The purpose of this note is to show that the category of internal groupoids in an arbitrary category is equivalent to the full subcategory of involutive-2-links that are unital and associative. An involutive-2-link consists of a morphism together with two intertwined involutions on its domain. This approach to internal groupoids contrasts with the one in which a groupoid is seen as a reflexive graph equipped with a multiplicative structure (as e.g. in [5]). Here, the underlying reflexive graph is seen as a unique structure (provided it exists) associated with an involutive-2-link, thus recapturing once again a geometric and differential perspective [3], as briefly explained in [6]. Moreover, since the ambient category is arbitrary, this approach is suitable to study Lie groupoids.

FormalPara Theorem 1

Let \({\textbf{C}}\) be any category. The category of internal groupoids is equivalent to the full subcategory of unital and associative involutive-2-links.

The category of involutive-2-links, internal to an arbitrary category \({\textbf{C}}\), consists of triples \((\theta ,\varphi ,m:{A\rightarrow B})\) where \(m:A\rightarrow B\) is any morphism in \({\textbf{C}}\), whereas \(\theta ,\varphi :{A\rightarrow A}\) are such that \(\theta ^2=\varphi ^2=1_A\) and \(\theta \varphi \theta =\varphi \theta \varphi \). A morphism of involutive-2-links, from \((\theta ,\varphi ,m:{A\rightarrow B})\) to \((\theta ',\varphi ',m':{A'\rightarrow B'})\), is a pair of morphisms \((f:{A\rightarrow A'},g:{B\rightarrow B'})\) such that \(f\theta =\theta ' f\), \(f\varphi =\varphi ' f\) and \(m'f=gm\).

FormalPara Definition 1

Let \({\textbf{C}}\) be any category. An involutive-2-link structure in \({\textbf{C}}\), say \((\theta ,\varphi ,m:C_2\rightarrow C_1)\), is said to be:

  1. (1)

    associative when the pair \((m\varphi ,m\theta )\) is bi-exact (see diagram (7) bellow with \(\pi _1=m\varphi \) and \(\pi _2=m\theta \)) and the induced morphisms \(m_1,m_2:C_3\rightarrow C_2\), determined by (see diagram (8))

    $$\begin{aligned} \pi _1m_1=mp_1,\quad \pi _2 m_1=\pi _2 p_2\\ \pi _1 m_2=\pi _1 p_1, \quad \pi _2 m_2=m p_2 \end{aligned}$$

    are such that \(mm_1=mm_2\).

  2. (2)

    unital when the two pairs of morphisms \((m,m\theta )\), \((m,m\varphi )\) are jointly monomorphic and there exist morphisms \(e_1,e_2:C_1\rightarrow C_2\) such that

    $$\begin{aligned}&me_1=1_{C_1}=me_2 \end{aligned}$$
    (1)
    $$\begin{aligned}&\theta e_2=e_2,\quad \varphi e_1=e_1 \end{aligned}$$
    (2)
    $$\begin{aligned}&m\theta \varphi e_2=m\varphi \theta e_1 \end{aligned}$$
    (3)
    $$\begin{aligned}&m\theta e_1 m\varphi =m\varphi e_2 m\theta \end{aligned}$$
    (4)
    $$\begin{aligned}&m\theta e_1 m=m\theta e_1 m\theta \end{aligned}$$
    (5)
    $$\begin{aligned}&\quad m\varphi e_2 m=m\varphi e_2 m\varphi . \end{aligned}$$
    (6)

A pair of parallel morphisms (or a graph) is said to be bi-exact if when considered as a span it can be completed into a commutative square which is both a pullback and a pushout and moreover, if considered as a cospan, it can be completed into another commutative square which is both a pullback and a pushout. In other words, a graph such as

(7)

is bi-exact precisely when there exist \(p_1,p_2,d,c\) as displayed

(8)

such that both squares are commutative and simultaneously a pullback and pushout. Such squares are also called exact squares, bicartesian squares, Dolittle diagrams or pulation squares [1].

The functor, say F, from the category of internal groupoids to the category of involutive-2-links, defined via the assignment

(9)

with \(\theta =\langle i\pi _1,m\rangle \), \(\varphi =\langle m,i\pi _2\rangle \) is full and faithful. This functor takes an internal groupoid (see e.g. [2], Section 7.1), forgets its underlying reflexive graph, keeps the multiplicative structure \(m:{C_2\rightarrow C_1}\) and contracts the morphisms \(i\pi _1\) and \(i\pi _2\) in the form of the two endomorphisms \(\theta ,\varphi :{C_2\rightarrow C_2}\). As a consequence,

$$\begin{aligned}&m\varphi =\pi _1,\quad m\theta =\pi _2 \end{aligned}$$
(10)
$$\begin{aligned}&\pi _1\varphi = m,\quad \pi _1\theta =i\pi _1 \end{aligned}$$
(11)
$$\begin{aligned}&\pi _2\varphi =i\pi _2,\quad \pi _2\theta = m. \end{aligned}$$
(12)

Conditions \(\theta ^2=\varphi ^2=1_{C_2}\) and \(\theta \varphi \theta =\varphi \theta \varphi \) are easily verified and it is easy to see that the functor is faithful. In order to see that the functor F is full let us consider two internal groupoids, say C and \(C'\), as illustrated in diagram (13) below. Let us assume the existence of a morphism of involutive-2-links from F(C) to \(F(C')\), that is, a pair of morphisms \(f_i:{C_i\rightarrow C'_i}\), with \(i=1,2\) such that \(\theta 'f_2=f_2\theta \), \(\varphi 'f_2=f_2\varphi \) and \(m'f_2=f_1m\), with \(\theta ,\varphi ,\theta ',\varphi '\) the respective involutions associated with F(C) and \(F(C')\). We need to show that the pair \((f_2,f_1)\) can be extended to a morphism of internal groupoids

(13)

First observe that \(f_2(x,y)=(f_1(x),f_1(y))\) since \(\pi '_1f_2=m'\varphi 'f_2=m'f_2\varphi =f_1m\varphi =f_1\pi _1\) and similarly \(\pi '_2f_2=f_1\pi _2\). This means that the hypotheses \(\theta 'f_2=f_2\theta \), \(\varphi 'f_2=f_2\varphi \) and \(m'f_2=f_1m\) are translated, respectively, as

$$\begin{aligned} (f_1(x)^{-1},f_1(x)f_1(y))&=(f_1(x^{-1}),f_1(xy))\\ (f_1(x)f_1(y),f_1(y)^{-1})&=(f_1(xy),f_1(y^{-1}))\\ f_1(x)f_1(y)&=f_1(xy) \end{aligned}$$

from which we conclude \(i'f_1=f_1i\). We also have \(f_1ed(x))=f_1(x^{-1}x)=f_1(x^{-1})f_1(x)=f_1(x)^{-1}f_1(x)=e'd'f_1(x)\) and \(f_1ec(x)=e'c'f_1(x)\), which give

$$\begin{aligned} \langle 1,e'd'\rangle f_1=f_2\langle 1,ed\rangle \\ \langle 1,e'c'\rangle f_1=f_2\langle 1,ec\rangle \end{aligned}$$

and permits the definition of \(f_0\) either as \(d'f_1e\) or as \(c'f_1e\). Hence, the triple \((f_2,f_1,f_0)\) is a morphism of internal groupoids from C to \(C'\), showing that the functor F is full. Note that this part of the proof is Yoneda invariant, that is, it would be sufficient to make it for ordinary groupoids.

The unitary and associativity conditions of Definition 1 characterize those involutive-2-links that are of the form F(C) for some internal groupoid C. Indeed, if \((\theta ,\varphi ,m)\) is a unital and associative involutive-2-link, then, the fact that the pairs \((m,m\theta )\) and \((m,m\varphi )\) are jointly monomorphic uniquely determines the morphisms \(e_1\) and \(e_2\) which are required to exist by the unitary property and must verify the axioms for an internal groupoid if interpreted as \(e_1(x)=(x,ed(x))\) and \(e_2(x)=(ec(x),x)\). Moreover, the morphism \(i:C_1\rightarrow C_1\) is obtained by condition (3) either as \(i=m\theta \varphi e_2\) or as \(i=m\varphi \theta e_1\). The morphism \(e:C_0\rightarrow C_1\) is uniquely determined by condition (4) as such that \(ed=m\theta e_1\) and \(ec=m\varphi e_2\) where d and c are obtained as in diagram (8) with \(\pi _1=m\varphi \) and \(\pi _2=m\theta \). Conditions (1), (5) and (6) assert the contractibility of the pairs \((m,m\theta )\) and \((m,m\varphi )\) in the sense of Beck (see [4], p. 150). Condition (2) is a central ingredient and gives \(e_1e=e_2e\) from which the conditions \(dm=d\pi _2\) and \(cm=c\pi _1\) are deduced, thus permitting to define the two morphisms \(m_1\) and \(m_2\) from the fact that the pair \((m\theta ,m\varphi )\) is bi-exact. Associativity then follows from \(mm_1=mm_2\). The remaining details of the proof are easily obtained and an example which extends the notion of a crossed module from groups to magmas is illustrated in [7].