category theory

# Contents

## Definition

A groupoid $G$ is a category in which all morphisms are isomorphisms. Often the requirement is made that $G$ is small.

A groupoid is tame if its cardinality is finite.

Groupoids form a $2$-category (in fact a $\left(2,1\right)$-category) Grpd.

## Notation

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 $G\left(x,y\right)$, or other notations for hom-sets. The set $G\left(x,x\right)$ (which is a group under composition) is also written $G\left(x\right)$ and called the vertex group of $G$ at $x$. It is also written ${\mathrm{Aut}}_{G}\left(x\right)$ and called the automorphism group of $x$ in $G$, or written ${\pi }_{1}\left(G,x\right)$ 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 $ab:x\to z$, although a more traditional notation would be $ba$. 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 $G\left(x,y\right)$ is nonempty for all $x,y\in \mathrm{Ob}\left(G\right)$; 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 ${\Pi }_{0}\left(G\right)$ (especially if you think of a groupoid as giving a homotopy 1-type).

## Examples

1. Any group $H$ gives rise to a groupoid, sometimes denoted $BH$ but often conflated with $H$ itself, which has exactly one object $*$ and with $BH\left(*,*\right)=H$. Any inhabited connected groupoid is equivalent to one arising in this way.

2. 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.

3. From any action of a group $H$ on a set $X$ we obtain an action groupoid or “weak quotient$X//H$. If $X=\left\{*\right\}$ this gives the groupoid $BH$, above.

4. A setoid, that is a set $X$ equipped with an equivalence relation $E$, becomes a groupoid with the multiplication $\left(x,y\right);\left(y,z\right)=\left(x,z\right)$ for all $\left(x,y\right),\left(y,z\right)\in E$. (This gives one reason for the forward notation for composition.) Such a groupoid is equivalent to the discrete category on the quotient set $X/E$.

5. In particular, if $E$ is the universal relation $X×X$, 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 $I=\left\{0,1{\right\}}^{2}$. 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 $\left[0,1\right]$ in topology.

6. Any topological space $X$ has a fundamental groupoid ${\Pi }_{1}\left(X\right)$ whose objects are the points of $X$ and whose arrows are (homotopy classes of) paths, with composition by concatenation of paths. Note that ${\Pi }_{0}\left({\Pi }_{1}\left(X\right)\right)={\Pi }_{0}\left(X\right)$ is the set of path components? of $X$, and for any $x\in X$, ${\pi }_{1}\left({\Pi }_{1}\left(X\right),x\right)={\pi }_{1}\left(X,x\right)$ 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.

7. More generally, if we choose some subset $S$ of the points of a space $X$, then we have a full subgroupoid of ${\Pi }_{1}\left(X\right)$ containing only those points in $S$, denoted ${\Pi }_{1}\left(X,S\right)$. This can result in much more manageable groupoids; for instance ${\Pi }_{1}\left(\left[0,1\right],\left\{0,1\right\}\right)$ is the groupoid $I$ considered above, while ${\Pi }_{1}\left(\left[0,1\right]\right)$ has uncountably many objects (but is equivalent to $I$).

8. If $\Gamma$ is a directed graph or quiver, then the free groupoid $F\left(\Gamma \right)$ 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 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 $u$.)

9. A paper by Živaljević gives examples of groupoids used in combinatorics.

## Properties

### Characterization as 2-coskeletal Kan complexes

Groupoids $K$ are equivalent to 1-hypergroupoids, which are in particular 2-coskeletal Kan complexes $N\left(K\right)$ – 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 $\left(f,g\right)↦g\circ f$ in the grouopoid is encoded in the 2-simplices of the Kan complex

$\begin{array}{ccc}& & y\\ & {}^{f}↗& =& {↘}^{g}\\ x& & \underset{g\circ f}{\to }& & z\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && y \\ & {}^{\mathllap{f}}\nearrow &=& \searrow^{\mathrlap{g}} \\ x &&\underset{g\circ f}{\to}&& z } \,.

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 unique filler. With the above identification of 2-simplices with composition operations, this means that the 2 ways

$\begin{array}{ccccc}y& \stackrel{g}{\to }& & \to & z\\ ↑& & & ↗& {↓}^{h}\\ {}^{f}↑& {}^{g\circ f}↗& & & ↓\\ x& \underset{h\circ \left(g\circ f\right)}{\to }& & \to & w\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{and}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccccc}y& \stackrel{g}{\to }& & \to & z\\ ↑& ↘& & & {↓}^{h}\\ {}^{f}↑& & & {↘}^{h\circ g}& ↓\\ x& \underset{\left(h\circ g\right)\circ f}{\to }& & \to & w\end{array}$\array{ y &\stackrel{g}{\to}& &\to& z \\ \uparrow && &\nearrow& \downarrow^{\mathrlap{h}} \\ {}^{\mathllap{f}}\uparrow &^{\mathllap{g \circ f}}\nearrow& && \downarrow \\ x &\underset{h \circ (g\circ f )}{\to}&&\to& w } \;\;\; and \;\;\; \array{ y &\stackrel{g}{\to}& &\to& z \\ \uparrow &\searrow& && \downarrow^{\mathrlap{h}} \\ {}^{\mathllap{f}}\uparrow && &\searrow^{\mathrlap{h\circ g}}& \downarrow \\ x &\underset{(h \circ g)\circ f}{\to}&&\to& w }

of composing a sequence of three composable morphisms are equal

$\begin{array}{ccccc}y& \to & & \to & z\\ ↓& & & ↗& ↓\\ ↓& ↗& & & ↓\\ x& \to & & \to & w\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\stackrel{=}{\to }\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccccc}y& \to & & \to & z\\ ↓& ↘& & & ↓\\ ↓& & & ↘& ↓\\ x& \to & & \to & w\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ y &\to& &\to& z \\ \downarrow && &\nearrow& \downarrow \\ \downarrow &\nearrow& && \downarrow \\ x &\to&&\to& w } \;\;\; \stackrel{=}{\to} \;\;\; \array{ y &\to& &\to& z \\ \downarrow &\searrow& && \downarrow \\ \downarrow && &\searrow& \downarrow \\ x &\to&&\to& w } \,.

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.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valueh-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

## References

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

• Alan Weinstein, Groupoids: Unifying Internal and External Symmetry – A Tour through some Examples, Notices of the AMS volume 43, Number 7 (pdf)

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.

• Philip Higgins, Presentations of Groupoids, with Applications to Groups, Proc. Camb. Phil. Soc., 60 (1964) 7–20.

• Ronnie Brown, Topology and groupoids, Booksurge, 2006.

• 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.

Revised on April 3, 2013 14:24:33 by Urs Schreiber (82.169.65.155)