A small groupoid $G$ consists of a
equipped with the following structure maps:
A pair of maps $s, t \colon G_1 \rightrightarrows G_0$ called source and target or domain and codomain. An element $g\in G_1$ is thought of as an arrow from $x=s(g)$ to $y=t(g)$.
A multiplication or composition map $m \colon G_1 \times_{s, G_0, t} G_1 \to G_1$, usually denoted as $g h$ for $m(g,h)$, which satisfies
$s(g h)=s(h)$, $t(g h)=t(g)$, and
associativity: $(g h)k=g(h k)$,
identity section: $e: G_0\to G_1$, such that $e(t(g))g=g=ge(s(g))$ (in particular, $s\circ e= t \circ e$),
inverse, $i: G_1 \to G_1$, also denoted by $i(g)=g^{-1}$, such that for all $g\in G$,
$g^{-1} g = e(s(g)), \quad g g^{-1} = e(t(g)).$
Hence a small groupoid is a small category in which all morphisms are isomorphisms.
A groupoid is called tame if its cardinality is finite.
Groupoids naturally form a $2$-category (in fact a $(2,1)$-category) Grpd. For more on this see also the lecture notes at geometry of physics -- homotopy types.
If $x,y$ are objects (also called vertices) of the groupoid $G$ then the set of morphisms (also called arrows) from $x$ to $y$ is written $G(x,y)$, or other notations for hom-sets. The set $G(x,x)$ (which is a group under composition) is also written $G(x)$ and called the vertex group of $G$ at $x$. It is also written $Aut_G(x)$ and called the automorphism group of $x$ in $G$, or written $\pi_1(G,x)$ and called the fundamental group of $G$ at $x$ (especially if you think of a groupoid as giving a homotopy 1-type).
As in any category, there is a question of notation for the composition, and in particular of the order in which to write things. It can be more convenient to write the composition of $a:x \to y$ and $b: y \to z$ as $a b:x \to z$, although a more traditional notation would be $b a$. The two conventions can be distinguished by writing $a; b$ or $b\circ a$ (which is the most traditional notation for categories). See composition for further discussion.
A groupoid $G$ is connected, or transitive?, if $G(x,y)$ is nonempty for all $x,y \in Ob(G)$; it is called inhabited?, or nonempty, if it has at least one object. A maximal inhabited connected subgroupoid? of $G$ is called a component of $G$, and $G$ is then the disjoint union (the coproduct in $\Grpd$) of its connected components. The set of components of $G$ is written $\pi_0(G)$ (especially if you think of a groupoid as giving a homotopy 1-type).
Any group $H$ gives rise to a groupoid, sometimes denoted $\mathbf{B}H$ but often conflated with $H$ itself, which has exactly one object $*$ and with $\mathbf{B}H(*,*) = H$. That is, there is an inclusion of categories $Group \to Groupoids$, and this functor has a left adjoint, giving the universal group of a groupoid. Any inhabited connected groupoid is equivalent to one arising in this way.
A disjoint union of (the one-object groupoids corresponding to) groups is naturally a groupoid, also called a bundle of groups. The axiom of choice is equivalent to the claim that any groupoid is equivalent to one of this form.
From any action of a group $H$ on a set $X$ we obtain an action groupoid or “weak quotient” $X//H$. This is also written $X \rtimes H$, a semidirect product, since it is a special case of the semidirext product of an action of a groupoid on a groupoid. If $X=\{*\}$ this gives the groupoid $\mathbf{B}H$, above.
A setoid, that is a set $X$ equipped with an equivalence relation $E$, becomes a groupoid with the multiplication $(x,y);(y,z) = (x,z)$ for all $(x,y), (y,z) \in E$. (This gives one reason for the forward notation for composition.) Such a groupoid is equivalent to the discrete category on the quotient set $X/E$.
In particular, if $E$ is the universal relation $X \times X$, then we get the square groupoid? $X^2$, also called the trivial groupoid? on $X$. Despite the latter name, there is an important special case, namely the groupoid $I= \{0,1\}^2$. This groupoid has non-identity elements $\iota:0 \to 1, \iota^{-1}: 1 \to 0$, and can be regarded as a groupoid model of the unit interval $[0,1]$ in topology.
Any topological space $X$ has a fundamental groupoid $\pi_1(X)$ whose objects are the points of $X$ and whose arrows are (homotopy classes rel end points of of) paths, with composition by concatenation of paths. Note that $\pi_0(\pi_1(X)) = \pi_0(X)$ is the set of path components of $X$, and for any $x\in X$, $\pi_1(\Pi_1(X),x) = \pi_1(X,x)$ is the fundamental group of $X$ with basepoint $x$. In theory one can obtain the higher homotopy groups in this way as well, by generalizing from the fundamental groupoid to the fundamental infinity-groupoid.
More generally, if we choose some subset $S$ of the points of a space $X$, then we have a full subgroupoid of $\pi_1(X)$ containing only those points in $S$, denoted $\pi_1(X,S)$. This can result in much more manageable groupoids; for instance $\Pi_1([0,1],\{0,1\})$ is the groupoid $I$ considered above, while $\Pi_1([0,1])$ has uncountably many objects (but is equivalent to $I$).
If $\Gamma$ is a directed graph or quiver, then the free groupoid $F(\Gamma)$ is well defined. It is the left adjoint functor to the forgetful functor from groupoids to directed graphs. This shows an advantage of groupoids: the notion of free equivalence relation or free action groupoid does not easily make sense. But we can still talk of a presentation of an equivalence relation or action groupoid by generators and relations, by considering presentations of groupoids instead.
Mike: It’s not clear to me that the notion of “free equivalence relation” doesn’t make sense. Can’t I talk about a left adjoint to the forgetful functor from equivalence relations to, say, directed graphs? Maybe sets-equipped-with-a-binary-relation would be more appropriate, but either one works fine.
Ronnie: Are you sure this forgetful functor equivalence relations to directed graphs has a left adjoint? Suppose the directed graph $\Gamma$ has one vertex $x$ and one loop $u:x \to x$. The free groupoid on $\Gamma$ is the group of integers, which as a groupoid is not an equivalence relation.
Toby: But there is still a free setoid (set equipped with an equivalence relation) on $\Gamma$; it is the point. As a groupoid, it is not the same as the free groupoid on $\Gamma$, although it is the same as the free setoid on the free groupoid on $\Gamma$. If there's an advantage to working with groupoids, perhaps it's that the free groupoid functor preserves distinctions that the free setoid functor forgets? (In this case, a distinction preserved or forgotten is that between $\Gamma$ and the point, which as a graph does not have $u$.)
A paper by Živaljević gives examples of groupoids used in combinatorics.
The book “Topology and Groupoids” listed below takes the view that 1-dimensional homotopy theory, including the Seifert-van Kampen Theorem, the theory of covering spaces, and the less well known theory of the fundamental group(oid) of an orbit space by a discontinuous group action, is best presented using the notion of groupoid rather than group as basic. This had led in the 1960s to the question of the prospective use of (strict) groupoids in higher homotopy theory. One answer is given in the book Nonabelian algebraic topology discussed elsewhere on the nlab.
Groupoids $K$ are equivalent to 1-hypergroupoids, which are in particular 2-coskeletal Kan complexes $N(K)$ – their nerves.
The objects of the groupoids are the 0-simplices and the morphisms of the groupoid are the 1-simplices of the Kan complex. The composition operation $(f,g) \mapsto g \circ f$ in the grouopoid is encoded in the 2-simplices of the Kan complex
The associativity condition on the composition is exhibited by the 3-coskeleton-property of the Kan complex. This says that every simplicial 2-sphere in the Kan complex has a unique filler. With the above identification of 2-simplices with composition operations, this means that the 2 ways
of composing a sequence of three composable morphisms are equal
For handling just groupoids exclusively their description in terms of Kan complexes may be a bit of an overkill, but the advantage is that it embeds groupoids naturally in the more general context of 2-groupoids, 3-groupoids and eventually ∞-groupoids. For instance a pseudo-functor out of an ordinary groupoid into a 2-groupoid is simply a homomorphism of the corresponding Kan complexes.
The disadvantage of the simplicial approach is the difficulty of describing multiple compositions in higher dimensions, an important idea which is quite conveniently handled cubically.
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | (-1)-groupoid/truth value | (0,1)-sheaf | mere proposition, h-proposition | |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
A motivation and introduction of the concept of groupoid and a tour of examples (including the refinement to topological groupoids and Lie groupoids) is in
A page Groupoids in Mathematics by Ronnie Brown includes the introductory text
Technical discussion is for instance in the following references.
Philip Higgins, 1971, Categories and Groupoids, van Nostrand, New York. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195. (download)
Philip Higgins, Presentations of Groupoids, with Applications to Groups, Proc. Camb. Phil. Soc., 60 (1964) 7–20.
Ronnie Brown, Topology and groupoids, Booksurge, 2006. (description)
Rade T. Živaljević, Groupoids in combinatorics—applications of a theory of local symmetries. Algebraic and geometric combinatorics, 305–324, Contemp. Math., 423, Amer. Math. Soc., Providence, RI, 2006.