.60).

**Definition 10.63.** A distortion $\mathfrak{b}$ of $\mathcal{C}/\mathcal{D}$ is called a *bidistortion* when the following properties are satisfied:

(b1) For sets $S \in \mathcal{P}_{\tilde{\mathcal{D}},1}$, 
$$ \sup \mathfrak{b}(\overline{S}^\times) \le \sup \mathfrak{b}(S^\times). $$
(b2) For finite sequences $(\sigma_1, \dots, \sigma_n)$ of parallel morphisms of $\tilde{\mathcal{D}}$ with arbitrary length $n$,
$$ \sup \mathfrak{b}(\{\sigma_1, \sigma_n\}^\times) \preccurlyeq \sup \mathfrak{b}(\{\sigma_1, \sigma_2\}^\times) \cup \{\sigma_2, \sigma_3\}^\times \cup \dots \cup \{\sigma_{n-1}, \sigma_n\}^\times). $$

(b3) For morphisms $\sigma \in \tilde{\mathcal{D}}_1$,
$$ \sup \mathfrak{b}(\{\bar{\sigma}\sigma, \mathrm{id}\}^\times) \preccurlyeq \mathfrak{b}(\sigma). $$

(b4) For morphisms $\sigma \in \tilde{\mathcal{D}}_1$,
$$ \sup \mathfrak{b}(\{\sigma\}^\times) \preccurlyeq \mathfrak{b}(\sigma). $$

(b5) For sets $S \in \mathcal{P}_{\tilde{\mathcal{D}},1}$,
$$ \sup \mathfrak{b}(S^\times) \preccurlyeq \sup \bigcup_{\sigma, \tau \in S} \mathfrak{b}(\{\sigma, \tau\}^\times). $$

(b6) For pairs $(S, T) \in \mathcal{P}_{\tilde{\mathcal{D}},2}$,
$$ \sup \mathfrak{b}((TS)^\times) \preccurlyeq \sup \mathfrak{b}(T^\times S^\times). $$

The pair $(\mathcal{C}/\mathcal{D}, \mathfrak{b})$ is called a *bidistortion functor*.

**Remarks.** (i) In (b4) and (b5), the reverse relation holds by (c3). In (b1), the reverse relation also holds:

$$ \sup \mathfrak{b}(S^\times) = \sup \mathfrak{b}(\overline{S}^\times) \preccurlyeq \sup \mathfrak{b}(\overline{S}^\times). $$

(ii) By (b4) and (b1),

$$ (10.31) \qquad \mathfrak{b}(\sigma) \approx \mathfrak{b}(\bar{\sigma}) $$

$$ \text{for } \sigma \in \tilde{\mathcal{D}}_1. \text{ So} $$

for sets $S \in \mathcal{P}_{\tilde{\mathcal{D}},1}$ by (b1).

(iii) For fixed $n$, the relation of (b2) is equivalent to

$$ \mathfrak{b}(\{\sigma_1, \sigma_n\}^\times) \preccurlyeq \sup \mathfrak{b}(\{\sigma_1, \sigma_2\}^\times) + \sup \mathfrak{b}(\{\sigma_2, \sigma_3\}^\times) + \dots + \sup \mathfrak{b}(\{\sigma_{n-1}, \sigma_n\}^\times). $$

But this relation is weaker when $n$ is arbitrary.

(iv) Observe that (b2) and (b5) yield

$$ \sup \mathfrak{b}((S_1 \cup \dots \cup S_n)^\times) \preccurlyeq \sup \mathfrak{b}(S_1^\times \cup \dots \cup S_n^\times) $$

for finite sequences $(S_1, \dots, S_n)$ in $\mathcal{P}_{\tilde{\mathcal{D}},1}$, with arbitrary length $n$, such that $S_i \cap S_{i+1} \neq \emptyset$ for all $i \in \{1, \dots, n-1\}$.
(v) The conditions (b2) and (b5) also yield

$$ \sup_{i} \mathfrak{b}((\bigcup_{i} S_{i})^{\times}) \preceq \sup_{i} \mathfrak{b}(\bigcup_{i} S_{i}^{\times}) $$

for any family {$S_i$} $\subset \mathcal{P}_{\tilde{D},1}$ with non-empty intersection.

(vi) If $\times$ is a pairing, then (b4) and (b5) are obvious. If $\times$ is semi-functorial, then (b6) is obvious. If $\times$ and $\mathfrak{b}$ are involutive, then (b1) is obvious; in fact, if $\times$ is involutive and $\mathfrak{b}$ satisfies (10.31), then (b1) follows. If $\times$ is trivial, then (b2) and (b4)-(b6) are obvious, and (b3) follows from (d2) and (b1).

(vii) By (b3) and (d1), if any $S \in \mathcal{P}_{\mathcal{C},1}$ with $\mathfrak{b}(S^\times) = \{0\}$ has at most one element, then $\mathcal{C}$ is a groupoid with the involution given by inversion.

(viii) In (b5), the relation is obvious if $S$ has one element, and it is enough to consider only the sets {$\sigma, \tau$}^x with $\sigma \neq \tau$ if $S$ has more than one element by (c3).

Let $(\mathcal{C}/\mathcal{D}, \mathfrak{b})$ be a bidistortion functor. For any $S \in \mathcal{P}_{\tilde{D},1}$, if $\sup_{i} \mathfrak{b}(S)$ is small, consider $\sup_{i} \mathfrak{b}(S^\times)$ as some kind of “approximate diameter” of $S$; this idea will be justified by Examples 10.102 and 10.103 below. Observe that the “approximate diameter” of $\emptyset$ is zero by (c1), and larger sets have larger “approximate diameter” by (c3). In particular, if $\mathfrak{b}(\sigma)$ and $\mathfrak{b}(\tau)$ are small for some parallel morphisms $\sigma$ and $\tau$ of $\tilde{D}$, then consider $\sup_{i} \mathfrak{b}(\{\sigma, \tau\}^\times)$ as some kind of “approximate distance” between them. By (b1), this “approximate diameter” or “approximate distance” is “almost invariant” by involution. By (b5) and (d2), the condition of having “small approximate diameter” is preserved by the operation of $\mathcal{P}_{\tilde{D}}$.

According to the above idea, (b2) looks like some version of the characteristic property of ultrametrics stronger than the triangle inequality for parallel morphisms of $\tilde{D}$. In the same sense, (b3) means that $\bar{\sigma}$ is “almost inverse” of $\sigma$ when $\mathfrak{b}(\sigma)$ is small. The conditions (b4) and (b5) mean that $\mathfrak{b}$ can be “approximately used” as if $\times$ were a pairing, and (b6) means that $\mathfrak{b}$ can be “approximately used” as if $\times$ were semi-functorial. According to the (10.31), $\mathfrak{b}$ can be also “approximately used” as if $\times$ were involutive.

The following notation will be used for the sake of simplicity. When there is no possibility of confusion, for any $S \in \mathcal{P}_{\tilde{D},1}$, $\sup_{i} \mathfrak{b}(S)$ may be simply denoted by $S$. Similarly, for any morphism $\sigma$ of $\tilde{D}$, $\mathfrak{b}(\sigma)$ may be simply denoted by $\sigma$.

**Lemma 10.64.** For sets $S \in \mathcal{P}_{\tilde{D},1}$ and morphisms $\sigma, \tau \in \tilde{D}_1$ such that $\tau S \sigma$ is defined,

$$ (\tau S \sigma)^{\times} \preceq S^{\times} \cup \{\sigma, \tau\}. $$

*Proof.* By (b6), (d2) and (b4), we get

$$ (\tau S \sigma)^{\times} \preceq \{\tau\}^{\times} S^{\times} \{\sigma\}^{\times} \preceq S^{\times} \cup \{\sigma\}^{\times} \cup \{\tau\}^{\times} \preceq S^{\times} \cup \{\sigma, \tau\}. \quad \square $$

**Corollary 10.65.** For parallel morphisms $\sigma, \tau \in \tilde{D}_1$,

$$ \{\bar{\tau}\sigma, \mathrm{id}\}^{\times} \preceq \{\sigma, \tau\}^{\times}. $$
*Proof.* By (b2), Lemma 10.64, (10.31) and (c2),

$$ \{\bar{\tau}\sigma, \mathrm{id}\}^\times \preceq \{\bar{\tau}\sigma, \bar{\tau}\tau\}^\times \cup \{\bar{\tau}\tau, \mathrm{id}\}^\times \preceq \{\sigma, \tau\}^\times. \quad \square $$

The relation of Corollary 10.65 means that, if the “aproximate distance” between $\sigma$ and $\tau$ is small, then the “aproximate distance” between $\bar{\tau}\sigma$ and $\mathrm{id}$ is also small.

**Lemma 10.66.** For diagrams in $\tilde{\mathcal{D}}$ of the form

<figure><img src="image_22.png" /></figure>

we have

$$ \begin{aligned} \{\gamma\bar{\sigma}, \tau\}^\times &\preceq \{\gamma, \tau\sigma\}^\times \cup \{\sigma, \tau\}, \\ \{\bar{\tau}\gamma, \sigma\}^\times &\preceq \{\gamma, \tau\sigma\}^\times \cup \{\sigma, \tau\}. \end{aligned} $$

*Proof.* The first relation of the statement follows from (b2), (b3), (10.31) and Lemma 10.64:

$$ \begin{aligned} \{\gamma\bar{\sigma}, \tau\}^\times &\preceq \{\gamma\bar{\sigma}, \tau\sigma\bar{\sigma}\}^\times \cup \{\tau\sigma\bar{\sigma}, \tau\}^\times \\ &\preceq \{\gamma, \tau\sigma\}^\times \cup \{\sigma\bar{\sigma}, \mathrm{id}_M\}^\times \cup \{\sigma, \tau\} \\ &\preceq \{\gamma, \tau\sigma\}^\times \cup \{\sigma, \tau\}. \end{aligned} $$

By applying this relation to the diagram

<figure><img src="image_23.png" /></figure>

and using (b1) and (10.31), we get

$$ \begin{aligned} \{\bar{\tau}\gamma, \sigma\}^\times &\preceq \{\bar{\tau}, \sigma\bar{\gamma}\}^\times \cup \{\bar{\gamma}, \sigma\} \\ &\approx \{\gamma\bar{\sigma}, \tau\}^\times \cup \{\gamma, \sigma\}, \end{aligned} $$

yielding the second relation of the statement by using the first one again.

**Corollary 10.67.** For diagrams in $\tilde{\mathcal{D}}$ of the form

<figure><img src="image_24.png" /></figure>
we have

$$
\begin{aligned}
\{\delta\gamma\bar{\sigma}, \tau\}^\times &\asymp \{\delta\gamma, \tau\sigma\}^\times \cup \{\sigma, \tau\}, \\
\{\gamma\bar{\sigma}, \bar{\delta}\tau\}^\times &\asymp \{\delta\gamma, \tau\sigma\}^\times \cup \{\gamma, \delta, \sigma, \tau\}.
\end{aligned}
$$

*Proof.* The first relation follows by applying the first relation of Lemma 10.66 to the diagram

<figure><img src="image_25.png" /></figure>

By using the first relation of Lemma 10.66 with the diagrams

<figure><img src="image_26.png" /></figure>

we get the relations

$$
\begin{aligned}
\{\gamma\bar{\sigma}, \bar{\delta}\tau\}^\times &\asymp \{\gamma, \bar{\delta}\tau\sigma\}^\times \cup \{\sigma, \bar{\delta}\tau\}, \\
\{\bar{\sigma}\bar{\tau}\delta, \bar{\gamma}\}^\times &\asymp \{\bar{\sigma}\bar{\tau}, \bar{\gamma}\bar{\delta}\}^\times \cup \{\bar{\sigma}\bar{\tau}, \bar{\delta}, \bar{\gamma}\},
\end{aligned}
$$

which yields the second relation of the statement by using (d2), (b1) and (10.31). $\square$

**Remark.** It is clear that Lemma 10.66 and Corollary 10.67 can be generalized to diagrams in $\tilde{D}$ with the shape of any polygon.

**Lemma 10.68.** For diagrams in $\tilde{D}$ of the form

<figure><img src="image_27.png" /></figure>

with arbitrary length $n$,

$$
\begin{aligned}
\{\gamma_n\sigma_n \dots \sigma_1, \tau_n \dots \tau_1\gamma_0\}^\times &\asymp \{\gamma_1\sigma_1, \tau_1\gamma_0\}^\times \cup \dots \cup \{\gamma_n\sigma_n, \tau_n\gamma_{n-1}\}^\times \\
&\phantom{\asymp} \cup \{\sigma_1 + \dots + \sigma_n, \tau_1 + \dots + \tau_n\}.
\end{aligned}
$$

*Proof.* By (b2),

$$
\begin{aligned}
\{\gamma_n \sigma_n \dots \sigma_1, \tau_n \dots \tau_1 \gamma_0\}^\times &\asymp \{\gamma_n \sigma_n \dots \sigma_1, \tau_n \gamma_{n-1} \sigma_{n-1} \dots \sigma_1\}^\times \\
&\phantom{\asymp} \cup \{\tau_n \gamma_{n-1} \sigma_{n-1} \dots \sigma_1, \tau_n \tau_{n-1} \gamma_{n-2} \sigma_{n-2} \dots \sigma_1\}^\times \\
&\phantom{\asymp} \cup \dots \cup \{\tau_n \dots \tau_2 \gamma_1 \sigma_1, \tau_n \dots \tau_1 \gamma_0\}^\times
\end{aligned}
$$

Then the result follows from Lemma 10.64 and (d2). $\square$
**Remark.** If $n$ is fixed, then the relation of Lemma 10.68 is equivalent to

$$ \{\gamma_n\sigma_n \dots \sigma_1, \tau_n \dots \tau_1\gamma_0\}^\times \preceq \{\gamma_1\sigma_1, \tau_1\gamma_0\}^\times \cup \dots \cup \{\gamma_n\sigma_n, \tau_n\gamma_{n-1}\}^\times \\ \cup \{\sigma_1, \dots, \sigma_n, \tau_1, \dots, \tau_n\}. $$

But this relation is stronger for arbitrary $n$.

## 10.8 A-functors and a-natural transformations

Let $F: \mathcal{J} \to \tilde{\mathcal{D}}$ be a weakly filtered stp-map (see Section 10.1).

**Definition 10.69.** It is said that:

(i) $F$ is *Cauchy* if

$$ \lim_{i,j \to \infty} F(\mathcal{J}(i,j)) = 0. $$

(ii) $F$ is *asymptotically commutative* (or simply *a-commutative*) if

$$ \lim_{i,j \to \infty} F(\mathcal{J}(i,j))^\times = 0. $$

(iii) $F$ is an *asymptotic functor* (or simply an *a-functor*) if

$$ \lim_{i,j,k \to \infty} \bigcup_{f \in \mathcal{J}(i,j), g \in \mathcal{J}(j,k)} \{F(gf), F(g)F(f)\}^\times = 0. $$

In this case, a *sub-a-functor* of $F$ is an *a-functor* which is a *sub-stp-map* of $F$.

**Remark.** If $F$ is a-commutative or a-functor, then it is Cauchy by (c2) and (c3).

The conditions of Definition 10.69 have the following intuitive interpretations. By (d1), $F$ is Cauchy when $F(\mathcal{J})$ is “almost contained” in $\mathcal{C}$ at the “infinity” of $\mathcal{J}_0$. $F$ is a-commutative when it is Cauchy and becomes “almost commutative” at the “infinity” of $\mathcal{J}_0$. Finally, $F$ is an a-functor when it is Cauchy and “almost becomes” a functor at the “infinity” of $\mathcal{J}_0$.

The properties of being a-functor, Cauchy and a-commutative are inherited by taking cofinal weakly filtered sub-stp-maps. Thus any cofinal weakly filtered sub-stp-map of an a-functor is a sub-a-functor.

**Lemma 10.70.** *F is an a-commutative a-functor if and only if*

$$ \lim_{i^* \to \infty} \bigcup_{i^* \le i, i' \atop i^* \le i''} \left( \bigcup_{i^* \le i, i' \atop i^* \le i''} (F(i'', i') F(i, i'') \cup \overline{F(i', i'')} F(i, i'') \cup F(i'', i') \overline{F(i'', i')}) \right)^\times = 0. $$

*Proof.* The “if” part follows by (c3). The “only if” part follows from (b5) and the following assertions.
**Claim 24.** For $i \le i_1'', i_2'' \le i'$ in $\mathcal{I}_0$, and for $f_1 \in \mathcal{I}(i, i_1'')$, $g_1 \in \mathcal{I}(i_1'', i')$, $f_2 \in \mathcal{I}(i, i_1'')$ and $g_2 \in \mathcal{I}(i_1'', i')$, we have $\{F(g_1) F(f_1), F(g_2) F(f_2)\}^\times \to 0$ as $i \to \infty$ uniformly on $i', i_1'', i_2''$, $f_1$, $g_1$, $f_2$ and $g_2$.

By (b2),

$$ \{F(g_1) F(f_1), F(g_2) F(f_2)\}^\times \\ \le \{F(g_1) F(f_1), F(g_1 f_1)\}^\times \cup \{F(g_1 f_1), F(g_2 f_2)\}^\times \cup \{F(g_2 f_2), F(g_2) F(f_2)\}^\times, $$

showing Claim 24 because $F$ is an a-commutative a-functor.

**Claim 25.** For $i, i' \le i_1'', i_2''$ in $\mathcal{I}_0$, and for $f_1 \in \mathcal{I}(i, i_1'')$, $f_2 \in \mathcal{I}(i, i_2'')$, $h_1 \in \mathcal{I}(i', i_1'')$ and $h_2 \in \mathcal{I}(i', i_2'')$, we have $\{\overline{F(h_1)} F(f_1), \overline{F(h_2)} F(f_2)\}^\times \to 0$ as $i, i' \to \infty$ uniformly on $i_1'', i_2''$, $f_1$, $h_1$, $f_2$ and $h_2$.

Take some $i'' \in \mathcal{I}_0$ such that $i_1'', i_2'' \le i''$. Then there are morphisms $e_1 \in \mathcal{I}(i_1'', i'')$ and $e_2 \in \mathcal{I}(i_2'', i'')$. By (b2), Lemma 10.64, (b1) and (10.31),

$$ \begin{aligned} \{\overline{F(h_1)} F(f_1), \overline{F(h_2)} F(f_2)\}^\times &\le \{\overline{F(h_1)} F(f_1), \overline{F(h_1)} \overline{F(e_1)} F(e_1) F(f_1)\}^\times \\ &\quad \cup \{\overline{F(h_1)} \overline{F(e_1)} F(e_1) F(f_1), \overline{F(h_1)} \overline{F(e_1)} F(e_2) F(f_2)\}^\times \\ &\quad \cup \{\overline{F(h_1)} \overline{F(e_1)} F(e_2) F(f_2), \overline{F(h_2)} \overline{F(e_2)} F(e_2) F(f_2)\}^\times \\ &\quad \cup \{\overline{F(h_2)} \overline{F(e_2)} F(e_2) F(f_2), \overline{F(h_2)} F(f_2)\}^\times \\ &\le \{\text{id}, \overline{F(e_1)} F(e_1)\}^\times \cup \{F(e_1) F(f_1), F(e_2) F(f_2)\}^\times \\ &\quad \cup \{\overline{F(h_1)} \overline{F(e_1)}, \overline{F(h_2)} \overline{F(e_2)}\}^\times \cup \{\overline{F(e_2)} F(e_2), \text{id}\}^\times \\ &\quad \cup \{F(e_1), F(e_2), F(f_1), F(f_2), F(h_1), F(h_2)\} \\ &\approx \{\text{id}, \overline{F(e_1)} F(e_1)\}^\times \cup \{F(e_1) F(f_1), F(e_2) F(f_2)\}^\times \\ &\quad \cup \{F(e_1) F(h_1), F(e_2) F(h_2)\}^\times \cup \{\overline{F(e_2)} F(e_2), \text{id}\}^\times \\ &\quad \cup \{F(e_1), F(e_2), F(f_1), F(f_2), F(h_1), F(h_2)\}, \end{aligned} $$

and Claim 25 follows by (b3) and Claim 24, and since $F$ is Cauchy.

**Claim 26.** For $i_1'', i_2'' \le i, i' \in \mathcal{I}_0$, and for $k_1 \in \mathcal{I}(i_1'', i)$, $k_2 \in \mathcal{I}(i_2'', i)$, $g_1 \in \mathcal{I}(i_1'', i')$ and $g_2 \in \mathcal{I}(i_2'', i')$, we have $\{F(g_1) \overline{F(k_1)}, F(g_2) \overline{F(k_2)}\}^\times \to 0$ as $i_1'', i_2'' \to \infty$ uniformly on $i, i', k_1$, $g_1$, $k_2$ and $g_2$.

By Corollary 10.67 and (10.31),

$$ \{F(g_1) \overline{F(k_1)}, F(g_2) \overline{F(k_2)}\}^\times \\ \le \{\overline{F(g_2)} F(g_1), \overline{F(k_2)} F(k_1)\}^\times \cup \{F(g_1), F(g_2), F(k_1), F(k_2)\}, $$

yielding Claim 26 by Claim 25.
**Claim 27.** For $i \le i_1'' \le i' \le i_2''$ in $\mathcal{J}_0$, and for $f_1 \in \mathcal{J}(i, i_1'')$, $g \in \mathcal{J}(i_1'', i')$, $f_2 \in \mathcal{J}(i, i_2'')$ and $h \in \mathcal{J}(i', i_2'')$, we have $\{F(g) F(f_1), \overline{F(h)} F(f_2)\}^\times \to 0$ as $i \to \infty$ uniformly on $i'$, $i_1''$, $i_2''$, $f_1$, $g$, $f_2$ and $h$.

By (b2),

$$ \{F(g) F(f_1), \overline{F(h)} F(f_2)\}^\times \asymp \{F(g) F(f_1), F(g f_1)\}^\times \cup \{F(g f_1), \overline{F(h)} F(f_2)\}^\times, $$

yielding Claim 27 because $F$ is an $a$-functor and by Claim 25.

With similar arguments, we also get the following.

**Claim 28.** For $i_2'' \le i \le i_1'' \le i'$ in $\mathcal{J}_0$, and for $f \in \mathcal{J}(i, i_1'')$, $g_1 \in \mathcal{J}(i_1'', i')$, $k \in \mathcal{J}(i_2'', i)$ and $g_2 \in \mathcal{J}(i_2'', i')$, we have $\{F(g_1) F(f), F(g_2) \overline{F(k)}\}^\times \to 0$ as $i_2'' \to \infty$ uniformly on $i$, $i'$, $i_1''$, $f$, $g_1$, $k$ and $g_2$.

**Claim 29.** For $i_2'' \le i, i' \le i_1''$ in $\mathcal{J}_0$, and for $f \in \mathcal{J}(i, i_1'')$, $h \in \mathcal{J}(i', i_1'')$, $k \in \mathcal{J}(i_2'', i)$ and $g \in \mathcal{J}(i_2'', i')$, we have $\{\overline{F(h)} F(f), F(g) \overline{F(k)}\}^\times \to 0$ as $i_2'' \to \infty$ uniformly on $i$, $i'$, $i_1''$, $f$, $g$, $h$ and $k$.

By Corollary 10.67, (b1) and (10.31),

$$ \{\overline{F(h)} F(f), F(g) \overline{F(k)}\}^\times \asymp \{\overline{F(g)} \overline{F(h)}, \overline{F(k)} \overline{F(f)}\}^\times \cup \{F(f), F(g), F(h), F(k)\} \\ \approx \{F(h) F(g), F(f) F(k)\}^\times \cup \{F(f), F(g), F(h), F(k)\}, $$

which yields Claim 29 by Claim 24 and since $F$ is Cauchy. $\square$

**Lemma 10.71.** For $(f_1, \dots, f_n) \in \mathcal{J}_\infty$,

$$ \{F(f_n \cdots f_1), F(f_n) \cdots F(f_1)\}^\times \\ \asymp \{F(f_n \cdots f_1), F(f_n \cdots f_2) F(f_1)\}^\times \cup \{F(f_n \cdots f_2), F(f_n \cdots f_3) F(f_2)\}^\times \cup \cdots \\ \cdots \cup \{F(f_n f_{n-1}), F(f_n) F(f_{n-1})\}^\times \cup \{F(f_1) + \cdots + F(f_{n-2})\}. $$

*Proof.* By (b2),

$$ \{F(f_n \cdots f_1), F(f_n) \cdots F(f_1)\}^\times \asymp \{F(f_n \cdots f_1), F(f_n \cdots f_2) F(f_1)\}^\times \\ \cup \{F(f_n \cdots f_2) F(f_1), F(f_n \cdots f_3) F(f_2) F(f_1)\}^\times \cup \cdots \\ \cdots \cup \{F(f_n f_{n-1}) F(f_{n-2}) \cdots F(f_1), F(f_n) \cdots F(f_1)\}^\times. $$

Then the result follows by Lemma 10.64 and (d2). $\square$

**Corollary 10.72.** For any fixed $n \in \mathbb{N}$, if $F$ is an $a$-functor, then

$$ \lim_{i_0, \dots, i_n \to \infty} \bigcup_{f_1 \in \mathcal{J}(i_0, i_1), \dots, f_n \in \mathcal{J}(i_{n-1}, i_n)} \{F(f_n \cdots f_1), F(f_n) \cdots F(f_1)\}^\times = 0. $$

Now, consider two parallel weakly filtered stp-maps $F, F' : \mathcal{J} \to \tilde{\mathcal{D}}$, and a transformation $t : F \to F'$.
**Definition 10.73.** It is said that:

(i) $t$ is asymptotically undistorted (or simply a-undistorted) if

$$ \lim_{i \to \infty} t_i = 0. $$

(ii) $t$ is asymptotically natural (or simply a-natural) if $F$ and $F'$ are Cauchy, and

$$ \lim_{i,j \to \infty} \bigcup_{f \in \mathcal{I}(i,j)} \{F'(f) t_i, t_j, F(f)\}^\times = 0. $$

**Remark.** If $t$ is a-natural, then it is a-undistorted by (c2).

The conditions of Definition 10.73 have the following intuitive interpretations. According to (d1), $t$ is a-undistorted when $t_i$ “almost belongs” to $\mathcal{C}$ as $i \to \infty$. On the other hand, $t$ is a-natural when it is a-undistorted and the diagrams (10.10) become “almost commutative” as $i, j \to \infty$ in $\mathcal{I}_0$.

The properties of being a-undistorted and a-natural are inherited by taking cofinal weakly filtered sub-transformations. If a weakly filtered stp-map is a-commutative, then its identity transformation is a-natural by (c3). The following result follows by applying Corollary 10.67 to the diagram (10.10).

**Lemma 10.74.** If $t$ is a-natural, then

$$ \lim_{i,j \to \infty} \bigcup_{f \in \mathcal{I}(i,j)} (\{\overline{F'(f)} t_j, t_i, \overline{F(f)}\}^\times \cup \{\overline{F'(f)} t_j, F(f), t_i\}^\times \\ \cup \{F(f) \bar{t}_i, \bar{t}_j, F'(f)\}^\times \cup \{t_j, F(f) \bar{t}_i, F'(f)\}^\times) = 0. $$

## 10.9 The completion defined by a bidistortion

The completion (or b-completion) of $\mathcal{C}$ is the category $\overline{\mathcal{C}}$ defined as follows. Its objects are the small a-commutative a-functors to $\tilde{\mathcal{D}}$. Given two objects $F : \mathcal{J} \to \tilde{\mathcal{D}}$ and $F' : \mathcal{J}' \to \tilde{\mathcal{D}}$ of $\overline{\mathcal{C}}$, consider small cofinal sub-a-functors with the same source category, $A : \mathcal{J} \to \tilde{\mathcal{D}}$ of $F$ and $A' : \mathcal{J} \to \tilde{\mathcal{D}}$ of $F'$. Then any a-natural transformation $t : A \to A'$ represents a morphism $F \to F'$ of $\overline{\mathcal{C}}$. But different a-natural transformations between different small cofinal sub-a-functors may represent the same morphism; precisely, an equivalence relation is introduced as follows. Consider other small cofinal sub-a-functors, $B : \mathcal{K} \to \tilde{\mathcal{D}}$ of $F$ and $B' : \mathcal{K} \to \tilde{\mathcal{D}}$ of $F'$, and another a-natural transformation $u : B \to B'$. Let $\mathbf{a} : \mathcal{J} \to \mathcal{J}$, $\mathbf{a}' : \mathcal{J}' \to \mathcal{J}'$, $\mathbf{b} : \mathcal{K} \to \mathcal{J}$ and $\mathbf{b}' : \mathcal{K} \to \mathcal{J}'$ be cofinal stp-reparametrizations defining $A, A', B$ and $B'$, respectively.

**Definition 10.75.** It is said that $t$ and $u$ are asymptotic, denoted by $t \sim u$, if

$$ \lim_{j,k \to \infty} \bigcup_{f \in \mathcal{I}(\mathbf{a}_j, \mathbf{b}_k), f' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)} \{F'(f') t_j, u_k F(f)\}^\times = 0, $$

where $(j, k)$ runs in the subset of $\mathcal{J}_0 \times \mathcal{K}_0$ determined by the conditions $\mathbf{a}_j \le \mathbf{b}_k$ and $\mathbf{a}'_j \le \mathbf{b}'_k$.
**Remark.** By (b5) and Lemma 10.64, $t \sim u$ yields

$$ \lim_{j,k \to \infty} (F'(\mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)) t_j \cup u_k F(\mathcal{I}(\mathbf{a}_j, \mathbf{b}_k)))^\times = 0 $$

for $(j, k)$ as above.

The condition of Definition 10.75 means that the diagrams of the form

$$ \begin{tikzcd} & F_{\mathbf{a}_j} \arrow[r, "t_j"] & F'_{\mathbf{a}'_j} \arrow[d, "F(f)"'] \arrow[d, "F'(f)"'] \\ (10.32) & F_{\mathbf{b}_k} \arrow[r, "u_k"] & F'_{\mathbf{b}'_k} \arrow[d, "F(f)"'] \arrow[d, "F'(f)"'] \end{tikzcd} $$

becomes “almost commutative” as $j, k \to \infty$ uniformly on $f$ and $f'$. By applying Corollary 10.67 to the diagram (10.32), we get the following.

**Lemma 10.76.** If $t \sim u$, then, with $(j, k)$ like in Definition 10.75,

$$ \lim_{j,k \to \infty} \bigcup_{f \in \mathcal{I}(\mathbf{a}_j, \mathbf{b}_k), f' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)} (\overline{\{F'(f') u_k, t_j F(f)\}}^\times \cup \overline{\{F'(f') u_k F(f), t_j\}}^\times) = 0. $$

The set of pairs $(j, k)$ considered Definition 10.75 is cofinal in $\mathcal{I}_0 \times \mathcal{K}_0$ by the cofinality of $\mathbf{a}, \mathbf{b}, \mathbf{a}'$ and $\mathbf{b}'$. The following result shows that indeed any other cofinal subset can be used.

**Lemma 10.77.** For any cofinal subset

$$ S \subset \{(j, k) \in \mathcal{I}_0 \times \mathcal{K}_0 \mid \mathbf{a}_j \le \mathbf{b}_k, \mathbf{a}'_j \le \mathbf{b}'_k\}, $$

if the convergence of Definition 10.75 holds with $(j, k)$ running in $S$, then $t \sim u$.

*Proof.* Take $(j, k) \in \mathcal{I}_0 \times \mathcal{K}_0$ satisfying $\mathbf{a}_j \le \mathbf{b}_k$ and $\mathbf{a}'_j \le \mathbf{b}'_k$, and let $f \in \mathcal{I}(\mathbf{a}_j, \mathbf{b}_k)$ and $f' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)$. By the cofinality of $S$, there is some $(j', k') \in S$ such that $(j, k) \le (j', k')$. Thus there are morphisms $h_1 \in \mathcal{I}(j, j')$, $h_2 \in \mathcal{K}(k, k')$, $g \in \mathcal{I}(\mathbf{a}'_j, \mathbf{b}'_k)$ and $g' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)$. We get the following diagram in $\tilde{\mathcal{D}}$:
<figure><img src="image_28.png" /></figure>

By applying Lemma 10.68 and Corollary 10.67 to the appropriate parts of the above diagram, and using (b2), Lemma 10.64, (c2), (10.31) and (d2), we get

$$
\begin{align*}
& \{u_k F(f), F'(f') t_j\}^\times \\
& \approx \{u_k F(f), \overline{F'\mathbf{b}'(h_2)} u_{k'} F(g) \overline{F\mathbf{a}(h_1)}\}^\times \\
& \phantom{\approx} \cup \{\overline{F'\mathbf{b}'(h_2)} u_{k'} F(g) \overline{F\mathbf{a}(h_1)}, \overline{F'\mathbf{b}'(h_2)} F'(g') t_{j'} \overline{F\mathbf{a}(h_1)}\}^\times \\
& \phantom{\approx} \cup \{\overline{F'\mathbf{b}'(h_2)} F'(g') t_{j'} \overline{F\mathbf{a}(h_1)}, F'(f') t_j\}^\times \\
& \approx \{\overline{F'\mathbf{b}'(h_2)} u_{k'} F(g), u_k F(f) \overline{F\mathbf{a}(h_1)}\}^\times \cup \{\overline{F'\mathbf{b}'(h_2)} F'(g') t_{j'}, F'(f') t_j \overline{F\mathbf{a}(h_1)}\}^\times \\
& \phantom{\approx} \cup \{u_{k'} F(g), F'(g') t_{j'}\}^\times \cup \{u_k F(f), F'\mathbf{b}'(h_2), \overline{F\mathbf{a}(h_1)}, F'(f') t_j\} \\
& \approx \{F'\mathbf{b}'(h_2) u_k F(f), u_{k'} F(g) \overline{F\mathbf{a}(h_1)}\}^\times \cup \{F'(g') t_{j'}, \overline{F'\mathbf{b}'(h_2)} F'(f') t_j\}^\times \\
& \phantom{\approx} \cup \{u_{k'} F(g), F'(g') t_{j'}\}^\times \cup \{u_k F(f), F'\mathbf{b}'(h_2), \overline{F\mathbf{a}(h_1)}, F'(f') t_j\} \\
& \approx \{\overline{F\mathbf{b}(h_2)} F(f), F(g) \overline{F\mathbf{a}(h_1)}\}^\times \cup \{\overline{F'\mathbf{b}'(h_2)} u_k, u_{k'} \overline{F\mathbf{b}(h_2)}\}^\times \\
& \phantom{\approx} \cup \{\overline{F'\mathbf{a}'(h_1)} t_j, t_{j'} \overline{F\mathbf{a}(h_1)}\}^\times \cup \{\overline{F'(g')} F'\mathbf{a}'(h_1), \overline{F'\mathbf{b}'(h_2)} F'(f')\}^\times \\
& \phantom{\approx} \cup \{u_{k'} F(g), F'(g') t_{j'}\}^\times \\
& \phantom{\approx} \cup \{t_j, t_{j'}, u_k, u_{k'}, F(f), F(g), F'(f'), F'(g'), \overline{F\mathbf{a}(h_1)}, \overline{F'\mathbf{b}'(h_2)}\},
\end{align*}
$$

which converges to zero uniformly on $f$ and $f'$ as $j, k \to \infty$ by Lemma 10.70, and because $F$ and $F'$ are a-commutative a-functors, $t$ and $u$ are a-natural, and $\mathcal{S}$, $\mathbf{a}$, $\mathbf{b}$, $\mathbf{a}'$ and $\mathbf{b}'$ are cofinal. $\square$

**Proposition 10.78.** "~" is an equivalence relation.

*Proof.* Consider the notation of Definition 10.75. To prove that this relation is reflexive, take $j, k \in \mathcal{J}_0$, $f \in \mathcal{I}(\mathbf{a}_j, \mathbf{a}_k)$ and $f' \in \mathcal{I}(\mathbf{a}'_j, \mathbf{a}'_k)$. Take also some $\ell \ge j, k$ in $\mathcal{J}_0$, and morphisms
$g \in \mathcal{J}(j, \ell)$ and $h \in \mathcal{J}(k, \ell)$. By applying Lemma 10.68 to the diagram

$$
\begin{tikzcd}
& & F_{a_j} \arrow[r, "F\mathbf{a}(g)"'] & F_{a_\ell} \arrow[r, "t_\ell"'] & F_{a'_\ell} \arrow[r, "\overline{F'\mathbf{a}'(g)}"'] & F_{a'_j} \\
F(f) \arrow[r, "\downarrow"'] & \id \arrow[r, "\downarrow"'] & \id \arrow[r, "\downarrow"'] & \id \arrow[r, "\downarrow"'] & F'(f') \arrow[r, "\downarrow"'] \\
F_{a_k} \arrow[r, "F\mathbf{a}(h)"'] & F_{a_\ell} \arrow[r, "t_\ell"'] & F_{a'_\ell} \arrow[r, "\overline{F'\mathbf{a}'(h)}"'] & F_{a'_k}
\end{tikzcd}
$$

and using (b4), (c2), (b1) and (10.31), we get

$$
\begin{align*}
& \{F'(f') \overline{F'\mathbf{a}'(g)} t_\ell F\mathbf{a}(g), \overline{F'\mathbf{a}'(h)} t_\ell F\mathbf{a}(h) F(f)\}^\times \\
& \asymp \{F\mathbf{a}(g), F\mathbf{a}(h) F(f)\}^\times \cup \{F'(f') \overline{F'\mathbf{a}'(g)}, \overline{F'\mathbf{a}'(h)}\}^\times \cup \{t_\ell, F'\mathbf{a}'(g), F\mathbf{a}(h)\} \\
& \approx \{F\mathbf{a}(g), F\mathbf{a}(h) F(f)\}^\times \cup \{F'\mathbf{a}'(g) \overline{F'(f')}, F'\mathbf{a}'(h)\}^\times \cup \{t_\ell, F'\mathbf{a}'(g), F\mathbf{a}(h)\}.
\end{align*}
$$

From (b2), Lemma 10.64 and the above relation, it follows that

$$
\begin{align*}
\{F'(f') t_j, t_k F(f)\}^\times &\asymp \{F'(f') t_j, F'(f') \overline{F'\mathbf{a}'(g)} t_\ell F\mathbf{a}(g)\}^\times \\
&\quad \cup \{F'(f') \overline{F'\mathbf{a}'(g)} t_\ell F\mathbf{a}(g), \overline{F'\mathbf{a}'(h)} t_\ell F\mathbf{a}(h) F(f)\}^\times \\
&\quad \cup \{\overline{F'\mathbf{a}'(h)} t_\ell F\mathbf{a}(h) F(f), t_k F(f)\}^\times \\
&\asymp \{t_j, \overline{F'\mathbf{a}'(g)} t_\ell F\mathbf{a}(g)\}^\times \cup \{F\mathbf{a}(g), F\mathbf{a}(h) F(f)\}^\times \\
&\quad \cup \{F'\mathbf{a}'(g) \overline{F'(f')}, F'\mathbf{a}'(h)\}^\times \cup \{\overline{F'\mathbf{a}'(h)} t_\ell F\mathbf{a}(h), t_k\}^\times \\
&\quad \cup \{F(f), F'(f'), F'\mathbf{a}'(g), F\mathbf{a}(h), t_\ell\},
\end{align*}
$$

which converges to zero uniformly on $f$ and $f'$ as $j, k \to \infty$ by Lemmas 10.70 and 10.74, by the a-naturality of $t$, since $F$ and $F'$ are a-commutative a-functors, and because $\mathbf{a}$ and $\mathbf{a}'$ are cofinal. This shows that $t \sim t$.

To prove that this relation is symmetric, suppose that $t \sim u$. We will also use that $u \sim u$
because the reflexivity of “∼” is already proved. Let $j \in \mathcal{J}_0$ and $k \in \mathcal{K}_0$ such that $\mathbf{a}_j \ge \mathbf{b}_k$
and $\mathbf{a}'_j \ge \mathbf{b}'_k$, and take $g \in \mathcal{J}(\mathbf{b}_k, \mathbf{a}_j)$ and $g' \in \mathcal{J}'(\mathbf{b}'_k, \mathbf{a}'_j)$. Since $\mathbf{b}$ and $\mathbf{b}'$ are cofinal, for some
$\ell \in \mathcal{K}_0$, there are morphisms $h_1 \in \mathcal{J}(\mathbf{a}_j, \mathbf{b}_\ell)$, $h_2 \in \mathcal{J}(\mathbf{b}_k, \mathbf{b}_\ell)$, $h'_1 \in \mathcal{J}(\mathbf{a}'_j, \mathbf{b}'_\ell)$ and $h'_2 \in \mathcal{J}(\mathbf{b}'_k, \mathbf{b}'_\ell)$.
By applying Lemma 10.68 to the diagram

$$
\begin{tikzcd}
& & F_{\mathbf{b}_k} \arrow[r, "F(h_2)"'] & F_{\mathbf{b}_\ell} \arrow[r, "\overline{F(h_1)}"'] & F_{\mathbf{a}_j} \\
u_k \arrow[r, "\downarrow"'] & \id \arrow[r, "\downarrow"'] & \id \arrow[r, "\downarrow"'] & t_j \arrow[r, "\downarrow"'] \\
F'_{\mathbf{b}'_k} \arrow[r, "F'(h'_2)"'] & F'_{\mathbf{b}'_\ell} \arrow[r, "\overline{F'(h'_1)}"'] & F'_{\mathbf{a}'_j}
\end{tikzcd}
$$
and using (10.31), (b2) and Lemma 10.64, it follows that

$$
\begin{align*}
\{t_j F(g), F'(g') u_k\}^\times &\asymp \{t_j F(g), t_j \overline{F(h_1)} F(h_2)\}^\times \cup \{t_j \overline{F(h_1)} F(h_2), \overline{F'(h'_1)} F'(h'_2) u_k\}^\times \\
&\phantom{\asymp} \cup \{\overline{F'(h'_1)} F'(h'_2) u_k, F'(g') u_k\}^\times \\
&\asymp \{F(g), \overline{F(h_1)} F(h_2)\}^\times \cup \{u_\ell F(h_2), F'(h'_2) u_k\}^\times \\
&\phantom{\asymp} \cup \{t_j \overline{F(h_1)}, \overline{F'(h'_1)} u_\ell\}^\times \cup \{\overline{F'(h'_1)} F'(h'_2), F'(g')\}^\times \\
&\phantom{\asymp} \cup \{F(h_1), F(h_2), u_k, t_j, F'(h'_1), F'(h'_2)\},
\end{align*}
$$

which converges to zero uniformly on $g$ and $g'$ as $j, k \to \infty$ by Lemmas 10.70 and 10.76, and because $F$ and $F'$ are a-commutative a-functors, $t$ and $u$ are a-natural, and $u \sim u \sim t$. This shows that $u \sim t$.

To prove that this relation is transitive, take other small cofinal sub-a-functors, $C : \mathcal{L} \to \tilde{\mathcal{D}}$ of $F$ and $C' : \mathcal{K} \to \tilde{\mathcal{D}}$ of $F'$, and an a-natural transformation $v : C \to C'$ such that $u \sim v$; so $v \sim u$ because the symmetry of “∼” is already proved. Let $\mathbf{c} : \mathcal{J} \to \mathcal{I}$ and $\mathbf{c}' : \mathcal{J} \to \mathcal{I}'$ be cofinal stp-reparametrizations defining $C$ and $C'$, respectively. Take also $j \in \mathcal{J}_0$, $\ell \in \mathcal{L}_0$, $f \in \mathcal{I}(\mathbf{a}_j, \mathbf{c}_\ell)$ and $f' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{c}'_\ell)$. Since $\mathbf{b}$ and $\mathbf{b}'$ are cofinal, there is some $k \in \mathcal{K}_0$ and morphisms $g \in \mathcal{I}(\mathbf{a}_j, \mathbf{b}_k)$, $h \in \mathcal{I}(\mathbf{c}_\ell, \mathbf{b}_k)$, $g' \in \mathcal{I}'(\mathbf{a}'_j, \mathbf{b}'_k)$ and $h' \in \mathcal{I}'(\mathbf{c}'_\ell, \mathbf{b}'_k)$. By arguing like in the proof of the symmetry of “∼” with the diagram

$$
\begin{tikzcd}
F_{\mathbf{a}_j} \arrow[r, "F(g)"'] \arrow[d, "t_j"] & F_{\mathbf{b}_k} \arrow[r, "\overline{F(h)}"'] \arrow[d, "u_k"] & F_{\mathbf{c}_\ell} \arrow[d, "v_\ell"] \\
F'_{\mathbf{a}'_j} \arrow[r, "F'(g')"'] & F'_{\mathbf{b}'_k} \arrow[r, "\overline{F'(h')}"'] & F'_{\mathbf{c}'_\ell}
\end{tikzcd}
$$

we get

$$
\{v_\ell F(f), F'(f') t_j\}^\times
\begin{align*}
&\asymp \{F(f), \overline{F(h)} F(g)\}^\times \cup \{u_k F(g), F'(g') t_j\}^\times \cup \{v_\ell \overline{F(h)}, \overline{F'(h')} u_k\}^\times \\
&\phantom{\asymp} \cup \{\overline{F'(h')} F'(g'), F'(f')\}^\times \cup \{F(h), F(g), t_j, v_\ell, F'(h'), F'(g')\},
\end{align*}
$$

which converges to zero uniformly on $f$ and $f'$ as $j, \ell \to \infty$ by Lemmas 10.70 and 10.76, and because $F$ and $F'$ are a-commutative a-functors, $t, u$ and $v$ are a-natural, and $t, v \sim u$. Hence $t \sim v$. $\square$

The quotient set of the above equivalence relation is the definition of $\bar{\mathcal{C}}(F, F')$. This set may not be small, but it is bijective to a small set by the axiom of choice, and thus we can assume that it is small after reparametrization. The equivalence class of each $t$ as above will be denoted by $[t]$. The following lemma will be used to define the operation of $\bar{\mathcal{C}}$.

**Lemma 10.79.** Let $\mathcal{I}$, $\mathcal{J}$ and $\mathcal{K}$ be small weakly filtered categories, and let $\mathbf{a} : \mathcal{J} \to \mathcal{I}$ and $\mathbf{b} : \mathcal{K} \to \mathcal{I}$ be cofinal stp-maps. Then there is a small weakly filtered category $\mathcal{X}$ such that there are cofinal functors $\mathbf{p} : \mathcal{X} \to \mathcal{J}$ and $\mathbf{q} : \mathcal{X} \to \mathcal{K}$, and there is a transformation $\xi : \mathbf{ap} \to \mathbf{bq}$.
Proof. The statement holds when $\mathcal{X}$ is the full subcategory of $\mathcal{J} \times \mathcal{K}$ with

$$ \mathcal{X}_0 = \{(j, k) \in \mathcal{J}_0 \times \mathcal{K}_0 \mid \mathbf{a}_j \le \mathbf{b}_k\}, $$

p and q are the restrictions of the factor projections of $\mathcal{J} \times \mathcal{K}$ to $\mathcal{J}$ and $\mathcal{K}$, respectively, and $\xi$ is defined by the choice of any morphism $\xi_x \in \mathcal{J}(j, k)$ for each $x = (j, k) \in \mathcal{X}$. $\square$

Consider two consecutive morphisms of $\bar{\mathcal{C}}$,

$$ F \xrightarrow{[t]} F' \xrightarrow{[u]} F'', $$

where $t: A \to A'$ and $u: B \to B'$ for cofinal sub-a-functors A of F, A' and B of F', and $B'$ of F''. Let $\mathcal{J}$, $\mathcal{J}'$ and $\mathcal{J}''$ be the source categories of F, F' and F'', and let $\mathbf{a}: \mathcal{J} \to \mathcal{J}$, $\mathbf{a}': \mathcal{J} \to \mathcal{J}'$, $\mathbf{b}: \mathcal{K} \to \mathcal{J}'$ and $\mathbf{b}': \mathcal{K} \to \mathcal{J}''$ be cofinal stp-reparametrizations defining A, A', B and B', respectively. By Lemma 10.79, there is a small weakly filtered category $\mathcal{X}$ so that there are cofinal functors $\mathbf{p}: \mathcal{X} \to \mathcal{J}$ and $\mathbf{q}: \mathcal{X} \to \mathcal{K}$, and there is an auxiliary transformation $\xi: \mathbf{a}'\mathbf{p} \to \mathbf{b}\mathbf{q}$. These stp-maps form the following diagram:

<figure><img src="image_29.png" /></figure>

Consider the cofinal sub-a-functors $P = \mathbf{F}\mathbf{a}\mathbf{p}$ of F and $Q = \mathbf{F}''\mathbf{b}'\mathbf{q}$ of F'', and the transformation $\alpha = u\mathbf{q} \cdot F'\xi \cdot t\mathbf{p}: P \to Q$. The a-naturality of $\alpha$ follows by applying Lemma 10.68 to the diagram

$$ \begin{array}{ccc} (\mathbf{F}\mathbf{a}\mathbf{p})_x & \xrightarrow{t_{\mathbf{p}x}} & (\mathbf{F}'\mathbf{a}'\mathbf{p})_x \xrightarrow{F'(\xi_x)} (\mathbf{F}'\mathbf{b}\mathbf{q})_x \xrightarrow{u_{\mathbf{q}x}} (\mathbf{F}''\mathbf{b}'\mathbf{q})_x \\ \mathbf{F}\mathbf{a}\mathbf{p}(f) \downarrow & & \mathbf{F}'\mathbf{a}'\mathbf{p}(f) \downarrow & \mathbf{F}'\mathbf{b}\mathbf{q}(f) \downarrow & \mathbf{F}''\mathbf{b}'\mathbf{q}(f) \downarrow \\ (\mathbf{F}\mathbf{a}\mathbf{p})_y & \xrightarrow{t_{\mathbf{p}y}} & (\mathbf{F}'\mathbf{a}'\mathbf{p})_y \xrightarrow{F'(\xi_y)} (\mathbf{F}'\mathbf{b}\mathbf{q})_y \xrightarrow{u_{\mathbf{q}y}} & (\mathbf{F}''\mathbf{b}'\mathbf{q})_y \end{array} $$

for any morphism $f: x \to y$ of $\mathcal{X}$, and using that $t$ and $u$ are a-natural, $F$, $F'$ and $F''$ are a-commutative, and $\mathbf{a}, \mathbf{a}', \mathbf{b}, \mathbf{b}', \mathbf{p}$ and $\mathbf{q}$ are cofinal. The details of this kind of argument have been already given in the proofs of Lemma 10.77 and Proposition 10.78. Then define the operation of $\bar{\mathcal{C}}$ by setting $[u][t] = [\alpha]$.

**Proposition 10.80.** The above operation is well defined.
Proof. With the above notation, suppose that $t \sim v$ and $u \sim w$ for a-natural transformations $v: C \to C'$ and $w: D \to D'$, and small cofinal sub-a-functors, $C$ of $F$, $C'$ and $D$ of $F'$, and $D'$ of $F''$. Let $c: \mathcal{M} \to \mathcal{J}$, $c': \mathcal{M} \to \mathcal{J}'$, $d: \mathcal{N} \to \mathcal{J}'$ and $d': \mathcal{N} \to \mathcal{J}''$ be cofinal stp-reparametrizations defining $C$, $C'$, $D$ and $D'$, respectively. Let $\mathcal{Y}$ be another small weakly filtered category such that there are cofinal functors $r: \mathcal{Y} \to \mathcal{M}$ and $s: \mathcal{Y} \to \mathcal{N}$, and there is an auxiliary transformation $\zeta: c'r \to ds$. Then consider the a-natural transformation

$$ \beta = ws \cdot F'\zeta \cdot vr : Fcr \to F''d's. $$

To check that $\alpha \sim \beta$, consider objects $x \in \mathcal{X}_0$ and $y \in \mathcal{Y}_0$ such that $ap(x) \le cr(y)$ in $\mathcal{J}_0$ and $b'q(x) \le d's(y)$ in $\mathcal{J}_0''$, and consider morphisms $f \in \mathcal{J}(ap(x), cr(y))$ and $f'' \in \mathcal{J}''(b'q(x), d's(y))$. Suppose also that $a'p(x) \le c'r(y)$ and $bq(x) \le ds(y)$ in $\mathcal{J}_0$; with these additional conditions, $(x, y)$ still runs in a cofinal subset of $\mathcal{X}_0 \times \mathcal{Y}_0$ by the cofinality of the involved stp-reparametrizations. Then there are morphisms $g \in \mathcal{J}(a'p(x), c'r(y))$ and $g' \in \mathcal{J}(bq(x), ds(y))$. By applying Lemma 10.68 to the diagram

$$ \begin{tikzcd} (Fap)_x \arrow[r, "t_{px}"] \arrow[d, "F(f)"'] & (F'a'p)_x \arrow[r, "F'(\xi_x)"] \arrow[d, "F'(g)"'] & (F'bq)_x \arrow[r, "u_{q_x}"] \arrow[d, "F'(g')"'] & (F''b'q)_x \arrow[d, "F''(f'')"'] \\ (Fcr)_y \arrow[r, "v_{ry}"] & (F'c'r)_y \arrow[r, "F'(\zeta_y)"] & (F'ds)_y \arrow[r, "w_{sy}"] & (F''d's)_y \end{tikzcd} $$

and using that $t \sim v$ and $u \sim w$, that $F$, $F'$ and $F''$ are a-commutative a-functors, that $t$, $u$, $v$ and $w$ are a-natural, and that the involved stp-reparametrizations are cofinal, we get

$$ \lim_{x,y \to \infty} \{F''(f'') \alpha_x, \beta_y F(f)\}^\times = 0 $$

uniformly on $f$ and $f'$ for $x$ and $y$ as above. Therefore $\alpha \sim \beta$ by Lemma 10.77. $\square$

**Lemma 10.81.** Let $A, A', A'': \mathcal{J} \to \tilde{\mathcal{D}}$ be small cofinal sub-a-functors of small a-commutative a-functors $F: \mathcal{J} \to \tilde{\mathcal{D}}$, $F': \mathcal{J}' \to \tilde{\mathcal{D}}$ and $F'': \mathcal{J}'' \to \tilde{\mathcal{D}}$, respectively, and let $t, u: A \to A'$ and $t', u': A' \to A''$ be a-natural transformations. If $t \sim u$ and $t' \sim u'$, then $t't \sim u'u$.

Proof. Let $a: \mathcal{J} \to \mathcal{J}$, $a': \mathcal{J} \to \mathcal{J}'$ and $a'': \mathcal{J} \to \mathcal{J}''$ be cofinal stp-reparametrizations defining $A, A'$ and $A''$, respectively. Take $j, k \in \mathcal{J}_0$ such that $a_j \le a_k$ and $a_j'' \le a_k''$. By Lemma 10.77, we can also assume that $a'_j \le a'_k$ since $a'$ is cofinal. Thus there are morphisms $f \in \mathcal{J}(a_j, a_k)$, $f' \in \mathcal{J}(a'_j, a'_k)$ and $f'' \in \mathcal{J}(a''_j, a''_k)$. The result follows by applying Lemma 10.68 to the diagram

$$ \begin{tikzcd} F_{a_j} \arrow[r, "t_j"] \arrow[d, "F(f)"'] & F'_{a'_j} \arrow[r, "t'_j"] \arrow[d, "F'(f')"'] & F''_{a''_j} \arrow[d, "F''(f'')"'] \\ F_{a_k} \arrow[r, "u_k"] & F'_{a'_k} \arrow[r, "u'_k"] & F''_{a''_k} \end{tikzcd} $$

and using the hypotheses. $\square$
**Proposition 10.82.** *The operation of $\bar{C}$ is associative.*

*Proof.* With the notation of the definition of this operation, take another morphism $[v] : F'' \to F'''$ of $\bar{C}$, where $v$ is an a-natural transformation $C \to C'$ for small cofinal sub-a-functors, $C : \mathcal{L} \to \tilde{\mathcal{D}}$ of $F''$ and $C' : \mathcal{L} \to \tilde{\mathcal{D}}$ of $F'''$, defined by cofinal stp-reparametrizations $\mathbf{c} : \mathcal{L} \to \mathcal{J}''$ and $\mathbf{c}' : \mathcal{L} \to \mathcal{J}'''$, respectively. By Lemma 10.79, there is a small weakly filtered category $\mathcal{Y}$ with cofinal functors $\mathbf{r} : \mathcal{Y} \to \mathcal{K}$ and $\mathbf{s} : \mathcal{Y} \to \mathcal{K}$, and an auxiliary transformation $\zeta : \mathbf{b}'\mathbf{r} \to \mathbf{cs}$. Similarly, there is a small weakly filtered category $\mathcal{Z}$ with cofinal functors $\mathbf{d} : \mathcal{Z} \to \mathcal{X}$ and $\mathbf{e} : \mathcal{Z} \to \mathcal{Y}$, and an auxiliary transformation $\theta : \mathbf{qd} \to \mathbf{re}$. These stp-maps form the diagram

<figure><img src="image_30.png" /></figure>

As we saw, the a-natural transformations

$$ \alpha = u\mathbf{q} \cdot F'\xi \cdot t\mathbf{p} : F\mathbf{ap} \to F''\mathbf{b}'\mathbf{q}, \quad \beta = v\mathbf{s} \cdot F''\zeta \cdot u\mathbf{r} : F'\mathbf{br} \to F'''\mathbf{c}'\mathbf{s} $$

are representatives of the products $[u][t]$ and $[v][u]$. Then, by using the auxiliary transformations

$$ \zeta\mathbf{e} \cdot \mathbf{b}'\theta : \mathbf{b}'\mathbf{q}\mathbf{d} \to \mathbf{c}\mathbf{s}\mathbf{e}, \quad \mathbf{b}\theta \cdot \xi\mathbf{d} : \mathbf{a}'\mathbf{p}\mathbf{d} \to \mathbf{b}\mathbf{r}\mathbf{e}, $$

it follows that the following a-natural transformations $F\mathbf{apd} \to F'''\mathbf{c}'\mathbf{s}\mathbf{e}$ respectively represent $[v]([u][t])$ and $([v][u])[t]$:

$$ v\mathbf{s}\mathbf{e} \cdot F''(\zeta\mathbf{e} \cdot \mathbf{b}'\theta) \cdot \alpha\mathbf{d} = v\mathbf{s}\mathbf{e} \cdot F''\zeta\mathbf{e} \cdot F''\mathbf{b}'\theta \cdot u\mathbf{q}\mathbf{d} \cdot F'\xi\mathbf{d} \cdot t\mathbf{p}\mathbf{d}, $$

$$ \beta\mathbf{e} \cdot F'(\mathbf{b}\theta \cdot \xi\mathbf{d}) \cdot t\mathbf{p}\mathbf{d} = v\mathbf{s}\mathbf{e} \cdot F''\zeta\mathbf{e} \cdot u\mathbf{r}\mathbf{e} \cdot F'\mathbf{b}\theta \cdot F'\xi\mathbf{d} \cdot t\mathbf{p}\mathbf{d}. $$

So, by Lemma 10.81, it is enough to prove that the a-natural transformations $\gamma = F''\mathbf{b}'\theta \cdot u\mathbf{q}\mathbf{d}$ and $\delta = u\mathbf{r}\mathbf{e} \cdot F'\mathbf{b}\theta$ are asymptotic. For this purpose, consider objects $z, z' \in \mathcal{Z}_0$ such that there are morphisms $f \in \mathcal{J}'((\mathbf{b}\mathbf{q}\mathbf{d})_z, (\mathbf{b}\mathbf{q}\mathbf{d})_{z'})$ and $f' \in \mathcal{J}'((\mathbf{b}'\mathbf{r}\mathbf{e})_z, (\mathbf{b}'\mathbf{r}\mathbf{e})_{z'})$. Then, by (b2)
and Lemma 10.64,

$$
\begin{align*}
\{F''(f') \gamma_z, \delta_{z'} F'(f)\}^\times &\asymp \{F''(f') F''\mathbf{b}'(\theta_z) u_{(\mathbf{q}\mathbf{d})_z}, F''(f' \mathbf{b}'(\theta_z)) u_{(\mathbf{q}\mathbf{d})_z}\}^\times \\
&\phantom{\asymp} \cup \{F''(f' \mathbf{b}'(\theta_z)) u_{(\mathbf{q}\mathbf{d})_z}, u_{(\mathbf{r}\mathbf{e})_{z'}}, F'(\mathbf{b}(\theta_{z'})) f\}^\times \\
&\phantom{\asymp} \cup \{u_{(\mathbf{r}\mathbf{e})_{z'}}, F'(\mathbf{b}(\theta_{z'})) f), u_{(\mathbf{r}\mathbf{e})_{z'}}, F'(\mathbf{b}(\theta_{z'})) F'(f)\}^\times \\
&\asymp \{F''(f') F''\mathbf{b}'(\theta_z), F''(f' \mathbf{b}'(\theta_z))\}^\times \\
&\phantom{\asymp} \cup \{F''(f' \mathbf{b}'(\theta_z)) u_{(\mathbf{q}\mathbf{d})_z}, u_{(\mathbf{r}\mathbf{e})_{z'}}, F'(\mathbf{b}(\theta_{z'})) f\}^\times \\
&\phantom{\asymp} \cup \{F'(\mathbf{b}(\theta_{z'})) f), F'(\mathbf{b}(\theta_{z'})) F'(f)\}^\times \cup \{u_{(\mathbf{q}\mathbf{d})_z}, u_{(\mathbf{r}\mathbf{e})_{z'}}\},
\end{align*}
$$

which converges to zero uniformly on $f$ and $f'$ as $z, z' \to \infty$ because $F'$ and $F''$ are a-commutative a-functors, $u \sim u$ (Proposition 10.78), and $\mathbf{q}, \mathbf{d}, \mathbf{r}$ and $\mathbf{e}$ are cofinal. $\square$

**Proposition 10.83.** With the above operation, $\bar{\mathcal{C}}$ is a groupoid.

*Proof.* Let $\mathcal{J} : \mathcal{J} \to \tilde{\mathcal{D}}$ and $\mathcal{F}' : \mathcal{J}' \to \tilde{\mathcal{D}}$ be objects of $\bar{\mathcal{C}}$, let $A$ and $A'$ be small cofinal sub-a-functors of $\mathcal{F}$ and $\mathcal{F}'$, respectively, with the same source category $\mathcal{J}$, and let $t : A \to A'$ be an a-natural transformation representing a morphism $\mathcal{F} \to \mathcal{F}'$ of $\bar{\mathcal{C}}$. Its involution $\bar{t} : A' \to A$ is a-natural by Lemma 10.74, and thus $\bar{t}$ represents a morphism $\mathcal{F}' \to \mathcal{F}$ of $\bar{\mathcal{C}}$.

Let $\mathbf{a} : \mathcal{J} \to \mathcal{J}$ be any cofinal stp-reparametrization defining $A$. For $i \in \mathcal{J}_0$, $j \in \mathcal{J}_0$ and $f, g \in \mathcal{J}(i, \mathbf{a}_j)$, by (b2), Lemma 10.64, (c2) and (b3),

$$
\begin{align*}
\{F(f), \bar{t}_j t_j F(g)\}^\times &\asymp \{F(f), F(g)\}^\times \cup \{F(g), \bar{t}_j t_j F(g)\}^\times \\
&\asymp \{F(f), F(g)\}^\times \cup \{\mathrm{id}_{F_{\mathbf{a}_j}}, \bar{t}_j t_j\}^\times \\
&\asymp \{F(f), F(g)\}^\times \cup \{t_j\},
\end{align*}
$$

which converges to zero as $i, j \to \infty$ uniformly on $f$ and $g$ because $t$ is a-undistorted, $\mathcal{F}$ is a-commutative and $\mathbf{a}$ is cofinal. It follows that $\bar{t}t$ is asymptotic to the identity transformation of $\mathcal{F}$, and thus $[\bar{t}] [t] = \mathrm{id}_\mathcal{F}$. Similarly, $[t] [\bar{t}] = \mathrm{id}_{\mathcal{F}'}$, obtaining that $[\bar{t}]$ is the inverse of $[t]$. $\square$

Let $\mathcal{O}$ be the category consisting only of one object $\mathcal{O}$ and the morphism $\mathrm{id}_\mathcal{O}$. For each $M \in \mathcal{C}_0$, let $\iota_M$ be the functor $\mathcal{O} \to \mathcal{C}$ defined by $\iota_M(\mathrm{id}_\mathcal{O}) = \mathrm{id}_M$; and, for each morphism $\phi : M \to N$ of $\mathcal{C}$, let $i(\phi)$ be the natural transformation $\iota_M \to \iota_N$ defined by $i(\phi)_{\mathcal{O}} = \phi$. By considering $\mathcal{C}$ as subcategory of $\tilde{\mathcal{D}}$, we get that $\iota_M \in \bar{\mathcal{C}}_0$ and $i(\phi)$ represents a morphism $\iota(\phi) : \iota_M \to \iota_N$ of $\bar{\mathcal{C}}$. This defines a canonical involutive functor $\iota : \mathcal{C} \to \bar{\mathcal{C}}$. The induced map $\bar{\iota} : \mathcal{C} \setminus \mathcal{C}_0 \to \bar{\mathcal{C}} \setminus \bar{\mathcal{C}}_0$ may not be injective because $\bar{\mathcal{C}}(\iota_M, \iota_N)$ may be larger than $\iota(\mathcal{C}(M, N))$.

## 10.10 Pseudo-metrics defined by a bidistortion

Consider the distortion $\hat{\mathfrak{b}}$ of $\mathcal{C}/\mathcal{D}$ induced by $\mathfrak{b}$ according to (10.14). In general, $\hat{\mathfrak{b}}$ is not a bidistortion, but at least we have the following.

**Lemma 10.84.** $\hat{\mathfrak{b}}(\bar{\sigma}) \approx \hat{\mathfrak{b}}(\sigma)$ for $\sigma \in \tilde{\mathcal{D}}_1$.