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Residually linear abstract groupoids K. Draoui, H. Choulli, H. Mouanis Mathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, University Sidi Mohamed Ben Abdellah Fez, Morocco
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    § 1. IntroductionCategory theory provides a unified framework for many areas of mathematics. Groupoids, as categories where all morphisms are isomorphisms, are used in abstract and geometric settings in diverse branches of mathematics and physics, see, for example, [1] for a survey. They play an essential role in the study of symmetry in differential geometry, physics, and algebraic topology, where the fundamental groupoid represents a very useful notion. In general, there are two approaches to introduce groupoids as a natural generalization of groups: one categorical and the other algebraic. Unlike groups, in a groupoid, the composition (respectively, the binary operation) is not always defined for all arrows (morphisms), which produces several identities; however, inverses exist for all arrows. Groupoids were introduced, in the algebraic approach, and named by Braudt [2], then developed by Higgins [3]. Subsequently, Ehresmann [4] continued the exploration and introduced topological and differential groupoids. Many authors investigated Lie groupoids in many fields such as differential geometry, quantum mechanics, gauge theories, etc. Groupoids find applications in various domains and directions, they serve, for example, in the simplification of proofs of many results in group theory [5], in the study of partial actions and representations of groups, etc. The generalization of the notions of group theory to the framework of groupoids is not a completely simple transition and it has its own challenges. In fact, while a groupoid in one of its simple facets can be seen, by applying a strong version of the axiom of choice, as a disjoint union of groups, this vision is not always valid. For example, for structured groupoids, a topological or differential groupoid is not a disjoint union of topological or differential groups. Studying the residual properties, for various classes of groups, provides important contributions in discrete group theory, see [6] for a discussion. For instance, for the residual finiteness property, introduced in 1940 by Mal’cev [7], and studied intensively later by many authors, we refer to [8], [9] for good surveys. Residual finiteness is used to establish some remarkable results such as the following [7], [8]. (1) Any finitely generated linear group is residually finite. (2) Any finitely generated residually finite group is Hopfian. (3) Any finitely presented residually finite group has a solvable word problem. Menal [10] in 1978 investigated the residual linearity property for the class of nilpotent groups. He determined sufficient conditions for a residually linear group to be residually finite. Precisely, he claimed that if the center and commutator subgroups of a nilpotent residually linear group are finitely generated, then this group must be residually finite. In the topological case, the famous Tannaka–Krein duality principle for compact groups states that we can reconstruct a compact Lie group $G$ from its category $\mathrm{rep}(G)$ of finite dimensional linear representations [11]. In this case, there is a mapping from $G$ to its associated character group $G_{\Bbbk}$ consisting of the unital algebra maps from the Hopf algebra of representative functions $\mathcal{R}_{\Bbbk}(G)$ of $G$ to the base field $\Bbbk$. In fact, this mapping is an isomorphism. Its injectivity is established through the Peter–Weyl theorem [12], Theorem 12, which implies that, for all $e \neq g \in G$, there exists $\rho \in \mathrm{rep}(G)$ such that $\rho(g) \neq 1$. The abstraction of this property precisely defines residual linearity in the discrete case [13]. Groupoid representations appear in various areas of mathematics. For instance, when a group acts on a space $X$, the equivariant vector bundles over $X$ correspond to representations of the associated action groupoid. Parallel to its importance in group theory, the representation theory of groupoids plays a crucial role in establishing categorical versions of classical theorems, including the Gelfand–Raikov and Peter–Weyl theorems, see [14]. When one considers topological or differential groupoids, the composition and inverse maps are asked to be continuous or smooth, respectively. Using representation theory for compact topological groupoids, Amini [14] generalized the Tannaka–Krein duality to this context in a similar procedure as for compact groups. To the best of our knowledge, such duality in the general case of abstract groupoids is still unexplored. In this paper, we introduce, in the abstract case, the notions of residual linearity and residual finiteness for groupoids, and provide some algebraic properties. We recover straightforwardly some results in the discrete group theory, concerning equivalent statements, stability properties and among others, the ones concerned with the relationship between residual finiteness and residual linearity such as their equivalence in the class of finitely generated groupoids. In addition, we define, analogously to groups, the notions of nilpotent groupoid, center and commutator subgroupoids, and use them to extend the aforementioned result of Menal [10], Theorem 1, to the context of groupoids. At the end, in § 4, we investigate the Hopf algebroid $\mathcal{R}_{\Bbbk}(\mathscr{G})$ of representative functions of a transitive groupoid $\mathscr{G}$ and its character groupoid $\mathscr{X(G)}$ constructed via Tannaka’s process. We show, as a usual case, that $\mathscr{G}$ is abelian if and only if $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is cocommutative, if and only if $\mathscr{X(G)}$ is abelian. As a last application, we prove in Theorem 4.5 that the character groupoid $\mathscr{X(G)}$ associated with a transitive groupoid $\mathscr{G}$ is always residually linear. This provides a necessary condition for the question of reconstructing $\mathscr{G}$ from $\mathscr{X(G)}$; however, its sufficiency has not been established in this work. § 2. PreliminariesFor a basic background on category theory language, we refer to [15]. In this section, we will fix some notation and basic material that will be needed in what follows. These materials are collected from the cited references and adapted to the terminology followed here. Throughout, $\Bbbk$ denotes an arbitrary base field and $G$ a discrete group with neutral element $e$. 2.1. Residual propertiesIn this part, we first briefly recall the notions of residual finiteness and residual linearity of an abstract group, along with some illustrative examples. We also recall the definition of the Hopf algebra of representative functions on a discrete group and a characterization of its elements. Definition 2.1 (see [8]). A group $G$ is said to be residually finite if, for every $e_{G} \neq g \in G$, there exists a finite group $D$ and a group homomorphism $f_g\colon G \to D$ such that $f_g(g) \neq e_{D}$. Mal’cev [7] described how to construct important examples of large residually finite groups classes as follows: the group generated by a set of invertible square matrices over a field is residually finite. Definition 2.2 (see [10]). A group $G$ is called residually linear if, for for all $e \neq g \in G$, there exist a finite dimensional $\Bbbk$-vector space $V$ and a representation $\rho_g$ of $G$ on $V$, namely, a group homomorphism $\rho_g\colon G \to \mathrm{GL}(V)$ such that $g \notin \mathrm{Ker}(\rho_g)$. We let $\mathrm{rep}(G)$ denote the category of finite dimensional representations of $G$ over $\Bbbk$. Important examples of residually linear groups are, of course, linear groups and also residually finite groups, which include finite, free, finitely generated abelian, polycyclic, and profinite groups, etc. [8]. Following [16], the Hopf algebra of representative functions $\mathcal{R}_{\Bbbk}(G)$ of a discrete group $G$ is the finite dual $\Bbbk[G]^{\circ}$ of its group algebra $\Bbbk[G]$. An element $f$ of $\Bbbk[G]^{\circ}$ is characterized as follows:
$$
\begin{equation}
f \in \mathcal{R}_{\Bbbk}(G) \quad \Longleftrightarrow \quad \exists \, f_i, g_i \in \Bbbk[G]^*,\quad f(xy)= \sum_{i=1}^n f_i(x) g_i(y)\quad \forall \, x, y \in \Bbbk[G].
\end{equation}
\tag{1}
$$
Recall that a category $\mathscr{C}$ is called: (a) locally small if the collection of morphisms $\mathrm{Mor}(x, y)$ between any pair of objects $x$, $y$ of $\mathscr{C}$ forms a set; (b) small if both the collection of objects $\mathrm{Ob}(\mathscr{C})$ and that of morphisms $\mathrm{Mor}(x, y)$ form sets for all $x, y \in \mathrm{Ob}(\mathscr{C})$; (c) finite if $\mathrm{Ob}(\mathscr{C})$ and $\mathrm{Mor}(x, y)$ are finite for all $x, y \in \mathrm{Ob}(\mathscr{C})$; (d) weakly finite if $\mathscr{C}$ has only finitely many isomorphism classes of objects and $\mathrm{Mor}(x, y)$ is finite for all $x, y \in \mathrm{Ob}(\mathscr{C})$. Making freely reference to the axiom of choice, recall from [15], Theorem 1, p. 93, that a functor between two categories is an equivalence if and only if it is fully faithful and essentially surjective. 2.2. Abstract groupoids and related notionsRecall [5] that a groupoid $\mathscr{G}:=(\mathscr{G}_0, \mathscr{G}_1, s, t, \iota)$ is a small category in which every morphism (arrow) is invertible, where $\mathscr{G}_0$, $\mathscr{G}_1$ denote, respectively, the sets of objects and arrows of $\mathscr{G}$. The structure maps $s, t\colon \mathscr{G}_1 \to \mathscr{G}_0$ and $\iota \colon \mathscr{G}_0 \to \mathscr{G}_1$ denote, respectively, the source, target and identity maps, and they are defined, respectively, as $s(f)=x$, $t(f)=y$, for every $f\colon x \to y$ in $\mathscr{G}_1$, and $\iota(x)=1_x$ the identity morphism on $x$, for every $x \in \mathscr{G}_0$. Two morphisms $f$, $g$ are composable with composite $f g \in \mathscr{G}_1$ whenever $s(f)=t(g)$. The isotropy group $\mathscr{G}^x$ of $\mathscr{G}$ at $x \in \mathscr{G}_0$ is defined as $\mathscr{G}^x=\{f \in \mathscr{G}_1, \, s(f)=t(f)=x\}$. The disjoint union of the isotropy groups is denoted by $\mathrm{Iso}(\mathscr{G})=\biguplus_{x \in \mathscr{G}_0} \mathscr{G}^x :=\{f \in \mathscr{G}_1, \, s(f)=t(f) \}$. A groupoid $\mathscr{G}$ in which any isomorphic objects are identical, or equivalently $\mathscr{G}(x, y)=\varnothing$ is the empty set for all $x\neq y$, is called totally disconnected, or a group bundle. In this case, we have $\mathscr{G}_1=\mathrm{Iso}(\mathscr{G})$. A groupoid $\mathscr{G}$ is called transitive if it has exactly one connected component, namely, for all $x, y \in \mathscr{G}_0$, there exists $f \in \mathscr{G}_1$ such that $s(f)=x$ and $t(f)=y$, or equivalently when the map $s \times t \colon \mathscr{G}_1 \to \mathscr{G}_0 \times \mathscr{G}_0$ sending $f \in \mathscr{G}_1$ to $(s(f), t(f))$ is surjective. By a finite dimensional $\Bbbk$-linear representation of a groupoid $\mathscr{G}$ (or simply a $\mathscr{G}$-representation), we mean a functor from $\mathscr{G}$ to the category $\mathrm{vect}_{\Bbbk}$ of finite dimensional vector spaces over $\Bbbk$. We denote by $\mathrm{rep}(\mathscr{G}):=\mathrm{Fun}(\mathscr{G}, \mathrm{vect}_{\Bbbk})$ the category whose objects are $\mathscr{G}$-representations. A groupoid morphism $\alpha\colon \mathscr{G} \to \mathscr{M}$ is nothing but a functor between the underlying categories. Definition 2.3 (see [17], [18]). A groupoid $\mathscr{G}$ is said to be abelian if all its isotropy groups are abelian. Example 2.4. Let $X$ be a topological space. The fundamental groupoid $\Pi(X)$ whose objects are the points of $X$ and morphisms are homotopy classes of paths is known to be an abelian groupoid if $X$ is a group [17], Example 1.1. A subgroupoid $\mathscr{D}$ of a groupoid $\mathscr{G}$ is a subcategory of it which is closed under inverses, namely $f^{-1} \in \mathscr{D}_1$, for every $f \in \mathscr{D}_1$. For $f \in \mathscr{G}_1$, $f \mathscr{D}$ is called the right coset of $\mathscr{D}$ in $\mathscr{G}$ containing $f$ and it is defined as $f \mathscr{D}= \{f d, \, d \in \mathscr{D}_1, \, s(f)=t(d) \}$. The left coset $\mathscr{D}f$ of $\mathscr{D}$ in $\mathscr{G}$ containing $f$ is defined similarly. The trivial subgroupoid of $\mathscr{G}$, denoted by $\mathcal{I}_{\mathscr{G}}$, is given by $(\mathcal{I}_{\mathscr{G}})_0=\mathscr{G}_0$, and $(\mathcal{I}_{\mathscr{G}})_1= \{ 1_x, \, x \in \mathscr{G}_0 \}$. The following definition is taken from [19], Definition 3.1, and adapted to our notation. Definition 2.5. A subgroupoid $\mathscr{N}$ of a groupoid $\mathscr{G}$ is said to be a normal subgroupoid, written $\mathscr{N} \triangleleft \mathscr{G}$, provided that: (a) $\mathscr{N}_0=\mathscr{G}_0$; (b) $f \mathscr{N} f^{-1} \subseteq \mathscr{N}_1 \cap \mathrm{Iso}(\mathscr{G})$ for every $f \in \mathscr{G}_1$. Definition 2.6 (see [5]). Let $\mathtt{F} \colon \mathscr{G} \to \mathscr{H}$ be a groupoid morphism. The kernel $\mathrm{Ker}(\mathtt{F})$ of $\mathtt{F}$ is the subgroupoid of $\mathscr{G}$ consisting of the same set of objects, but containing only those morphisms whose images under $\mathtt{F}$ are identities. Namely: (a) $\mathrm{Ker}(\mathtt{F})_0=\mathscr{G}_0$; (b) $\mathrm{Ker}(\mathtt{F})_1=\{ f \in \mathscr{G}_1, \, \mathtt{F}(f)=1_{\mathtt{F}(s(f))}=1_{\mathtt{F}(t(f))} \}$. Let $\mathscr{N}$ be a normal subgroupoid of a groupoid $\mathscr{G}$. Following [19], Lemmas 3.3 and 3.4, we can define a congruence relation $\sim$ as follows: $f \sim g$ if and only if $s(f)=s(g)$ and $f g^{-1} \in \mathscr{N}_1$. The relation $\sim$ is defined on objects as follows: $x \sim y$ if and only if there exists a morphism $f\colon x \to y$ in $\mathscr{G}_1$ connecting $x$ and $y$. We can now define the structure of a quotient groupoid, where we identify the set of objects of $\mathscr{G} / \mathscr{N}$ with that of $\mathscr{G}$ (see [19], Definition 3.5). Definition 2.7. Let $\mathscr{N}$ be a normal subgroupoid of $\mathscr{G}$. The quotient groupoid $\mathscr{G} / \mathscr{N}$ is the groupoid defined by (a) $(\mathscr{G} / \mathscr{N})_0=\mathscr{G}_0 / \sim$ , (b) $(\mathscr{G} / \mathscr{N})_1= \mathscr{G}_1 / \sim$ and the canonical projection $\pi\colon \mathscr{G} \to \mathscr{G} / \mathscr{N}$ is a groupoid morphism. The source, target, and identity maps as well as the inverse and composition are understood from the congruence relation and explicitly induced from $\pi$. Note that, for a normal subgroupoid $\mathscr{N}$, an element of $(\mathscr{G} / \mathscr{N})_1$ is the right (or left, by [18], Remark 3.10) coset $f \mathscr{N}$ of $\mathscr{N}$ in $\mathscr{G}$ containing some $f \in \mathscr{G}_1$, see also [20], § 4. On the other hand, $(\mathscr{G} / \mathscr{N})_0$ can be identified with $\mathscr{G}_0$ through $\pi$ and if $\mathscr{G}$ is totally disconnected, then $(\mathscr{G} / \mathscr{N})_0=\mathscr{G}_0$. 2.3. Character groupoid of the Hopf algebroid of representative functions of a transitive groupoidFrom a given transitive groupoid $\mathscr{G}$, one can construct an associated commutative Hopf algebroid $\mathcal{R}_{\Bbbk}(\mathscr{G})$ over a commutative base algebra, termed in [21] the Hopf algebroid of representative functions on $\mathscr{G}$, as a universal object obtained by applying the Tannakian formalism on the category of finite dimensional representations of $\mathscr{G}$, with the fibre functor constituting together a Tannakian linear category; this is mentioned also in [22]. This construction recovers the one in the discrete group case of the commutative Hopf algebra of representative functions. The Hopf algebroid structures are explicitly described in [21]. We only sketch here the general setup needed in what follows, and refer to [21] and [23] for a detailed exposition. Let $\mathscr{G}=(\mathscr{G}_0, \mathscr{G}_1)$ be a transitive groupoid and $\mathtt{F}$ be a $\mathscr{G}$-representation. Then, for every $x \in \mathscr{G}_0$, there is a linear isomorphism $V_x:=\mathtt{F}(x) \to V$ for a given $n$-dimensional $\Bbbk$-vector space $V$. The vector space $V_x$ is called the fibre of $\mathtt{F}$ at $x$ and the disjoint union $X:=\biguplus_{x \in \mathscr{G}_0} V_x$ is called the vector $\mathscr{G}$-bundle of the representation $\mathtt{F}$, and $n$ the rank of $X$. Let $\pi\colon X \to \mathscr{G}_0$ be the canonical projection; we also set
$$
\begin{equation*}
\Gamma(\mathtt{F})=\{\phi \colon \mathscr{G}_0 \to X, \, \pi \circ \phi = 1 \}.
\end{equation*}
\notag
$$
Let $\mathtt{B}$ be the base algebra of $\mathscr{G}$ consisting of all maps: $\mathscr{G}_0 \to \Bbbk$. Following the notation adopted in [23], Notation, p. 66, and § 3.1, p. 79, a description of $\mathcal{R}_{\Bbbk}(\mathscr{G})$ as a finite dual coring is introduced, which coincides with the usual finite dual in the case of a Dedekind domain (in particular, a field as it is the case here). To be more precise, given a representation $\mathtt{F}$ of $\mathscr{G}$, an element of $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is of the form $\overline{\varphi \otimes_{T_V} p}$, represented by the element $\varphi \otimes_{T_V} p \in \Gamma(\mathtt{F})^* \otimes_{T_V} \Gamma(\mathtt{F})$, where $V$ is the vector space underlying the representation $\mathtt{F}$, $\Gamma(\mathtt{F})^*=\mathrm{Hom}(\Gamma(\mathtt{F}), \mathtt{B})$ and the $\Bbbk$-algebra $T_V$ is the endomorphism algebra of the finite dimensional representation $\mathtt{F}$, which naturally acts on the left of the finitely generated and projective $\mathtt{B}$-module $\Gamma(\mathtt{F})$ of the global sections of the $\mathscr{G}$-equivariant vector bundle $X$ underlying $\mathtt{F}$. Under this notation, $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is the quotient
$$
\begin{equation}
\mathcal{R}_{\Bbbk}(\mathscr{G}):= \frac{\bigoplus_{\mathtt{F} \in \mathrm{rep(\mathscr{G})}} \Gamma(\mathtt{F}^*) \otimes_{T_V} \Gamma(\mathtt{F}) }{\mathscr{J}},
\end{equation}
\tag{2}
$$
where $\Gamma(\mathtt{F})^*$ is identified with $\Gamma(\mathtt{F}^*)$, as they are isomorphic, $\mathtt{F}^*$ is the dual $\mathscr{G}$-representation of $\mathtt{F}$, $\mathscr{J}$ is the ideal
$$
\begin{equation*}
\mathscr{J}= \langle \varphi \otimes_{T_{W}} \lambda p - \varphi \lambda \otimes_{T_V} p,\ \varphi \in \Gamma(\mathtt{G}^*),\, p \in \Gamma(\mathtt{F}),\, \lambda \in T_{V, W} \rangle,
\end{equation*}
\notag
$$
$V$, $W$ are the vector spaces underlying the $\mathscr{G}$-representations $\mathtt{F}$, $\mathtt{G}$, respectively, and $T_{V, W}=\mathrm{Hom}_{\mathrm{rep}(\mathscr{G})}(V, W)$. The couple $(\mathtt{B},\mathcal{R}_{\Bbbk}(\mathscr{G}))$ constitutes in fact a commutative Hopf algebroid with structures inherited from those of $\mathscr{G}$ as displayed explicitly in [21], p. 10, see also [23], p. 66. $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is injected into its total algebra $\mathrm{M}_{\Bbbk}(\mathscr{G}_1)$ of all maps: $\mathscr{G}_1 \to \Bbbk$, via a $(\mathtt{B} \otimes_{\Bbbk} \mathtt{B})$-algebra map given by
$$
\begin{equation}
\zeta\colon \mathcal{R}_{\Bbbk}(\mathscr{G}) \hookrightarrow \mathrm{M}_{\Bbbk}(\mathscr{G}_1), \qquad \overline{\psi \otimes_{T_V} p} \mapsto \bigl[ g \mapsto \psi(t(g)) \bigl( \mathtt{F}(g)(p(s(g)))\bigr) \bigr],
\end{equation}
\tag{3}
$$
where $\mathtt{F}$ is the $\mathscr{G}$-representation corresponding to the vector space $V$. Let $\mathscr{G}$ be a transitive groupoid. The character groupoid associated with the Hopf algebroid $(\mathtt{B}, \mathcal{R}_{\Bbbk}(\mathscr{G}))$ of representative functions of $\mathscr{G}$ is $\mathscr{X}(\mathscr{G}):=(\mathtt{B}(\Bbbk), \mathcal{R}_{\Bbbk}(\mathscr{G})(\Bbbk))$, where $\mathtt{B}(\Bbbk)$ (respectively, $\mathcal{R}_{\Bbbk}(\mathscr{G})(\Bbbk))$ is the set of all $\Bbbk$-algebra maps from $\mathtt{B}$ (respectively, $\mathcal{R}_{\Bbbk}(\mathscr{G}))$ to $\Bbbk$. Let $\mathtt{s}$, $\mathtt{t}$, $\varepsilon$, $\Delta$, $\iota$ denote the source, target, counit, coproduct, and antipode maps of $(\mathtt{B}, \mathcal{R}_{\Bbbk}(\mathscr{G}))$, respectively. Then precisely $\mathscr{X}(\mathscr{G})=(\mathtt{B}(\Bbbk), \mathcal{R}_{\Bbbk}(\mathscr{G})(\Bbbk), \mathtt{s}^*, \mathtt{t}^*, \varepsilon^*)$, where $\mathtt{s}^*$, $\mathtt{t}^*$ and $\varepsilon^*$ are the dualizing maps of $\mathtt{s}$, $\mathtt{t}$ and $\varepsilon$, respectively. In particular, the inverse of a morphism $\Phi$ is given by $\iota^*$, that is, $\Phi^{-1}=\iota^*(\Phi)= \Phi \circ \iota$, and the composition of morphisms is defined by the convolution product defined as follows: for all $\Phi, \Psi \in \mathscr{X}(\mathscr{G})_1$,
$$
\begin{equation*}
\Phi * \Psi (\overline{\varphi \otimes_{T_V} p})= \sum_{i=1}^n \Phi( \overline{\varphi \otimes_{T_V} s_i}) \Psi(\overline{s_i^* \otimes_{T_V} p})
\end{equation*}
\notag
$$
for every $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$ with $\Delta(\overline{\varphi \otimes_{T_V} p}):= \sum_{i=1}^n \overline{\varphi \otimes_{T_V} s_i} \otimes_{\mathtt{B}} \overline{s_i^* \otimes_{T_V} p} $. Here, $\{s_i, s_i^*\}_{1 \leqslant i \leqslant n}$ denotes the dual basis for the “global section” $\Gamma(\mathtt{F})$ in the direct sum expressed in (2), where $\mathtt{F}$ is the $\mathscr{G}$-representation underlying $V$ and $n$, the rank of the associated $\mathscr{G}$-vector bundle.
§ 3. Residually linear groupoidsWe will start this section by introducing the notions of residual linearity and residual finiteness for groupoids. Before moving on to the next definition, we briefly comment on equality in categories: let $x$, $y$, $v$, $w$ be objects of a locally small category $\mathscr{C}$ and $f\colon x \to y$, $g\colon v \to w$ be morphisms of $\mathscr{C}$. Then $f$ and $g$ are equal if we have equality on objects: $x=v$ and $y=w$, and if $f=g$ in the set of morphisms $\mathrm{Mor}_{\mathscr{C}}(x, y)=\mathrm{Mor}_{\mathscr{C}}(v, w)$. Definition 3.1. A groupoid $\mathscr{G}$ is said to be residually finite if, for all morphisms $f, g \in \mathscr{G}_1$, where $s(f)=s(g)$, $t(f)=t(g)$ and $f \neq g$, there exist a finite groupoid $\mathscr{D}$ and morphism of groupoids $\mathtt{F}\colon \mathscr{G} \to \mathscr{D}$ such that $\mathtt{F}(f) \neq \mathtt{F}(g)$. Definition 3.2. A groupoid $\mathscr{G}$ is said to be residually linear if, for all morphisms $f, g \in \mathscr{G}_1$, where $s(f)=s(g)$, $t(f)=t(g)$ and $f \neq g$, there exists a $\mathscr{G}$-representation $\mathtt{F}$ such that $\mathtt{F}(f) \neq \mathtt{F}(g)$. It is clear from Definitions 3.1 and 3.2 that if two groupoids are equivalent, then one is residually linear (respectively, finite) if and only if the other one is also residually linear (respectively, finite). Proposition 3.3. Let $(\mathscr{G}_i)_{i \in I}$ be a family of residually linear groupoids. Then the direct product $\prod_{i \in I} \mathscr{G}_i$ is a residually linear groupoid. Proof. Let $f=(f_i)$ and $g=(g_i)$ be morphisms in $\mathscr{G} := \prod_{i \in I} \mathscr{G}_i$ such that $f \neq g$, $s_i(f_i)=s_i(g_i)$ and $t_i(f_i)=t_i(g_i)$, where $s_i$, $t_i$ denote, respectively, the source and target maps of $\mathscr{G}_i$, for every $i \in I$. Then there exists $i_0 \in I$ such that $f_{i_0} \neq g_{i_0}$. But $f_{i_0}$ and $g_{i_0}$ are morphisms of $\mathscr{G}_{i_0}$ which are residually linear, and hence there exists a representation $\mathtt{F} \in \mathrm{rep}(\mathscr{G}_{i_0})$ such that $\mathtt{F}(f_{i_0}) \neq \mathtt{F}(g_{i_0})$. Consider the composite
$$
\begin{equation*}
\mathtt{G}:=\mathtt{F} \circ \pi_{i_0} \colon \mathscr{G} \to \mathscr{G}_{i_0} \to \mathrm{vect}_{\Bbbk},
\end{equation*}
\notag
$$
where $\pi_{i_0}$ is the canonical projection. Then $\mathtt{G} \in \mathrm{rep}(\mathscr{G})$ and $\mathtt{G}(f) \neq \mathtt{G}(g)$, proving Proposition 3.3. Denote by $\mathtt{Rvect}_{\Bbbk}$ the frame groupoid of $(\mathrm{vect}_{\Bbbk}, \pi)$, where $\pi\colon \mathrm{vect}_{\Bbbk} \to (\mathrm{vect}_{\Bbbk})_0$ is the canonical projection, see [5], Example 2.5. Note that, for a groupoid $\mathscr{G}$, the two categories $\mathrm{Fun}(\mathscr{G}, \mathtt{Rvect}_{\Bbbk})$ and $\mathrm{Fun}(\mathscr{G}, \mathrm{vect}_{\Bbbk})$ coincide. Definition 3.4. By a linear groupoid we mean a groupoid $\mathscr{G}$ along with a faithful representation $\mathtt{F} \colon \mathscr{G} \to \mathtt{Rvect}_{\Bbbk}$. From this definition it follows immediately that every linear groupoid is residually linear. Remark 3.5. If $\mathscr{G}$ is a groupoid with one object, then the notion in Definition 3.2 of residual linearity of $\mathscr{G}$ coincides for it with the usual one of groups. Proposition 3.6. Let $\mathscr{G}$ be a groupoid. Then (a) $\mathbb{I}_{\mathscr{G}}:= \bigcap_{\mathtt{F} \in \mathrm{rep}(\mathscr{G})} \mathrm{Ker}(\mathtt{F})$ is a normal subgroupoid of $\mathscr{G}$, (b) the following sequence of groupoids is a “short exact sequence” in the sense that the image of $J$ coincides with the kernel of $\pi$, where $\mathscr{G}/ \mathbb{I}_{\mathscr{G}}$ is the quotient groupoid and $J$ and $\pi$ are the canonical injection and projection, respectively. Proof. (a) It is not difficult to see that the kernel of a groupoid morphism is a normal subgroupoid, and the intersection of a family of normal subgroupoids is again a normal subgroupoid.
Assertion (b) is clear since $\mathbb{I}_{\mathscr{G}}$ is a normal subgroupoid of $\mathscr{G}$ and constitutes the kernel of $\pi$. Proposition 3.6 is proved. Recall (see [15], § IV.4) that a skeleton of a category $\mathscr{C}$ is a full subcategory $\mathbf{C}$ such that every object of $\mathscr{C}$ is isomorphic in $\mathscr{C}$ exactly to one object of $\mathbf{C}$. Applying a strong version of the axiom of choice, any groupoid can be shown as being equivalent to a totally disconnected groupoid, which is any of its skeletons [15]. Consequently, most of the results displayed for totally disconnected groupoids in what follows are in fact valid for any groupoid. Proposition 3.7. For a totally disconnected groupoid $\mathscr{G}$, the following conditions are equivalent: (a) $\mathscr{G}$ is residually linear; (b) for every $x \in \mathscr{G}_0$ and $1_x \neq f \in \mathscr{G}^x$, there is a $\mathscr{G}$-representation $\mathtt{F}_f$ such that $\mathtt{F}_f(f) \neq 1$; (c) for every $x \in \mathscr{G}_0$ and $1_x \neq f \in \mathscr{G}^x$, there exists a normal subgroupoid $\mathscr{N}_f$ of $\mathscr{G}$ not containing $f$ such that $\mathscr{G} / \mathscr{N}_f$ is linear; (d) the intersection of all normal subgroupoids $\mathscr{N}$ of $\mathscr{G}$ for which $\mathscr{G} / \mathscr{N}$ is linear is trivial. Proof. Implication (a) $\Rightarrow$ (b) is straightforward.
(b) $\Rightarrow$ (a). It suffices to note that, for any $f, g \in \mathscr{G}^x$ with $x \in \mathscr{G}_0$, we have $f \neq g$ if and only if $f g^{-1} \neq 1_x$. (a) $\Rightarrow$ (c). Let $x \in \mathscr{G}_0$ and $1_x \neq f \in \mathscr{G}^x$, then there exists $\mathtt{F}_f \in \mathrm{rep}(\mathscr{G})$ such that $\mathtt{F}_f(f) \neq 1$. Let $\mathscr{N}_f:= \mathrm{Ker}(\mathtt{F}_f)$. Then, by the universal property of quotient groupoids related to $\mathtt{F}_f$, there exists a faithful functor $\mathscr{G} / \mathscr{N}_f \to \mathtt{Rvect}_{\Bbbk}$. (c) $\Rightarrow$ (a). Let $1 \neq f \in \mathscr{G}_1$, we pick any faithful functor $\mathtt{F}_f\colon \mathscr{G} / \mathscr{N}_f \to \mathtt{Rvect}_{\Bbbk}$ and consider its composition with the canonical projection. (c) $\Rightarrow$ (d). Let $f \in \mathscr{N}$ for any $\mathscr{N} \triangleleft \mathscr{G}$ such that $\mathscr{G} / \mathscr{N}$ is linear. Assume $f \neq 1$. By the hypothesis, there exists $f \notin \mathscr{N}_f \triangleleft \mathscr{G}$ such that $\mathscr{G} / \mathscr{N}_f$ is linear. This contradiction shows that $f=1$. (d) $\Rightarrow$ (c). Assume that there exists $1 \neq f \in \mathscr{G}_1$ such that, for all $\mathscr{N} \triangleleft \mathscr{G}$ with $\mathscr{G} / \mathscr{N}$ is not linear, $f \notin \mathscr{N}$. Then every $\mathscr{N} \triangleleft \mathscr{G}$ such that $\mathscr{G} / \mathscr{N}$ is linear. Hence $f \in \mathscr{N}$, but their intersection is trivial by hypothesis. Contradiction with $1 \neq f$. Proposition 3.7 is proved. Proposition 3.8. Let $\mathscr{G}$ be a groupoid. Then $\mathscr{G}$ is residually linear if and only if $\mathbb{I}_{\mathscr{G}}= \mathcal{I}_{\mathscr{G}}$. The proof is straightforward from the definition of a residually linear groupoid. Proposition 3.9. Let $\mathscr{G}$ be a groupoid. Then $\mathscr{G}/ \mathbb{I}_{\mathscr{G}}$ is residually linear. Proof. This is due to the universal property of quotient groupoids, see [5], Proposition 2.16, which we recall here: let $\alpha \colon \mathscr{G} \to \mathscr{H}$ be a morphism of groupoids and $\mathscr{N}$ a normal subgroupoid of $\mathscr{G}$ such that, $\mathscr{N} \subseteq \mathrm{Ker}(\alpha)$. Then $\alpha$ factors uniquely as $\overline{\alpha} \circ \pi = \alpha$, where $\pi \colon \mathscr{G} \twoheadrightarrow \mathscr{G}/ \mathscr{N}$ is the canonical projection and $\overline{\alpha} \colon \mathscr{G}/ \mathscr{N} \to \mathscr{H}$ is defined on objects by $\overline{\alpha}(\overline{x})=\alpha(x)$, and on morphisms by $\overline{\alpha}(\overline{f})=\alpha(f)$, for every $\overline{x} \in (\mathscr{G}/ \mathscr{N})_0$ and $\overline{f} \in (\mathscr{G}/ \mathscr{N})_1$. Let us now prove the proposition: let $f \in \mathscr{G}_1$ such that its image $\overline{f}$ under the canonical projection is not an identity. This means that $f \notin ( \mathbb{I}_{\mathscr{G}})_1$ and so there exists a representation $\mathtt{F} \in \mathrm{rep}(\mathscr{G})$ such that $\mathtt{F}(f) \neq 1$. Since $\mathbb{I}_{\mathscr{G}} \subseteq \mathrm{Ker}(\mathtt{F})$, we can apply the universal property of quotient groupoids and obtain a representation $\overline{\mathtt{F}} \in \mathrm{rep}(\mathscr{G}/ \mathbb{I}_{\mathscr{G}})$ such that $\overline{\mathtt{F}}(\overline{f}) \neq 1$. Now the required result follows from Proposition 3.7. Proposition 3.9 is proved. Proposition 3.10. Let $\alpha\colon \mathscr{G} \to \mathscr{H}$ be a faithful groupoid morphism. If $\mathscr{H}$ is residually linear, then so is $\mathscr{G}$. The converse statement holds if in addition $\alpha(\mathbb{I}_{\mathscr{G}}) \supseteq \mathbb{I}_{\mathscr{H}}$. Proof. Note that $\alpha$ is faithful if and only if $\mathrm{Ker}(\alpha)=\mathcal{I}_{\mathscr{G}}$. The first result follows by taking a composition with $\alpha$. For the second one, it suffices to show that $\mathbb{I}_{\mathscr{H}}=\mathcal{I}_{\mathscr{H}}$ in view of Proposition 3.8. Since $\mathscr{G}$ is residually linear, we have $\mathbb{I}_{\mathscr{G}}=\mathcal{I}_{\mathscr{G}}$ by Proposition 3.8. On the other hand, $\alpha$ is faithful, and so $\alpha(\mathcal{I}_{\mathscr{G}})=\mathcal{I}_{\mathscr{H}}$. Since $\alpha(\mathbb{I}_{\mathscr{G}}) \supseteq \mathbb{I}_{\mathscr{H}}$, we obtain $\mathbb{I}_{\mathscr{H}} \subseteq \mathcal{I}_{\mathscr{H}}$. But we already have $\mathcal{I}_{\mathscr{H}} \subseteq \mathbb{I}_{\mathscr{H}}$. This completes the proof. Proposition 3.11. Let $\mathscr{G}$ be a groupoid. Then $\mathscr{G}$ is residually linear if and only if $\mathscr{G}$ is equivalent to a subgroupoid of a product $\prod_{i \in I} \mathtt{Rvect}_{\Bbbk}$ of $\mathtt{Rvect}_{\Bbbk}$. Proof. Clearly, every subgroupoid of a residually linear groupoid is also residually linear. Hence, if $\mathscr{G}$ is equivalent to a subgroupoid of $\prod_{i \in I} \mathtt{Rvect}_{\Bbbk}$, which is residually linear by Proposition 3.3, then $\mathscr{G}$ is residually linear as well.
Conversely, let $\mathscr{G}$ be a residually linear groupoid. Then, for all $f, g \in \mathscr{G}_1$, $f \neq g$, there exists $\mathtt{F}_{f, g} \in \mathrm{rep}(\mathscr{G})$ such that $\mathtt{F}_{f, g}(f) \neq \mathtt{F}_{f, g}(g)$. Consider $\mathtt{F}:=\prod_{f, g \in \mathscr{G}_1, f \neq g} \mathtt{F}_{f, g} \colon \prod_{f, g \in \mathscr{G}_1, f \neq g} \mathscr{G} \to \prod_{f, g \in \mathscr{G}_1, f \neq g} \mathtt{Rvect}_{\Bbbk}$. Then $\mathtt{F} \circ J$ is clearly faithful since so is $\mathtt{F}$ (in addition, $\mathtt{F} \circ J$ is injective on objects, and hence it is an embedding), where $J \colon \mathscr{G} \to \prod_{f, g \in \mathscr{G}_1, f \neq g} \mathscr{G}$ is the canonical injection. Finally, $\mathscr{G}$ is residually linear since it is equivalent to its image under $\mathtt{F} \circ J$. Proposition 3.11 is proved. Proposition 3.12. Let $\mathscr{G}$ be a totally disconnected groupoid. Then $\mathscr{G}$ is residually linear if and only if each of its isotropy groups is residually linear. Proof. Let $x \in \mathscr{G}_0$ and $1 \neq f \in \mathscr{G}^x$. Since $\mathscr{G}$ is residually linear, there exists a representation $\mathtt{F}_f \in \mathrm{rep}(\mathscr{G})$ such that $\mathtt{F}_f(f) \neq 1$. Let $\rho_f^x \colon \mathscr{G}^x \to \mathrm{GL}(\mathtt{F}_f(x))$, $g \mapsto \mathtt{F}_f(g)$. Then $\rho_f^x \in \mathrm{rep}(\mathscr{G}^x)$ and we have $\rho_f^x(f) \neq 1$.
Conversely, assume that for every $x \in \mathscr{G}_0$, $\mathscr{G}^x$ is a residually linear group. Then, for every $1 \neq f \in \mathscr{G}^x$, there exists a representation $\rho_f \in \mathrm{rep}(\mathscr{G}^x)$ on a finite dimensional $\Bbbk$-vector space $V^f$ such that $\rho_f(f) \neq 1$. Consider the functor $\mathtt{F}\colon \mathscr{G} \to \prod_{g \in \mathscr{G}_1} \mathtt{Rvect}_{\Bbbk}$ defined on objects as follows: for every $y \in \mathscr{G}_0$, $\mathtt{F}(y)= \prod_{g \in \mathscr{G}_1} W^g$, where $W^g= V^g$ if $g \in \mathscr{G}^y$, $g \neq 1_{y}$ and $W^g=\Bbbk$ otherwise, and it is defined on morphisms as follows: for every $h \in \mathscr{G}^y$, $\mathtt{F}(h)= \prod_{g \in \mathscr{G}_1} \rho(h)$, where $\rho(h)= \rho_g (h)$ if $g \in \mathscr{G}^y$, $g \neq 1$ and $\rho(h)= 1_{\Bbbk}$ otherwise. Hence, $\mathtt{F}$ is a well defined faithful groupoid morphism. Now $\mathscr{G}$ is residually linear by Proposition 3.10. Proposition 3.12 is proved. Corollary 3.13. Let $\mathscr{G}$ be a totally disconnected groupoid. The following conditions are equivalent: (a) $\mathscr{G}$ is residually linear; (b) $\mathbb{I}_{\mathscr{G}}= \mathcal{I}_{\mathscr{G}}$; (c) $\mathscr{G}^x$ is residually linear for every $x \in \mathscr{G}_0$; (d) $I_{\mathscr{G}^x}=\lbrace 1_x \rbrace$, for every $x \in \mathscr{G}_0$, where $I_{\mathscr{G}^x}=\bigcap_{\sigma \in \mathrm{rep}(\mathscr{G}^x)} \mathrm{Ker}(\sigma)$. This result follows by combining Propositions 3.8 and 3.12. Example 3.14. Let $H$ be a non residually linear group, for instance the counter linear groups, see [24] for some of such examples. Consider the one-object associated groupoid $\mathscr{H}$. Then $\mathscr{H}$ is a non residually linear groupoid by Corollary 3.13. Definition 3.15. Let $\mathscr{G}$ be a groupoid and $\mathscr{P}$ a subgroupoid of $\mathscr{G}$. We say that $\mathscr{P}$ is a residually linear subgroupoid of $\mathscr{G}$ if, for all $f, g \in \mathscr{P}_1$ with $s(f)=s(g)$, $t(f)=t(g)$ and $f \neq g$, there exists a representation $\mathtt{F}_{f, g} \in \mathrm{rep}(\mathscr{G})$ such that $\mathtt{F}_{f, g}(f) \neq \mathtt{F}_{f, g}(g)$. Remark 3.16. Let $\mathscr{G}$ be a groupoid. (a) If $\mathscr{G}$ is residually linear, then every subgroupoid of it is residually linear. (b) Let $\mathscr{P}$ be a subgroupoid of $\mathscr{G}$. Then $\mathscr{P}$ is a residually linear subgroupoid if and only if $\mathscr{P} \cap \mathbb{I}_{\mathscr{G}}= \mathcal{I}_{\mathscr{G}}$. Lemma 3.17. Let $\mathscr{G}$ be a totally disconnected groupoid and $\mathscr{N} \triangleleft \mathscr{G}$. If $\mathscr{G} / \mathscr{N}$ is residually linear, then $\mathbb{I}_{\mathscr{G}} \subseteq \mathscr{N}$. Proof. Let $f \in \mathscr{G}_1 \setminus \mathscr{N}_1$, then $\pi(f)$ is not identity in $\mathscr{G} / \mathscr{N}$, where $\pi$ is the canonical projection. So, there exists $\mathtt{F}_f \in \mathrm{rep}(\mathscr{G} / \mathscr{N})$ such that $\mathtt{F}(\pi(f)) \neq 1$, the result required. Lemma 3.17 is proved. Corollary 3.18. Let $\mathscr{G}$ be a totally disconnected groupoid, and $\mathscr{N}_1$, $\mathscr{N}_2$ be two normal subgroupoids of $\mathscr{G}$ such that $\mathscr{G} / \mathscr{N}_1$ and $\mathscr{G} / \mathscr{N}_2$ are residually linear and $\mathscr{N}_1 \cap \mathscr{N}_2 = \mathcal{I}_{\mathscr{G}}$. Then $\mathscr{G}$ is residually linear. Moreover, if the condition $\mathscr{N}_1 \cap \mathscr{N}_2 = \mathcal{I}_{\mathscr{G}}$ is dropped, then we only have $\mathscr{G} / (\mathscr{N}_1 \cap \mathscr{N}_2)$ is residually linear. Proof. By Lemma 3.17, $\mathbb{I}_{\mathscr{G}} \subseteq \mathscr{N}_1$ and $\mathbb{I}_{\mathscr{G}} \subseteq \mathscr{N}_2$, hence $\mathbb{I}_{\mathscr{G}} \subseteq \mathscr{N}_1 \cap \mathscr{N}_2$, as required by Proposition 3.8. For the second claim, remark that $\mathscr{G} / (\mathscr{N}_1 \cap \mathscr{N}_2)$ can be embedded canonically in the product $(\mathscr{G} / \mathscr{N}_1) \times (\mathscr{G} / \mathscr{N}_2)$. Now the result is secured by Proposition 3.10 in light of Remark 3.16 (a). Corollary 3.18 is proved. Corollary 3.19. Let $\mathscr{G}$ be a totally disconnected groupoid, and $\mathscr{N}$ be a proper minimal normal subgroupoid of $\mathscr{G}$ (by this we mean that if $\mathscr{M}$ is another proper normal subgroupoid of $\mathscr{G}$ such that $\mathscr{M} \subseteq \mathscr{N}$, then $\mathscr{M} = \mathscr{N})$ such that $\mathscr{G} / \mathscr{N}$ is residually linear. Then either $\mathscr{G}$ is residually linear, or $\mathbb{I}_{\mathscr{G}}$ is exactly the minimal subgroupoid $\mathscr{N}$. The proof is immediate from Lemma 3.17 since $\mathscr{N}$ is minimal. Lemma 3.20. Let $\mathscr{G}$ be a totally disconnected groupoid, and $\mathscr{N} \triangleleft \mathscr{G}$ be such that $\mathscr{N}$ and $\mathscr{G} / \mathscr{N}$ are residually linear. Then $\mathscr{G}$ is residually linear. Proof. $\mathscr{G} / \mathscr{N}$ being residually linear implies by Lemma 3.17 that $\mathbb{I}_{\mathscr{G}} \subseteq \mathscr{N}$. But $\mathscr{N}$ is a residually linear subgroupoid, hence by Remark 3.16 (b), $\mathscr{N} \cap \mathbb{I}_{\mathscr{G}}= \mathcal{I}_{\mathscr{G}}$. Thus, $\mathbb{I}_{\mathscr{G}}= \mathcal{I}_{\mathscr{G}}$. Lemma 3.20 is proved. Theorem 3.21. Let $\mathscr{G}$ be a totally disconnected groupoid. The following conditions are equivalent: (a) $\mathscr{G}$ is residually linear; (b) there exists a subnormal series of subgroupoids of $\mathscr{G}$ such that $\mathscr{N}_{i+1} / \mathscr{N}_i$ is residually linear for all $1 \leqslant i \leqslant n -1$, $n \in \mathbb{N^*}$. Proof. (a) $\Rightarrow$ (b). It suffices to consider the sequence $\mathcal{I}_{\mathscr{G}} \subseteq \mathscr{G}$.
(b) $\Rightarrow$ (a). Let $\mathcal{I}_{\mathscr{G}}= \mathscr{N}_1 \subseteq \mathscr{N}_2 \subseteq \dots \subseteq \mathscr{N}_n = \mathscr{G}$ be such a sequence. The sequence $\mathcal{I}_{\mathscr{G}}$ is residually linear in $\mathscr{N}_2$, but $\mathscr{N}_2 / \mathcal{I}_{\mathscr{G}} \simeq \mathscr{N}_2$ and $\mathscr{N}_2 / \mathcal{I}_{\mathscr{G}}$ is residually linear by hypothesis. Hence by Lemma 3.20, $\mathscr{N}_2$ is residually linear. Proceeding similarly continuing, we find that $\mathscr{N}_{n-1}$ and $\mathscr{G} / \mathscr{N}_{n-1}$ are residually linear, which implies again by Lemma 3.20 that $\mathscr{G}$ is residually linear. Theorem 3.21 is proved. Proposition 3.22. The following conditions are equivalent: (a) every finite groupoid is residually linear; (b) every residually finite groupoid is residually linear. Proof. (a) $\Rightarrow$ (b). Let $\mathscr{G}$ be a residually finite groupoid, and $f, g \in \mathscr{G}_1$ be such that $s(f)=s(g)$, $t(f)=t(g)$ and $f \neq g$. Then there exists a finite groupoid $\mathscr{D}$ and a groupoid morphism $\mathtt{F}'_{f, g}\colon \mathscr{G} \to \mathscr{D}$ such that $\mathtt{F}'_{f, g}(f) \neq \mathtt{F}'_{f, g}(g)$. By the hypothesis, there exists a functor $\mathtt{F}_{f, g}\colon \mathscr{D} \to \mathtt{Rvect}_{\Bbbk}$ such that $\mathtt{F}_{f, g} (\mathtt{F}'_{f, g}(f)) \neq \mathtt{F}_{f, g} (\mathtt{F}'_{f,g}(g))$. It now suffices to consider the composite $\mathtt{F}_{f, g} \circ \mathtt{F}'_{f, g}$.
The implication (b) $\Rightarrow$ (a) is straightforward as every finite groupoid is residually finite. Proposition 3.22 is proved. Proof. Let $\mathscr{G}$ be a finite groupoid and $\mathbf{G}$ be a skeleton of $\mathscr{G}$. Since $\mathbf{G}$ is a totally disconnected finite groupoid equivalent to $\mathscr{G}$, it suffices then to show that $\mathbf{G}$ is residually linear. As all isotropy groups $\mathbf{G}^x$, $x \in \mathbf{G}_0$, are finite, they are residually linear by the Peter–Weyl theorem [12], as any finite group is a compact Lie group with its discrete topology. Let $1 \neq f \in \mathbf{G}_1$, then $1 \neq f \in \mathbf{G}^x$ for some $x \in \mathbf{G}_0$ and there exists a representation $\rho_f$ of $\mathbf{G}^x$ on a finite dimensional $\Bbbk$-vector space $V^f$ such that $\rho_f(f) \neq 1$. Consider the functor $\mathtt{F}\colon \mathbf{G} \to \mathtt{Rvect}_{\Bbbk}$ defined as follows: for every $x \in \mathbf{G}_0$, $\mathtt{F}(x)= \prod_{g \in \mathbf{G}_1} W^g$, where $W^g=V^g$ if $1 \neq g \in \mathbf{G}^x$ and $W^g\,{=}\,\Bbbk$ otherwise, and for every $h \in \mathbf{G}^y$, $\mathtt{F}(h)= \prod_{g \in \mathbf{G}_1} \rho(h)$, where $\rho(h)=\rho_g(h)$ if $1 \,{\neq}\, g \,{\in}\, \mathbf{G}^y$, and $\rho(h)=1_{\Bbbk}$ otherwise. Hence, $\mathtt{F}$ is a faithful groupoid morphism. Now the result follows from Proposition 3.10. Proposition 3.23 is proved. The proof is a direct consequence of Propositions 3.22 and 3.23. This result is secured by Proposition 3.23 and since every weakly finite category is equivalent to a finite category. Proof. We proceed as in Proposition 3.3. Let $(\mathscr{G}_i)_{i \in I}$ be a family of residually finite groupoids. Let $f=(f_i)$ and $g=(g_i)$ be morphisms of $\mathscr{G} := \prod_{i \in I} \mathscr{G}_i$ such that $s(f_i)=s(g_i)$, $t(f_i)=t(g_i)$ for every $i \in I$ and $f \neq g$. Then there exists some $i_0 \in I$ such that $f_{i_0} \neq g_{i_0}$ in $\mathscr{G}_{i_0}$. But $\mathscr{G}_{i_0}$ is residually finite. Hence there exist a finite groupoid $\mathscr{D}_{i_0}$ and a functor $\mathtt{F}_{i_0} \in \mathrm{Fun}(\mathscr{G}_{i_0}, \mathscr{D}_{i_0})$ such that $\mathtt{F}_{i_0}(f_{i_0}) \neq \mathtt{F}_{i_0}(g_{i_0})$. We set
$$
\begin{equation*}
\mathtt{F}:=\mathtt{F}_{i_0} \circ \pi_{i_0} \colon \mathscr{G} \to \mathscr{G}_{i_0} \to \mathscr{D}_{i_0}.
\end{equation*}
\notag
$$
Then $\mathtt{F} \in \mathrm{Fun}(\mathscr{G}, \mathscr{D}_{i_0})$ and $\mathtt{F}(f) \neq \mathtt{F}(g)$. Proposition 3.26 is proved. Proposition 3.27. A totally disconnected groupoid is residually finite if and only if each of its isotropy groups is residually finite. Proof. Let $\mathscr{G}$ be totally disconnected groupoid, $x \in \mathscr{G}_0$ and $1 \neq f \in \mathscr{G}^x$. Then there exist a finite groupoid $\mathscr{D}_f$ and a functor $\mathtt{F}_f \in \mathrm{Fun}(\mathscr{G}, \mathscr{D}_f)$ such that $\mathtt{F}_f(f) \neq 1$. Consider $\mathtt{F}'_f\colon \mathscr{G}^x \to \mathscr{D}_f^{\mathtt{F}_f(x)}$, $g \mapsto \mathtt{F}'_f(f)=\mathtt{F}_f(g)$. Thus, $\mathscr{D}_f^{\mathtt{F}_f(x)}$ is a finite group, $\mathtt{F}'_f$ is a group homomorphism and $\mathtt{F}'_f(f) \neq 1$.
Conversely, let $x \in \mathscr{G}_0$ and $1_x \neq f \in \mathscr{G}^x$. Then there exists a finite group $D_f$ and a group homomorphism $\mathtt{F}_f\colon \mathscr{G}^x \to D_f$ such that $\mathtt{F}_f(f) \neq 1$. Consider the associated one-object (we denote it, for example, by $*$) groupoid $\mathscr{D}_f$ of $D_f$. It easy to see that $\mathtt{F}_f$ extends to a groupoid morphism $\overline{\mathtt{F}}_f\colon (x, \mathscr{G}^x) \to \mathscr{D}_f$ by setting $\overline{\mathtt{F}}_f(x)=*$, and $\overline{\mathtt{F}}_f(g)=\mathtt{F}_f(g)$ for every $g \in \mathscr{G}^x$. Let $\pi_x\colon \mathscr{G} \to (x, \mathscr{G}^x)$ be the canonical projection (that is, the functor associating $x$ to every $y \in \mathscr{G}_0$, and $\pi_x(f)=f$ if $f \in \mathscr{G}^x$ and $1_x$ otherwise). Now, we set $\mathtt{G}_f=\overline{\mathtt{F}}_f \circ \pi_x$. Then $\mathtt{G}_f$ is a groupoid morphism to a finite groupoid $\mathscr{D}_f$ and satisfies $\mathtt{G}_f(f) \ne 1$. Proposition 3.27 is proved. Definition 3.28. Let $\mathscr{G}$ be a groupoid. The center $Z(\mathscr{G})$ of $\mathscr{G}$ is the subgroupoid defined by (a) $Z(\mathscr{G})_0=\mathscr{G}_0$; (b) $Z(\mathscr{G})_1= \{ f \in \mathscr{G}^x \ \forall\, g \in \mathscr{G}^x, \, fg=gf \}_{x \in \mathscr{G}_0}$. Proposition 3.29. $Z(\mathscr{G})$ is a normal subgroupoid of $\mathscr{G}$. In addition, $Z(\mathscr{G})_1=\biguplus_{x \in \mathscr{G}_0} Z(\mathscr{G}^x)$. The proof is immediate from Definition 2.5. Definition 3.30. Let $\mathscr{G}$ be a groupoid. The commutator $\mathscr{G}'$ of $\mathscr{G}$ is the subgroupoid defined by (a) $\mathscr{G}'_0=\mathscr{G}_0$; (b) $\mathscr{G}'_1= \{ f_1 g_1f_1^{-1} g_1^{-1} \cdots f_n g_nf_n^{-1} g_n^{-1}, \ f_1, \dots, f_n, g_1, \dots, g_n \in \mathscr{G}^x, \, n \in \mathbb{N}^* \}_{x \in \mathscr{G}_0}$. Proposition 3.31. $\mathscr{G}'$ is a normal subgroupoid of $\mathscr{G}$. In addition, $\mathscr{G}'_1=\biguplus_{x \in \mathscr{G}_0} \mathscr{G}^{x'}$, where $\mathscr{G}^{x'}$ is the commutator of $\mathscr{G}^x$ for every $x \in \mathscr{G}_0$. The proof is immediate from Definition 2.5. Definition 3.32. Let $\mathscr{G}$ be a groupoid and $\mathscr{D} \subseteq \mathscr{G}$ a subgroupoid. We say that $\mathscr{D}$ is finitely generated if there exists a finite family $F \subseteq \mathscr{G}_1$ of morphisms such that $F$ generates $\mathscr{D}^x$ as a group for every $x \in \mathscr{D}_0$. In particular, the isotropy groups $\mathscr{D}^x$ are finitely generated for every $x \in \mathscr{D}_0$, and if, in addition, $\mathscr{D}$ is normal in $\mathscr{G}$, then the set of objects $\mathscr{G}_0$ is finite. Proposition 3.33. Let $\mathscr{M}$ and $\mathscr{G}$ be equivalent groupoids such that $\mathscr{M}$ is finitely generated. Then $\mathscr{G}$ is finitely generated. Proof. Let $\alpha \colon \mathscr{M} \to \mathscr{G}$ be such an equivalence and $F$ be a finite generating set of $\mathscr{M}$. Let $x \in \mathscr{G}_0$ and $f \in \mathscr{G}^x$. Since $\alpha$ is essentially surjective, there exists $x' \in \mathscr{M}_0$ such that $f \in \mathscr{G}^{\alpha(x')}$. The fullness of $\alpha$ implies that there exists $g \in \mathscr{M}^{x'}$ such that $\alpha(g)=f$, but $g=g_1^{\epsilon_1} \cdots g_n^{\epsilon_n}$ by the assumption with $\{ g_1, \dots, g_n \} \subseteq F$, $\epsilon_i \in \{ -1, 1 \}$ for every $1 \leqslant i \leqslant n$, and $n \in \mathbb{N^*}$. So, $f=( \alpha(g_1))^{\epsilon_1} \cdots ( \alpha(g_n))^{\epsilon_n}$ with $\{ ( \alpha(g_1))^{\epsilon_1}, \dots, ( \alpha(g_n))^{\epsilon_n} \} \subseteq \alpha(F) \cap \mathscr{G}^x$. Hence $\mathscr{G}$ is generated by $\alpha(F)$ and is finite. Proposition 3.33 is proved. For groups, residual finiteness of finitely generated linear groups was established by Mal’cev [7]. We extend this result to groupoids. Proposition 3.34. Every finitely generated linear and totally disconnected groupoid is residually finite. Proof. Let $\mathscr{G}$ be a finitely generated linear and totally disconnected groupoid. Then its isotropy groups are also finitely generated. On the other hand, linearity of $\mathscr{G}$ implies that these groups are linear as well. Hence, by the above mentioned result of Mal’cev [7], these groups are residually finite. Using now Proposition 3.27, the groupoid $\mathscr{G}$ becomes residually finite. Proposition 3.34 is proved. Theorem 3.35. Let $\mathscr{G}$ be a finitely generated, totally disconnected groupoid. Then $\mathscr{G}$ is residually finite if and only if it is residually linear. Proof. The necessity is secured by the general case considered in Corollary 3.24.
Conversely, assume that $\mathscr{G}$ is a residually linear groupoid. It suffices to show that its isotropy groups are residually finite, by Proposition 3.27. The isotropy groups of $\mathscr{G}$ are finitely generated, whereas, Proposition 3.12 implies that they are residually linear. According to Mal’cev [7], finitely generated residually linear groups are residually finite, hence the isotropy groups of $\mathscr{G}$ are residually finite. Finally, $\mathscr{G}$ is residually finite. Theorem is proved. Lemma 3.36. Let $\mathscr{G}$ be a totally disconnected groupoid. Then $Z(\mathscr{G})$ (respectively, $\mathscr{G'})$ is finitely generated if and only if $Z(\mathscr{G}^x)$ (respectively, $(\mathscr{G}^x)')$ is finitely generated, for every $x \in \mathscr{G}^x$, where $Z(\mathscr{G}^x)$ (respectively, $(\mathscr{G}^x)')$ is the centre (respectively, the commutator) subgroup of $\mathscr{G}^x$. Proof. Let $F$ be a finite generating set of $Z(\mathscr{G})$. Then, clearly, $F \cap \mathscr{G}^x$ is a finite generating set of $Z(\mathscr{G}^x)$ for every $x \in \mathscr{G}_0$.
Conversely, let $F_x$ be a finite generating set of $Z(\mathscr{G}^x)$ for every $x \in \mathscr{G}_0$. Then the disjoint union $F= \biguplus_{x \in \mathscr{G}_0} F_x$ is a finite generating set of $Z(\mathscr{G})$. The same is also true for $\mathscr{G'}$. This proves the lemma. Definition 3.37. A groupoid $\mathscr{G}$ is said to be nilpotent if there is a series of normal subgroupoids of $\mathscr{G}$, where $Z_1 = Z(\mathscr{G})$ and $Z_{i+1}$ is such that $Z_{i+1} / Z_i = Z(\mathscr{G} / Z_i)$, for every $0 \leqslant i \leqslant n -1$ and $n \in \mathbb{N^*}$. Lemma 3.38. Let $\mathscr{G}$ be a groupoid and $\mathscr{N} \triangleleft \mathscr{G}$. Then $(\mathscr{N} \cap \mathscr{G}^x) \triangleleft \mathscr{G}^x$ for every $x \in \mathscr{G}_0$. Proof. First, by $\mathscr{N} \cap \mathscr{G}^x$ we mean their intersection as subcategories of $\mathscr{G}$, which results in a subcategory with one object $x$ and with morphisms that are both in $\mathscr{G}^x$ and $\mathscr{N}$, hence $\mathscr{N} \cap \mathscr{G}^x$ is a subgroupoid of $\mathscr{G}^x$, which keeps its normality structure. The result holds then by simply considering $\mathscr{N} \cap \mathscr{G}^x$ and $\mathscr{G}^x$ as groups. This proves the lemma. Lemma 3.39. Let $\mathscr{G}$ be a groupoid and $\mathscr{N}$ a subgroupoid of $\mathscr{G}$. Then $Z( \mathscr{N} \cap \mathscr{G}^x) = Z(\mathscr{N}) \cap \mathscr{G}^x$, for every $x \in \mathscr{G}_0$. The proof is immediate. Proposition 3.40. Let $\mathscr{G}$ be a nilpotent groupoid. Then the isotropy groups $\mathscr{G}^x$, $x \in \mathscr{G}_0$, are all nilpotent. Proof. Let $\mathcal{I}_{\mathscr{G}}= Z_0 \subseteq Z_1 \subseteq \dots \subseteq Z_n = \mathscr{G}$ be a series as in Definition 3.37, then for every $x \in \mathscr{G}_0$, it induces a series $\lbrace e \rbrace = Z_0 \cap \mathscr{G}^x \subseteq Z_1 \cap \mathscr{G}^x \subseteq \dots \subseteq Z_n \cap \mathscr{G}^x = \mathscr{G}^x$ in $\mathscr{G}^x$, making it into a nilpotent group in light of Lemmas 3.38 and 3.39. Proposition 3.40 is proved. Theorem 3.41. Let $\mathscr{G}$ be a residually linear nilpotent totally disconnected groupoid such that the center $Z(\mathscr{G})$ and commutator $\mathscr{G}'$ subgroupoids are finitely generated. Then $\mathscr{G}$ is residually finite. Proof. Combining Lemma 3.36 and Proposition 3.40, all isotropy groups of $\mathscr{G}$ are nilpotent and having finitely generated centers and commutators. By the result of Menal (see [10], Theorem 1), all isotropy groups of $\mathscr{G}$ are residually finite. Applying Proposition 3.27, $\mathscr{G}$ is then residually finite. This proves the theorem. The automorphisms group of a finitely generated residually finite (respectively, linear) group is also residually finite (respectively, linear) [13]. In the next proposition, we provide a similar result for groupoids, which need not be finitely generated. Let $\mathscr{G}$ be a groupoid. Denote by $\mathtt{End}(\mathscr{G})$ the category whose objects are endofunctors on $\mathscr{G}$, and morphisms are natural transformations between them. Note that all natural transformations are natural isomorphisms in this case. Denote by $\mathtt{end}(\mathscr{G})$ the full subcategory of $\mathtt{End}(\mathscr{G})$ (possibly empty) whose objects $\mathtt{F}, \mathtt{G} \colon \mathscr{G} \to \mathscr{G}$ satisfy: $\mathtt{F} \neq \mathtt{G}$ implies that there is some $u \in \mathscr{G}_0$ such that $\mathtt{F}_0(u) \neq \mathtt{G}_0(u)$ for every $\mathtt{F}, \mathtt{G} \in \mathtt{end}(\mathscr{G})_0$. Proposition 3.42. Let $\mathscr{G}$ be a residually linear (respectively, finite) totally disconnected groupoid. Then $\mathtt{end}(\mathscr{G})$ is residually linear (respectively, finite). Proof. By definition of its objects, $\mathtt{end}(\mathscr{G})$ is a totally disconnected groupoid. By Proposition 3.12, it suffices to show that the isotropy groups $\mathtt{end}(\mathscr{G})^{\mathtt{F}}$, $\mathtt{F} \colon \mathscr{G} \to \mathscr{G}$, are residually linear. For every $\mathtt{F} \colon \mathscr{G} \to \mathscr{G}$, consider the corresponding isotropy group $\mathtt{end}(\mathscr{G})^{\mathtt{F}}:= \lbrace \beta \colon \mathtt{F} \to \mathtt{F} \rbrace := \{ (\beta)_x \colon \mathtt{F}(x) \to \mathtt{F}(x) \}_{x \in \mathscr{G}_0}$. Thus, it amounts to show that, for every $x \in \mathscr{G}_0$, the group $\mathtt{I}_x:=\{ (\beta)_x \colon \mathtt{F}(x) \to \mathtt{F}(x) \}_{\beta\colon \mathtt{F} \to \mathtt{F}}$ is residually linear. But $\mathtt{I}_x$ is a subgroup of $\mathscr{G}^{\mathtt{F}(x)}$, which is residually linear by the hypothesis in view of Proposition 3.12. Hence $\mathtt{I}_x$ is residually linear as well, which completes the proof for residual linearity. The same procedure can be followed to prove the statement for residual finiteness. Proposition 3.42 is proved. More generally, similarly to above, denote by $\mathrm{fun}(\mathscr{G}, \mathscr{M})$ the full subcategory of the functor category $\mathrm{Fun}(\mathscr{G}, \mathscr{M})$ between two groupoids $\mathscr{G}$ and $\mathscr{M}$, whose objects are such that, for every $\mathtt{F}, \mathtt{G} \colon \mathscr{G} \to \mathscr{M}$ such that $\mathtt{F} \neq \mathtt{G}$, we have $\mathtt{F}_0(u) \neq \mathtt{G}(u)$, for some $u \in \mathscr{G}_0$. Corollary 3.43. Let $\mathscr{M}$ and $\mathscr{G}$ be two groupoids where $\mathscr{G}$ is totally disconnected and residually linear. Then $\mathrm{fun}(\mathscr{M}, \mathscr{G})$ is a residually linear groupoid. § 4. Application to the character groupoidTo get a straight generalization of the construction of the universal object $\mathcal{R}_{\Bbbk}(\mathscr{G})$ from a given groupoid $\mathscr{G}$ in the same fashion as for the case of groups, we shall work (in this section) with a transitive groupoid $\mathscr{G}$. In particular, the vector $\mathscr{G}$-bundle of any $\mathscr{G}$-representation has constant rank, namely, all fibres have equal dimensions. Lemma 4.1 (see [21], Lemma 3.8). Let $\mathscr{G}$ be a groupoid. Then the assignment establishes a groupoid morphism, where $\zeta$ is the injection defined in (3). Proposition 4.2. Assume that $\mathscr{G}$ is residually linear. Then $\mathtt{F}$ is a faithful groupoid morphism. Proof. In view of Lemma 4.1 we need only to show that $\mathtt{F}$ is faithful. Let $x, y \in \mathscr{G}_0$ and $f, g \in \mathscr{G}_1$ such that $s(f)=s(g)=x$, $t(f)=t(g)=y$ and $f \neq g$. Assume that $\mathtt{F}(f)=\mathtt{F}(g)$, then $\mathtt{F}(f g^{-1})=\varepsilon$, and so, for every $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$, $\mathtt{F}(f g^{-1})(\overline{\varphi \otimes_{T_V} p}) = \varepsilon (\overline{\varphi \otimes_{T_V} p})$. Since $\mathscr{G}$ is residually linear, there exists then a $\mathscr{G}$-representation $\mathtt{G}$ such that $\mathtt{G}(f g^{-1}) \neq 1_{W}$ for some $\Bbbk$-vector space $W$ with $\mathrm{dim}_{\Bbbk}(W)=n$. Let $\mathtt{G}(f g^{-1})=(\mathtt{G}^{i, j})_{i, j}$, $i, j \in \{ 1, \dots, n \}$. Note that, for every $h \in \mathscr{G}_1$, $\mathtt{G}^{i, j} (h)= \zeta(\overline{s_i^* \otimes_{T_V} s_i}) (h)$, where $\{s_i, s_i^*\}$ is the corresponding dual basis of $W$. Hence $\mathtt{G}^{i, j}= \zeta(\overline{s_i^* \otimes_{T_V} s_i})$ for any $i, j \in \{ 1, \dots, n \}$. Now applying $\mathtt{F}(f g^{-1})$ and $\varepsilon$ on the elements $\overline{s_i^* \otimes_{T_V} s_i}$, we obtain $\mathtt{G}^{i, j}(f g^{-1})= \overline{s_i^* \otimes_{T_V} s_i} (1_{y})= \delta_{i, j}. 1_{\Bbbk}$ for all $i, j \in \{ 1, \dots, n \}$. This contradicts $\mathtt{G}(f g^{-1}) \neq 1_{W}$. Hence $\mathtt{F}(f)\neq \mathtt{F}(g)$, and now the faithfulness holds. Proposition 4.2 is proved. Proposition 4.3. Let $\mathscr{G}$ be a residually linear groupoid. The following conditions are equivalent: (a) $\mathscr{X}(\mathscr{G})$ is abelian; (b) $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is cocommutative; (c) $\mathscr{G}$ is abelian. Proof. We shall proceed as follows: $(\mathrm{c}) \Rightarrow (\mathrm{b}) \Rightarrow (\mathrm{a}) \Rightarrow (\mathrm{c})$.
(c) $\Rightarrow$ (b). Let $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$ and $f, g \in \mathscr{G}_1$. We have
$$
\begin{equation*}
\begin{aligned} \, &(\zeta \otimes \zeta)\biggl( \sum_{i=1}^n \overline{\varphi \otimes_{T_V} s_i} \otimes_{\mathtt{B}} \overline{s_i^* \otimes_{T_V} p}\biggr) (f \otimes g)= \sum_{i=1}^n \zeta(\overline{\varphi \otimes_{T_V} s_i})(f) \, \zeta(\overline{s_i^* \otimes_{T_V} p})(g) \\ &= \zeta(\overline{\varphi \otimes_{T_V} p})(fg) = \zeta(\overline{\varphi \otimes_{T_V} p})(gf) = \sum_{i=1}^n \zeta(\overline{\varphi \otimes_{T_V} s_i})(g)\, \zeta(\overline{s_i^* \otimes_{T_V} p})(f) \\ &= \sum_{i=1}^n \zeta(\overline{s_i^* \otimes_{T_V} p})(f)\, \zeta(\overline{\varphi \otimes_{T_V} s_i})(g) =(\zeta \otimes \zeta)\biggl( \sum_{i=1}^n \overline{s_i^* \otimes_{T_V} p} \otimes_{\mathtt{B}} \overline{\varphi \otimes_{T_V} s_i} \biggr) (f \otimes g). \end{aligned}
\end{equation*}
\notag
$$
On the other hand, $\zeta \otimes \zeta$ is injective since so is $\zeta$, and now the implication holds.
(b) $\Rightarrow$ (a). Since $\mathcal{R}_{\Bbbk}(\mathscr{G})$ is cocommutative, we have, for every $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$,
$$
\begin{equation*}
\sum_{i=1}^n \overline{\varphi \otimes_{T_V} s_i} \otimes_{\mathtt{B}} \overline{s_i^* \otimes_{T_V} p} = \sum_{i=1}^n \overline{s_i^* \otimes_{T_V} p} \otimes_{\mathtt{B}} \overline{\varphi \otimes_{T_V} s_i}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, \Phi * \Psi (\overline{\varphi \otimes_{T_V} p}) &= \sum_{i=1}^n \Phi ( \overline{\varphi \otimes_{T_V} s_i}) \Psi (\overline{s_i^* \otimes_{T_V} p}) \\ &=m_{\Bbbk} (\Phi \otimes \Psi) \biggl( \sum_{i=1}^n \overline{\varphi \otimes_{T_V} s_i} \otimes_{\mathtt{B}} \overline{s_i^* \otimes_{T_V} p} \biggr) \\ &= m_{\Bbbk} (\Phi \otimes \Psi) \biggl( \sum_{i=1}^n \overline{s_i^* \otimes_{T_V} p} \otimes_{\mathtt{B}} \overline{\varphi \otimes_{T_V} s_i} \biggr)= \Psi * \Phi (\overline{\varphi \otimes_{T_V} p}), \end{aligned}
\end{equation*}
\notag
$$
where $m_{\Bbbk}$ denotes the product of $\Bbbk$ (which is commutative).
(a) $\Rightarrow$ (c). Let $x \in \mathscr{G}_0$ and $f, g \in \mathscr{G}^x$. For every $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$, we have
$$
\begin{equation*}
\begin{aligned} \, \mathtt{F}(fg) &=\zeta(\overline{\varphi \otimes_{T_V} p})(fg) = \sum_{i=1}^n \zeta(\overline{s_i^* \otimes_{T_V} p})(f) \, \zeta(\overline{s_i^* \otimes_{T_V} p})(g) \\ &= \sum_{i=1}^n \mathtt{F}(f)(\overline{s_i^* \otimes_{T_V} p}) \, \mathtt{F}(g)(\overline{s_i^* \otimes_{T_V} p}) \\ &= \mathtt{F}(f) * \mathtt{F}(g) \, (\overline{\varphi \otimes_{T_V} p}) = \mathtt{F}(g) * \mathtt{F}(f) \, (\overline{\varphi \otimes_{T_V} p}) \\ &= \sum_{i=1}^n \zeta(\overline{s_i^* \otimes_{T_V} p})(g) \, \zeta(\overline{s_i^* \otimes_{T_V} p})(f) = \zeta(\overline{\varphi \otimes_{T_V} p})(gf) = \mathtt{F}(gf). \end{aligned}
\end{equation*}
\notag
$$
Since $\mathtt{F}$ is faithful by Proposition 4.2, we get that $fg=gf$, and this completes the proof. Proposition 4.4. Consider the following map: Then $\mathrm{Im}(\Theta) \subseteq \mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$, and thus, $\mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$ is a subalgebra of the total algebra $\mathrm{M}_{\Bbbk}(\mathscr{X}(\mathscr{G})_1)$. Proof. Let $\overline{\varphi \otimes_{T_V} p} \in \mathcal{R}_{\Bbbk}(\mathscr{G})$ be such that $\Delta(\overline{\varphi \otimes_{T_V} p}):= \!\sum_{i=1}^n \overline{\varphi \otimes_{T_V} s_i} \otimes_{\mathtt{B}} \overline{s_i^* \otimes_{T_V} p}$ and $\alpha, \alpha' \in \mathscr{X}(\mathscr{G})$. Then
$$
\begin{equation*}
\begin{aligned} \, \Theta(\overline{\varphi \otimes_{T_V} p})( \alpha * \alpha') &= \sum_{i=1}^n \alpha(\overline{\varphi \otimes_{T_V} s_i}) \alpha' ( \overline{s_i^* \otimes_{T_V} p}) \\ &= \sum_{i=1}^n \Theta (\overline{\varphi \otimes_{T_V} s_i})(\alpha) \Theta ( \overline{s_i^* \otimes_{T_V} p})(\alpha'), \end{aligned}
\end{equation*}
\notag
$$
whence, on the one hand,
$$
\begin{equation*}
\Delta' (\Theta(\overline{\varphi \otimes_{T_V} p}))=\sum_{i=1}^n \Theta (\overline{\varphi \otimes_{T_V} s_i}) \otimes_{\mathtt{B}(\Bbbk)} \Theta ( \overline{s_i^* \otimes_{T_V} p}),
\end{equation*}
\notag
$$
and on the other hand, $\Theta(\overline{\varphi \otimes_{T_V} p}) \in \mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$ due to the characterization of the elements of $\mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$ as in (1), where $\Delta'$ denotes the coproduct of the character groupoid $\mathscr{X}(\mathscr{G})$. Proposition 4.4 is proved. Theorem 4.5. Let $\mathscr{G}$ be a groupoid. Then the character groupoid $\mathscr{X}(\mathscr{G})$ is residually linear. Proof. Let $\alpha, \alpha' \in \mathscr{X}(\mathscr{G})_1$ be such that $\mathtt{s}^*(\alpha)=\mathtt{s}^*(\alpha')$, $\mathtt{t}^*(\alpha)=\mathtt{t}^*(\alpha')$ and $\alpha \neq \alpha'$. Then there exists $\theta \in \mathcal{R}_{\Bbbk}(\mathscr{G})$ such that $\alpha(\theta) \neq \alpha'(\theta)$. This means that But $\Theta(\theta) \in \mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$ by Proposition 4.4. Hence it is of the form $\Theta(\theta)=\overline{\psi \otimes_{T_V} q}$, for some finite dimensional $\Bbbk$-vector space $V$. As a result, $\overline{\psi \otimes_{T_V} q} (\alpha) \neq \overline{\psi \otimes_{T_V} q} (\alpha')$, namely
$$
\begin{equation*}
\psi(\mathtt{t}^*(\alpha)) [\mathtt{F}(\alpha)(q(\mathtt{s}^*(\alpha)))] \neq \psi(\mathtt{t}^*(\alpha')) [\mathtt{F}(\alpha')(q(\mathtt{s}^*(\alpha')))],
\end{equation*}
\notag
$$
where $\mathtt{F}$ is the $\mathscr{X}(\mathscr{G})$-representation underlying $V$. Consequently, $\mathtt{t}^*(\alpha)=\mathtt{t}^*(\alpha')$ implies that $\mathtt{F}(\alpha)(q(s(x))) \neq \mathtt{F}(\alpha')(q(s(x)))$ and $\mathtt{s}^*(\alpha)=\mathtt{s}^*(\alpha')$ implies that $\mathtt{F}(\alpha) \neq \mathtt{F}(\alpha')$. Theorem 4.5 is proved. Remark 4.6. Notice that (4) shows that the Hopf algebroid $\mathcal{R}_{\Bbbk}(\mathscr{X}(\mathscr{G}))$ of representative functions of the character groupoid $\mathscr{X}(\mathscr{G})$ separates its morphisms, which recovers the same fact in the case of a compact Lie group [12]. AcknowledgementThe authors would like to thank the referees for their helpful comments and suggestions, which have enhanced the readability of the paper.
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\Bibitem{DraChoMou24} |
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