nLab groupoid (changes)

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This page is about the notion of groupoids prevalent in category theory and homotopy theory, also known as Brandt groupoids (Brandt (1927)). For the notion involving a globally defined binary operation, see magma.

Context

Algebra

Homotopy theory

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism = \linebreak propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Category theory

\section{Idea}

Where a group may be thought of as a group of symmetry transformations that isomorphically relates one object to itself (the symmetries of one object, such as the isometries of a polyhedron) a groupoid is a collection of symmetry transformations acting between possibly more than one object.

Hence a groupoid consists of a set of objects $x, y, z, \cdots$ and for each pair of objects $(x,y)$ there is a set of transformations, usually denoted by arrows

x \overset{f}{\longrightarrow} y

which may be composed if they are composable (i.e. if the first ends where the second starts)

\array{ && y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ x && \underset{g \circ f}{\longrightarrow} && z }

such that this composition is associative

\begin{imagefromfile} “file_name”: “AssociativityDiagram.png”, “width”: 400 \end{imagefromfile}

and such that for each object $x$ there is identity transformation $x \overset{id_X}{\longrightarrow} x$ in that this is a neutral element for the composition of transformations, whenever defined.

So far this structure is what is called a small category. What makes this a (small) groupoid is that all these transformations are to be “symmetries” in that they are invertible morphisms meaning that for each transformation $x \overset{f}{\longrightarrow} y$ there is a transformation the other way around $y \overset{f^{-1}}{\longrightarrow} x$ such that

f^{-1} \circ f = id_x \phantom{AAAA} f \circ f^{-1} = id_y \,.

If there is only a single object $x$ , then this definition reduces to that of a group, and in this sense groupoids are “groups with many objects”. Conversely, given any groupoid $\mathcal{G}$ and a choice of one of its objects $x$ , then the subcollection of transformations from and to $x$ is a group, sometimes called the automorphism group $Aut_{\mathcal{G}}(x)$ of $x$ in $\mathcal{G}$ .

Just as for groups, the “transformations” above need not necessarily be given by concrete transformations (say by bijections between objects which are sets). Just as for groups, such a concrete realization is always possible, but is an extra choice (called a representation of the groupoid). Generally one calls these “transformations” morphisms: $x \overset{f}{\longrightarrow} y$ is a morphism with “source” $x$ and “codomain” $y$ .

An archetypical example of a groupoid is the fundamental groupoid $\Pi_1(X)$ of a topological space (def. 1 below, for introduction see here): For $X$ a topological space, this is the groupoid whose

objects are the points $x \in X$ ;
morphisms $x \overset{[\gamma]}{\longrightarrow} y$ are the homotopy relative boundary-equivalence classes $[\gamma]$ of paths $\gamma \colon [0,1] \to X$ in $X$ , with $\gamma(0) = x$ and $\gamma(1) = y$ ;

and composition is given, on representatives, by concatenation of paths. Here the class of the reverse path $\bar\gamma \;\colon\; t \mapsto \gamma(1-t)$ constitutes the inverse morphism, making this a groupoid.

If one chooses a point $x \in X$ , then the corresponding group at that point is the fundamental group $\pi_1(X,x) \coloneqq Aut_{\Pi_1(X)}(x)$ of $X$ at that point.

This highlights one of the reasons for being interested in groupoids over groups: Sometimes this allows to avoid unnatural ad-hoc choices and it serves to streamline and simplify the theory.

A homomorphism between groupoids is the obvious: a function between their underlying objects together with a function between their morphisms which respects source and target objects as well as composition and identity morphisms. If one thinks of the groupoid as a special case of a category, then this is a functor. Between groupoids with only a single object this is the same as a group homomorphism.

For example if $f \;\colon\; X \to Y$ is a continuous function between topological spaces, then postcomposition of paths with this function induces a groupoid homomorphism $f_\ast \;\colon\; \Pi_1(X) \longrightarrow \Pi_1(Y)$ between the fundamental groupoids from above.

Groupoids with groupoid homomorphisms (functors) between them form a category Grpd (def. 1 below) which includes the categeory Grp of groups as the full subcategory of the groupoids with a single object. This makes precise how groupoid theory is a generalization of group theory.

However, for groupoids more than for groups one is typically interested in “conjugation actions” on homomorphisms. These are richer for groupoids than for groups, because one may conjugate with a different morphism at each object. If we think of groupoids as special cases of categories, then these “conjugation actions on homomorphisms” are natural transformations between functors.

For examples if $f,g \;\colon\; X \longrightarrow Y$ are two continuous functions between topological spaces, and if $\eta \;\colon\; f \Rightarrow g$ is a homotopy from $f$ to $g$ , then the homotopy relative boundary classes of the paths $\eta(x,-) \;\colon\; [0,1] \to Y$ constitute a natural transformation between $f_*, g_\ast \;\colon\; \Pi_1(X) \to \Pi_1(Y)$ in that for all paths $x_1 \overset{[\gamma]}{\longrightarrow} x_2$ in $X$ we have the “conjugation relation”

[\eta(x_1,-)] \cdot [f\circ\gamma] = [g \circ \gamma] \cdot [\eta(x_2,-)] \phantom{AAAA} \text{i.e.} \phantom{AAAA} \array{ f(x_1) &\overset{[\eta(x_1,-)]}{\longrightarrow}& g(x_1) \\ {}^{\mathllap{[f \circ \gamma]}}\downarrow && \downarrow^{\mathrlap{[g \circ \gamma]}} \\ f(x_2) &\underset{[\eta(x_2,-)]}{\longrightarrow}& g(x_2) } \,.

One may take care of the existence of these conjugation actions/natural transformation in two ways:

If one quotients them out, i.e. if one identifies two groupoid homomorphisms that differ by a conjugation action, then the resulting category of groupoids and classes of homomorphisms is called the homotopy category $Ho(Grpd)$ of Grpd (def. 5 below). This is equivalent to the full subcategory of the classical homotopy category of topological spaces on those that are (CW-complexes and) homotopy 1-types, i.e. those for which the fundamental groupoid is the only homotopy invariant, with all higher homotopy groups being trivial:

$Ho(Grpd) \simeq Ho(Top^{CW}_{\leq 1}) \hookrightarrow Ho(Top^{CW}) \,.$

This means that the concept of groupoids may be regarded as a combinatorial model for homotopy 1-types in homotopy theory, in contrast to the models by topological spaces given by topological homotopy theory. This equivalence generalizes to homotopy n-types as one passes to n-groupoids and eventually to all homotopy types as one passes to infinity-groupoids.
Instead of forgetting all the information encoded in the conjugations/natural transformations, one may also collect it all into the structure of a 2-category Grpd (in fact a (2,1)-category). As such this is the sub-2-category of Cat on those (small) categories all whose morphisms are invertible morphism.

For more introduction on this see at geometry of physics -- homotopy types.

In either of these two cases, beware that the category Grp of groups is not a full subcategory either of the homotopy category $Ho(Grpd)$ or of the (2,1)-category Grpd , because conjugation action is is not part of the definition of $Grp$ . Instead in homotopy theory it is pointed one-object groupoids which are equivalent to groups

Grp \hookrightarrow Grpd^{\ast/} \,.

(Even though there is a unique choice of point for a one-object groupoid, the respect for this (unique) choice forces the conjugation actions on groupoids to be trivial. ) This simple statement is in fact a special case of the May recognition theorem (see there) which holds for more general homotopy types and even for directed homotopy types.

The equivalence of groupoids with homotopy 1-types shows immediately that, with homotopy taken into account, the difference between groupoids and groups is rather mild, after all: Every groupoid is equivalent (isomorphic in $Ho(Grpd)$ ) to a disjoint union of groupoids with a single object (prop. 4 below). In particular groupoid representations are equivalently just tuples of group representations (prop. 6 below.) This is the reason why the competition between general topology textbooks that do and that do not prefer the fundamental groupoid over the fundamental group remains inconclusive.

However, this equivalence of groupoids with disjoint unions of groups depends on the axiom of choice. Even if this is assumed in the underlying foundations/set theory, it breaks once one considers groupoids equipped with geometric structure:

Just as one may consider topological groups and Lie groups, etc., there are the evident concepts of topological groupoids and Lie groupoids etc. (generally: internal groupoids). Their theory turns out to be genuinely richer than that of their group analogs, due to the interaction between the geometry and the homotopy theory (“higher geometry”).

The correct concept of homomorphisms between Lie groupoids for instance goes by many names, including Morita morphisms, Hilsum-Skandalis morphisms and bibundles. The general way to understand this and all other geometric groupoid theory is via the concept of stacks of groupoids. For more pointers on this see at higher geometry and for introduction in the case of higher differential geometry see at geometry of physics -- smooth homotopy types.

\section{Definition}

\subsection{Groupoids}

Definition

(groupoid – homotopily typed definition)

A small groupoid $\mathcal{G}$ is a space or homotopy type such that for all points $a$ and $b$ in $\mathcal{G}$ , all paths $c$ and $d$ in the path space (type) $a = b$ , all homotopies (2-paths) $e$ and $f$ in the path space type $c = d$ , the path space type $e = f$ is contractible.

Definition

(groupoid – dependently typed definition)

A small groupoid $\mathcal{G}$ is

a set $X$ , to be called the set of objects;
for all pairs of objects $(x,y) \in X \times X$ a set $Hom(x,y)$ , to be called the set of morphisms with domain or source $x$ and codomain or target $y$ ;
for all triples of objects $(x,y,z) \in X \times X \times X$ a function

$\circ_{x,y,z} \;\colon\; Hom(y,z) \times Hom(x,y) \longrightarrow Hom(x,z)$

to be called composition
for all objects $x \in X$ an element

$id_x \in Hom(x,x)$

to be called the identity morphism on $x$ ;
for all pairs $(x,y) \in X \times X$ of objects a function

$(-)^{-1} \;\colon\; Hom(x,y) \longrightarrow Hom(y,x)$

to be called the inverse-assigning function

such that

(associativity) for all quadruples of objects $x_1, x_2, x_3, x_4 \in X$ and all triples of morphisms $f \in Hom(x_1,x_2)$ , $g \in Hom(x_2,x_3)$ and $h \in Hom(x_3,x_4)$ an equality

$h \circ (g \circ f) \;=\; (h \circ g) \circ f$
(unitality) for all pairs of objects $x,y \in X$ and all moprhisms $f \in Hom(x,y)$ equalities

$id_y \circ f = f \phantom{AAAA} f \circ id_x = f$
(invertibility) for all pairs of objects $x,y \in X$ and every morphism $f \in Hom(x,y)$ equalities

$f^{-1}\circ f = id_{x} \phantom{AAAA} f \circ f^{-1} = id_y \,.$

If $\mathcal{G}_1, \mathcal{G}_2$ are two groupoids, then a homomorphism or functor between them, denoted

F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

a function $F_0 \;\colon\; X_1 \longrightarrow X_2$ between the respective sets of objects;
for each pair $x,y \in X_1$ of objects a function

$F_{x,y} \;\colon\; Hom_{\mathcal{G}_1}(x,y) \longrightarrow Hom_{\mathcal{G}_2}(F_0(x), F_0(y))$

between sets of morphisms

such that

(respect for composition) for all triples $x,y,z \in X_1$ and all $f \in Hom(x,y)$ and $g \in Hom(y,z)$ an equality

$F_{y,z}(g) \circ_2 F_{x,y}(f) \;=\; F_{x,z}(g\circ_1 f)$
(respect for identities) for all $x \in X$ an equality

$F_{x,x}(id_x) = id_{F_0(x)} \,.$

For $\mathcal{G}_1, \mathcal{G}_2$ two groupoids, and for $F,G \;\colon\; \mathcal{G}_1 \to \mathcal{G}_2$ two groupoid homomorphisms/functors, then a conjugation or homotopy or natural transformation (necessarily a natural isomorphism)

\eta \;\colon\; F \Rightarrow G

for each object $x \in X_1$ of $\mathcal{G}_1$ a morphism $\eta_{x} \in Hom_{\mathcal{G}_2}(F(x), G(x))$

such that

for all $x,y \in X_1$ and $f \in Hom_{\mathcal{G}_1}(x,y)$ an equality

$\eta_y \circ_2 F(f) = G(f) \circ \eta_x \phantom{AAAAAA} \array{ F(x) &\overset{\eta_x}{\longrightarrow}& G(x) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(y) &\underset{\eta_y}{\longrightarrow}& G(y) }$

For $\mathcal{G}_1, \mathcal{G}_2$ two groupoids and $F, G, H \colon \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ three functors between them and $\eta_1 \;\colon\; F \Rightarrow G$ and $\eta_2 \;\colon\; G \Rightarrow H$ conjugation actions/natural isomorphisms between these, there is the composite

\eta_2 \circ \eta_1 \;\colon\; F \Rightarrow H

with components the composite of the components

(\eta_2 \circ \eta_1)(x) \coloneqq \eta_2(x) \circ \eta_1(x) \,.

This yields for any two groupoid a hom-groupoid

Hom_{Grpd}(\mathcal{G}_1, \mathcal{G}_2)

whose objects are the groupoid homomorphisms / functors, and whose morphisms are the conjugation actions / natural transformations.

Remark

(groupoids are special cases of dagger categories)

A small groupoid (def. 2) is equivalently a small dagger category? in which all morphisms are unitary.

Remark

(groupoids are special cases of categories)

A small groupoid (def. 2) is equivalently a small category in which all morphisms are isomorphisms.

Remark

(Higher groupoid theory vs higher category theory) Due to the two preceding remarks, groupoid theory may be regarded as a special case of dagger category theory? or category theory, it is noteworthy that the two theories are quite different in character. For example higher groupoid theory is homotopy theory which is rich but quite tractable, for instance via tools such as simplicial homotopy theory or homotopy type theory, while higher category theory and higher dagger category theory is intricate and becomes tractable mostly by making recourse to higher groupoid theory in the guise of (infinity,1)-category theory and (infinity,n)-categories.

Definition

(groupoid – global definition)

A small groupoid $\mathcal{G}$ is

a set $G_0$ called its set of objects;
a set $G_1$ called its set of morphisms or set of arrows
a pair of functions

$s, t \;\colon\; G_1 \rightrightarrows G_0$

called source and target or domain and codomain

(an element $g\in G_1$ is denoted by $s(g) \overset{g}{\longrightarrow} t(g)$ );
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
1. $s(g h)=s(h)$ , $t(g h)=t(g)$ , and
2. associativity: $(g h)k=g(h k)$ ,
3. identity section: $e: G_0\to G_1$ , such that $e(t(g))g=g=ge(s(g))$ (in particular, $s\circ e= t \circ e$ ),
4. inverse, $i: G_1 \to G_1$ , also denoted by $i(g)=g^{-1}$ , such that for all $g\in G$ ,
  
  $g^{-1} g = e(s(g)), \quad g g^{-1} = e(t(g)).$

Remark

(Notation and Terminology)

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).

Definition

(tame groupoids)

A groupoid is called tame if its groupoid cardinality is finite.

\subsection{Categories of groupoids}

Remark

(1-category of groupoids)

From def. 2 we see that there is a category whose

objects are the small groupoids;
morphisms are the groupoid homomorphisms (functors).

But since this 1-category does not reflect the existence of homotopies/natural isomorphisms between homomorphsims/functors of groupoids (def. 2) this 1-category is not what one is interested in when considering homotopy theory/higher category theory.

In order to obtain the right notion of category of groupoids that does reflect homotopies, we first consider now the horizontal composition of homotopies/natural transformations.

Lemma

(horizontal composition of homotopies with morphisms)

Let $\mathcal{G}_1$ , $\mathcal{G}_2$ , $\mathcal{G}_3$ , $\mathcal{G}_4$ be groupoid and let

\mathcal{G}_1 \overset{F_1}{\longrightarrow} \mathcal{G}_2 \underoverset {\underset{\phantom{AA}F_2\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F^{'}_2\phantom{AA}}{\longrightarrow}} {\Downarrow{\mathrlap{\eta}}} \mathcal{G}_3 \overset{F_3}{\longrightarrow} \mathcal{G}_4

be morphisms and a homotopy $\eta$ . Then there is a homotopy

\mathcal{G}_1 \phantom{AAAA} \underoverset {\underset{F_3 \circ F_2\circ F_1}{\longrightarrow}} {\overset{F_3 \circ F^{'}_2\circ F_1}{\longrightarrow}} {\Downarrow{\mathrlap{ F_3 \cdot \eta_ \cdot F_1 }}} \phantom{AAAA}\mathcal{G}_2

between the respective composites, with components given by

(F_3 \cdot \eta \cdot F_1)(x) \;\coloneqq\; F_3(\eta(F_1(x))) \,.

This operation constitutes a groupoid homomorphism/functor

F_3\cdot (-)\cdot F_1 \;\colon\; Hom_{Grpd}(\mathcal{G}_2, \mathcal{G}_3) \longrightarrow Hom_{Grp}(\mathcal{G}_1, \mathcal{G}_4) \,.

Proof

The respect for identities is clear. To see the respect for composition, let

\mathcal{G}_2 \array{ \overset{F}{\longrightarrow} \\ \Downarrow \eta_1 \\ \overset{G}{\longrightarrow} \\ \Downarrow \eta_2 \\ \underset{H}{\longrightarrow} } \mathcal{G}_3

be two composable homotopies. We need to show that

F_3 \cdot (\eta_2 \circ \eta_1) \cdot F_1 = ( F_3 \cdot \eta_2 \cdot F_1 ) \circ ( F_3 \cdot \eta_1 \cdot F_1 ) \,.

Now for $x$ any object of $\mathcal{G}_1$ we find

\begin{aligned} (F_3 \cdot (\eta_2 \circ \eta_1) \cdot F_1)(x) & \coloneqq F_2((\eta_2 \circ \eta_1)(F_1(X))) \\ & \coloneqq F_3( \eta_2(F_1(x)) \circ \eta_1 (F_1(x))) \\ &= F_2( \eta_2(F_1(x)) ) \circ F_2( \eta_1(F_1(X)) ) \\ & = (( F_3 \cdot \eta_2 \cdot F_1 ) \circ ( F_3 \cdot \eta_1 \cdot F_1 ))(x) \end{aligned} \,.

Here all steps are unwinding of the definition of horizontal and of ordinary (vertical) composition of homotopies, except the third equality, which is the functoriality of $F_2$ .

Lemma

(horizontal composition of homotopies)

Consider a diagram of groupoids, groupoid homomorphsims (functors) and homotopies (natural transformations) as follows:

\mathcal{G}_1 \underoverset {\underset{\phantom{AA}F'_1\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_1\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_1}} \mathcal{G}_2 \underoverset {\underset{\phantom{AA}F'_2\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_2\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_2}} \mathcal{G}_3

The horizontal composition of the homotopies to a single homotopy of the form

\mathcal{G}_1 \underoverset {\underset{F'_2 \circ F'_1}{\longrightarrow}} {\overset{F_2\circ F_1}{\longrightarrow}} {\Downarrow \eta_2 \cdot \eta_1} \mathcal{G}_3

may be defined in temrs of the horizontal composition of homotopies with morphisms (lemma 1) and the (“vertical”) composition of homotopies with themselves, in two different ways, namely by decomposing the above diagram as

\array{ \mathcal{G}_1 \underoverset {\underset{\phantom{AA}F'_1\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_1\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_1}} \mathcal{G}_2 \underoverset {} {\overset{\phantom{AA}F_2\phantom{AA}}{\longrightarrow}} {} \mathcal{G}_3 \\ \mathcal{G}_1 \underoverset {\underset{\phantom{AA}F'_1\phantom{AA}}{\longrightarrow}} {} {} \mathcal{G}_2 \underoverset {\underset{\phantom{AA}F'_2\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_2\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_2}} \mathcal{G}_3 }

or as

\array{ \mathcal{G}_1 \underoverset {} {\overset{\phantom{AA}F_1\phantom{AA}}{\longrightarrow}} {} \mathcal{G}_2 \underoverset {\underset{\phantom{AA}F'_2\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_2\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_2}} \mathcal{G}_3 \\ \mathcal{G}_1 \underoverset {\underset{\phantom{AA}F'_1\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}F_1\phantom{AA}}{\longrightarrow}} {\Downarrow {\eta_1}} \mathcal{G}_2 \underoverset {\underset{\phantom{AA}F'_2\phantom{AA}}{\longrightarrow}} {} {} \mathcal{G}_3 }

In the first case we get

\eta_2 \cdot \eta_1 \;\coloneqq\; (\eta_2 \cdot F'_1) \circ (F_2 \cdot \eta_1)

while in the second case we get

\eta_2 \cdot \eta_1 \;\coloneqq\; ( F'_2 \cdot \eta_1 ) \circ (\eta_2 \cdot F_1) \,.

These two definitions coincide.

Proof

For $x$ an object of $\mathcal{G}_1$ , then we need that the following square diagram commutes in $\mathcal{G}_3$

\array{ F_2(F_1(x)) &\overset{ (F_2\cdot \eta_1)(x) }{\longrightarrow}& F_2(F'_1(x)) \\ {}^{\mathllap{ (\eta_2 \cdot F_1)(x) }}\downarrow && \downarrow^{\mathrlap{ (\eta_2\cdot F'_1)(x) }} \\ F'_2(F_1(x)) & \underset{ (F'_2 \cdot \eta_1)(x) }{\longrightarrow} & F'_2(F'_1(y)) } \phantom{AAAA} = \phantom{AAAA} \array{ F_2(F_1(x)) &\overset{F_2(\eta_1(x))}{\longrightarrow}& F_2(F'_1(x)) \\ { }^{\mathllap{\eta_2(F_1(x))}}\downarrow && \downarrow^{\mathrlap{ \eta_2(F'_1(x)) }} \\ F'_2(F_1(x)) & \underset{F'_2(\eta_1(x))}{\longrightarrow} & F'_2(F'_1(y)) } \,.

But the ommutativity of the square on the right is the defining compatibility condition on the components of $\eta_2$ applied to the morphism $\eta_1(x)$ in $\mathcal{G}_2$ .

Proposition

(horizontal composition with homotopy is natural transformation)

Consider groupoids, homomorphisms and homotopies of the form

\mathcal{G}_1 \underoverset {\underset{F'_1}{\longrightarrow}} {\overset{F_1}{\longrightarrow}} {\Downarrow \eta_1} \mathcal{G}_2 \phantom{AAAAAAA} \mathcal{G}_3 \underoverset {\underset{F'_3}{\longrightarrow}} {\overset{F_3}{\longrightarrow}} {\Downarrow \eta_3} \mathcal{G}_4 \,.

Then horizontal composition with the homotopies (lemma 2) constitutes a natural transformation between the functors of horizontal composition with morphisms (lemma 1)

( \eta_3\cdot (-)\cdot \eta_1 ) \;\colon\; ( F_3 \cdot (-)\cdot F_1 ) \;\Rightarrow\; ( F'_3 (-) \cdot F'_1 ) \;\colon\; Hom_{Grpd}(\mathcal{G}_2,\mathcal{G}_3) \longrightarrow Hom_{Grpd}(\mathcal{G}_1, \mathcal{G}_4) \,.

Proof

By lemma 2.

It first of all follows that the following makes sense

Definition

(homotopy category of groupoids)

There is also the homotopy category $Ho(Grpd)$ whose

objects are small groupoids;
morphisms are equivalence classes of groupoid homomorphisms modulo homotopy (i.e. functors modulo natural transformations).

This is usually denoted $Ho(Grpd)$ .

Of course what the above really means is that, without quotienting out homotopies, groupoids form a 2-category, in fact a (2,1)-category, in fact an enriched category which is enriched over the naive 1-category of groupoids from remark 1, hence a strict 2-category with hom-groupoids.

\subsection{Equivalences of groupoids}

Definition

(equivalence of groupoids)

Given two groupoids $\mathcal{G}_1$ and $\mathcal{G}_2$ , then a homomorphism

F\;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

is an equivalence if it is an isomorphism in the homotopy category $Ho(Grpd)$ (def. 5), hence if there exists a homomorphism the other way around

G \;\colon\; \mathcal{G}_2 \longrightarrow \mathcal{G}_1

and a homotopy/natural transformations of the form

G \circ F \simeq id_{\mathcal{G}_1} \phantom{AAAA} F \circ G \simeq id_{\mathcal{G}_2} \,.

Definition

(connected components of a groupoid)

Given a groupoid $\mathcal{G}$ with set of objects $X$ , then the relation “there exists a morphism from $x$ to $y$ ”, i.e.

(x\sim y) \;\coloneqq\; \left( Hom(x,y) \neq \emptyset \right)

is clearly an equivalence relation on $X$ . The corresponding set of equivalence classes is denoted

\pi_0(\mathcal{G})

and called the set of connected components of $\mathcal{G}$ .

Definition

(automorphism groups)

Given a groupoid $\mathcal{G}$ and an object $x$ , then under composition the set $Hom_{\mathcal{G}}(x,x)$ forms a group. This is called the automorphism group $Aut_{\mathcal{G}}(x)$ or vertex group or isotropy group of $x$ in $\mathcal{G}$ .

Definition

(weak homotopy equivalence of groupoids)

Let $\mathcal{G}_1$ and $\mathcal{G}_2$ be groupoids. Then a morphism (functor)

F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

is called a weak homotopy equivalence if

it induces a bijection on connected components (def. 7):

$\pi_0(F) \;\colon\; \pi_0(\mathcal{G}_1) \overset{\simeq}{\longrightarrow} \pi_0(\mathcal{G}_2)$
for each object $x$ of $\mathcal{G}_1$ the morphism

$F_{x,x} \;\colon\; Aut_{\mathcal{G}_1}(x) \overset{\simeq}{\longrightarrow} Aut_{\mathcal{G}_2}(F_0(x))$

is an isomorphism of automorphism groups (def. 8)

\subsection{Groupoid representations}

Definition

(groupoid representation)

Let $\mathcal{G}$ be a groupoid. Then:

A linear representation of $\mathcal{G}$ is a groupoid homomorphism (functor)

\rho \;\colon\; \mathcal{G} \longrightarrow Core(Vect)

to the groupoid core of the category Vect of vector spaces (example 8). Hence this is

For each object $x$ of $\mathcal{G}$ a vector space $V_x$ ;
for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a linear map $\rho(f) \;\colon\; V_x \to V_y$

such that

(respect for composition) for all composable morphisms $x \overset{f}{\to}y \overset{g}{\to} z$ in the groupoid we have an equality

$\rho(g) \circ \rho(f) = \rho(g \circ f)$
(respect for identities) for each object $x$ of the groupoid we have an equality

$\rho(id_x) = id_{V_x} \,.$

Similarly a permutation representation of $\mathcal{G}$ is a groupoid homomorphism (functor)

\rho \;\colon\; \mathcal{G} \longrightarrow Core(Set)

to the groupoid core of Set. Hence this is

For each object $x$ of $\mathcal{G}$ a set $S_x$ ;
for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a function $\rho(f) \;\colon\; S_x \to S_y$

such that composition and identities are respected, as above.

For $\rho_1$ and $\rho_2$ two such representations, then a homomorphism of representations

\phi \;\colon\; \rho_1 \longrightarrow \rho_2

is a natural transformation between these functors, hence is

for each object $x$ of the groupoid a (linear) function

$(V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x$
such that for all morphisms $x \overset{f}{\longrightarrow} y$ we have

$\phi(y) \circ \rho_1(f) = \rho_2(x) \circ \phi(x) \phantom{AAAAAA} \array{ (V_1)_x &\overset{\phi(x)}{\longrightarrow}& (V_2)_x \\ {}^{\mathllap{\rho_1(f)}}\downarrow && \downarrow^{\mathrlap{\phi_2(f)}} \\ (V_1)_y &\underset{\phi(y)}{\longrightarrow}& (V_2)_y }$

Representations of $\mathcal{G}$ and homomorphisms between them constitute a category, called the representation category $Rep_{Grpd}(\mathcal{G})$ .

\section{Examples}

Example

(fundamental groupoid)

Let $X$ be a topological space. For $x,y \in X$ two points, write $P_{x,y}X$ for the set of paths in $X$ from $x$ to $y$ . Consider the equivalence relation “homotopy relative boundary” on this set and write

Hom(x,y) \coloneqq (P_{x,y}X)/\simeq

for the set of equivalence classes under this relation. The concatenation of paths descends to these equivalence classes. This yields a groupoid with set of objects the set $X$ of points in the topological space. This is the called the fundamental groupoid $\Pi_1(X)$ of $X$ .

This construction extends to a functor

\Pi_1 \;\colon\; Top \longrightarrow Grpd_1

from the 1-category Top to the 1-category Grpd. In fact it extends to a functor

\Pi_1 \;\colon\; Ho(Top) \longrightarrow Ho(Grpd)

on homotopy categories.

Example

((2,1)-functoriality of fundamental groupoid)

If $X$ and $Y$ are topological spaces and $f \;\colon\; X \longrightarrow Y$ is a continuous function between them, then this induces a groupoid homomorphism (functor) between the respective fundamental groupoids (def. 1)

F_f \;\colon\; \Pi_1(X) \longrightarrow \Pi_1(Y)

given on objects by the underlying function of $f$

(F_f)_0 \coloneqq f

and given on the class of a path by the evident postcomposition with $f$

(F_f)_{x,y} \;\colon\; (x \overset{[\gamma]}{\longrightarrow} y) \;\mapsto\; (f(x) \overset{[f \circ \gamma]}{\longrightarrow} f(y) ) \,.

This construction clearly respects identity morphisms and composition and hence is itself a functor of the form

\Pi_1 \;\colon\; Top \longrightarrow Grpd_1

from the category Top of topological space to the 1-category Grpd of groupoids.

But more is true: If $f,g \;\colon\; X \longrightarrow Y$ are two continuous function and

\eta \;\colon\; f \Rightarrow g

is a left homotopy between them, hence a continuous function

\eta \;\colon\; X \times [0,1] \longrightarrow Y

such that $\eta(-,0) = f$ and $\eta(-,1) = g$ , then this induces a homotopy between the above groupoid homomorphisms (a natural transformation of functors).

This shows that the fundamental groupoid functor in fact descends to homotopy categories

\Pi_1 \;\colon\; Ho(Top) \longrightarrow Ho(Grpd) \,.

(In fact this means it even extends to a (2,1)-functor from the (2,1)-category of topological spaces, continuous functions, and higher homotopy-classes of left homotopues, to that of groupoids.)

As a direct consequence it follows that if there is a homotopy equivalence

X \simeq_h Y

between topological spaces, then there is an induced equivalence of groupoids betwee their fundamental groupoids

\Pi_1(X) \simeq \Pi_1(Y) \,.

Hence the fundamental groupoid is a homotopy invariant of topological spaces. Of course by prop. 4 the fundamental groupoid is equivalent, as a groupoid, to the disjoint union of the deloopings of all the fundamental groups of the given topological spaces, one for each connected component, and hence this is equivalently the statement that the set of connected components and the fundamental groups of a topological space are homotopy invariants.

Example

(discrete groupoid)

For $X$ any set, there is the discrete groupoid $Disc(X)$ , whose set of objects is $X$ and whose only morphisms are identity morphisms.

This is also the fundamental groupoid (example 1) of the discrete topological space on the set $X$ .

Example

(codiscrete groupoid)

For $X$ any set, there is the codiscrete groupoid $Codisc(X)$ , whose set of objects is $X$ and whose homsets are singletons. In other words, it is a groupoid where every object is uniquely isomorphic to every object.

Example

(delooping of a group)

Let $G$ be a group. Then there is a groupoid, denoted $B G$ , with a single object $p$ , with morphisms

Hom_{B G}(p,p) \coloneqq G

the elements of $G$ , with composition the multiplication in $G$ , with identity morphism the neutral element in $G$ and with inverse morphisms the inverse elements in $G$ .

This is also called the delooping of $G$ (because the loop space object of $B G$ at the unique point is the given group: $\Omega B G \simeq G$ ).

For $G_1, G_2$ two groups, then there is a natural bijection between group homomorphisms $\phi \colon G_1 \to G_2$ and groupoid homomorphisms $B_ G_1 \to B_ G_2$ : the latter are all of the form $B \phi$ , with $(B \phi)_0$ uniquely fixed and $(B \phi)_{p,p} = \phi$ .

This means that the construction $B(-)$ is a fully faithful functor

B(-) \;\colon\; Grp \hookrightarrow Grpd_1

into from the category Grp of groups to the 1-category of groupoids.

But beware that this functor is not fully faithful when homotopies of groupoids are taken into acount, because there are in general non-trivial homotopies between morphims of the form

B \phi_1, B \phi_2 \;\colon\; B G \longrightarrow B H

By definition, such a homotopy (natural transformation) $\eta \;\colon\; B \phi_1 \Rightarrow B \phi_2$ is a choice of a single elemet $\eta_p \in H$ such that for all $g \in G$ we have

\phi_2(g) = h \cdot \phi_1(g) \cdot h^{-1} \phantom{AAAAAAAAA} \array{ p &\overset{h}{\longrightarrow}& p \\ {}^{\mathllap{\phi_1(g)}}\downarrow && \downarrow^{\mathrlap{\phi_2(g)}} \\ p &\underset{h}{\longrightarrow}& p }

hence such that

\phi_2 = Ad_h \circ \phi_1 \,.

Therefore notably the induced functor

B(-) \;\colon\; Grp \longrightarrow Ho(Grp)

to the homotopy category of groupoids is not fully faithful.

But since $B G$ is canonically a pointed object in groupoids, we may also regard delooping as a functor

B(-) \;\colon\; Grp \longrightarrow Grpd^{\ast/}

to the category of pointed objects of Grpd. Since groupoid homomorphisms $B G_1 \to B G_2$ necessarily preserve the basepoint, this makes no difference at this point. But as we now pass to the homotopy category

B(-) \;\colon\; Grp \hookrightarrow Ho(Grpd^{\ast/})

then also the homotopies are required to preserve the basepoint, and for homotopies between homomorphisms between delooped groups this means, since there only is a single point, that these homotopies are all trivial. Hence regarded this way the functor is a fully faithful functor again, hence an equivalence of categories onto its essential image. By prop. 4 below this essential image consists precisely of the (pointed) connected groupoids:

Groups are equivalently pointed connected groupoids.

Example

(disjoint union/coproduct of groupoids)

Let $\{\mathcal{G}_i\}_{i \in I}$ be a set of groupoids. Then their disjoint union (coproduct) is the groupoid

\underset{i \in I}{\sqcup} \mathcal{G}_i

whose set of objects is the disjoint union of the sets of objects of the summand groupoids, and whose sets of morphisms between two objects is that of $\mathcal{G}_i$ if both objects are form this groupoid, and is empty otherwise.

Example

(disjoint union of delooping groupoids)

Let $\{G_i\}_{i \in I}$ be a set of groups. Then there is a groupoid $\underset{i \in I}{\sqcup} B G_i$ which is the disjoint union groupoid (example 6) of the delooping groupoids $B G_i$ (example 5).

Its set of objects is the index set $I$ , and

Hom(i,j) = \left\{ \array{ G_i &\vert& i = j \\ \emptyset &\vert& \text{otherwise} } \right.

Example

(groupoid core of a category)

For $\mathcal{C}$ any (small) category, then there is a maximal groupoid inside

Core(\mathcal{C}) \hookrightarrow \mathcal{C}

sometimes called the core of $\mathcal{C}$ . This is obtained from $\mathcal{C}$ simply by discarding all those morphisms that are not isomorphisms.

For instance

For $\mathcal{C} =$ Set then $Core(Set)$ is the goupoid of sets and bijections between them.

For $\mathcal{C}$ FinSet then the skeleton of this groupoid (prop. 4) is the disjoint union of deloopings (example 7) of all the symmetric groups:

$Core(FinSet) \simeq \underset{n \in \mathbb{N}}{\sqcup} \Sigma(n)$
For $\mathcal{C} =$ Vect then $Core(Vect)$ is the groupoid of vector spaces and linear bijections between them.

For $\mathcal{C} =$ FinDimVect then the skeleton of this groupoid is the disjoint union of delooping of all the general linear groups

$Core(FinDimVect) \simeq \underset{n \in \mathbb{N}}{\sqcup} GL(n) \,.$

Example

(groupoid representation of delooping groupoid is group representation)

If $B G$ is the delooping groupoid of a group $G$ (example 5), then a groupoid representation of $B G$ according to def. 9 is equivalently a group representation of the group $G$ :

Rep_{Grpd}(B G) \simeq Rep(G) \,.

$\,$

Here is some further examples that should be merged into the above text.

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 semidirect product of an action of a groupoid on a groupoid. If $X=\{*\}$ this gives the groupoid $\mathbf{B}H$ , above.
A symmetric proset, 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.
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.
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.

\section{Properties}

\subsection{Equivalences of groupoids}

Lemma

(automorphism group depends on basepoint only up to conjugation)

For $\mathcal{G}$ a groupoid, let $x$ and $y$ be two objects in the same connected component (def. 7). Then there is a group isomorphism

Aut_{\mathcal{G}}(x) \simeq Aut_{\mathcal{G}}(y)

between their automorphism groups (def. 8).

Proof

By assumption, there exists some morphism from $x$ to $y$

x \overset{f}{\longrightarrow} y \,.

The operation of conjugation with this morphism

\array{ Aut_{\mathcal{G}}(x) &\overset{Ad_{f}}{\longrightarrow}& Aut_{\mathcal{G}}(y) \\ g &\mapsto& f^{-1} \circ g \circ f }

is clearly a group isomorphism as required.

Lemma

(equivalences between disjoint unions of delooping groupoids)

Let $\{G_i\}_{i \in I}$ and $\{H_j\}_{j \in J}$ be sets of groups and consider a homomorphism (functor)

F \;\colon\; \underset{i \in I}{\sqcup} G_i \longrightarrow \underset{j \in J}{\sqcup} H_j

between the corresponding disjoint unions of delooping groupoids (example 5).

Then the following are equivalent:

$F$ is an equivalence of groupoids (def. 6);
$F$ is a weak homotopy equivalence (def. 3).

Proof

The implication 2) $\Rightarrow 1)$ is immediate.

In the other direction, assume that $F$ is an equivalence of groupoids, and let $G$ be an inverse up to natural isomorphism. It is clear that both induces bijections on connected components. To see that both are isomorphisms of automorphisms groups, observe that the conditions for the natural isomorphisms

\alpha \;\colon\; G \circ F \Rightarrow id \phantom{AAAA} \beta \;\colon\; F \circ G \Rightarrow id

are in each separate delooping groupoid $B H_j$ of the form

\array{ \ast &\overset{\alpha}{\longrightarrow}& \ast \\ {}^{\mathllap{G_{F_0(i),F_0(i)}(F_{i,i}(f))}}\downarrow && \downarrow^{\mathrm{id}} \\ \ast &\underset{\alpha}{\longrightarrow}& \ast } \phantom{AAAAAAAAAAAAAAAAAAAAA} \array{ \ast &\overset{\beta}{\longrightarrow}& \ast \\ {}^{\mathllap{F_{G_0(j),G_0(j)}(G_{j,j}(f))}}\downarrow && \downarrow^{\mathrm{id}} \\ \ast &\underset{\beta}{\longrightarrow}& \ast }

since there is only a single object. But this means $F_{i,i}$ and $F_{j,j}$ are group isomorphisms.

Proposition

(every groupoid is equivalent to a disjoint union of group deloopings)

Assuming the axiom of choice, then:

For $\mathcal{G}$ any groupoid, then there exists a set $\{G_i\}_{i \in I}$ of groups and an equivalence of groupoids (def. 6)

\mathcal{G} \simeq \underset{i \in I}{\sqcup} B G_i

between $\mathcal{G}$ and a disjoint union of delooping groupoids (example 7). This is called a skeleton of $\mathcal{G}$ .

Concretely, this exists for $I = \pi_0(\mathcal{G})$ the set of connected components of $\mathcal{G}$ (def. 7) and for $G_i \coloneqq Aut_{\mathcal{G}}(x)$ the automorphism group (def. 8) of any object $x$ in the given connected component.

Proof

Using the axiom of choice we may find a set $\{x_i\}_{i \in \pi_0(\mathcal{G})}$ of objects of $\mathcal{G}$ , with $x_i$ being in the connected component $i \in \pi_0(\mathcal{G})$ .

This choice induces a functor

inc \;\colon\; \underset{i \in \pi_0(\mathcal{G})}{\sqcup} Aut_{\mathcal{G}}(x_i) \longrightarrow \mathcal{G}

which takes each object and morphism “to itself”.

Now using the axiom of choice once more, we choose in each connected component $i \in \pi_0(\mathcal{G})$ and for each object $y$ in that connected component a morphism

x_i \overset{f_{x_i,y}}{\longrightarrow} y \,.

Using this we obtain a functor the other way around

p \;\colon\; \mathcal{G} \longrightarrow \underset{i \in \pi_0(\mathcal{G})}{\sqcup} Aut_{\mathcal{G}}(x_i)

which sends each object to its connected component, and which for pairs of objects $y$ , $z$ of $\mathcal{G}$ is given by conjugation with the morphisms choosen above:

\array{ Hom_{\mathcal{G}}(y,z) &\overset{p_{y,z}}{\longrightarrow}& & Aut_{\mathcal{G}}(x_i) & \\ \\ y && y & \overset{f_{x_i,y}}{\longleftarrow}& x_i \\ {}^{\mathllap{f}} \downarrow &\mapsto& {}^{\mathllap{f}}\downarrow \\ z && z & \underset{f_{x_i,z}^{-1}}{\longrightarrow} & x_i } \,.

It is now sufficient to show that there are conjugations/natural isomorphisms

p \circ inc \simeq id \phantom{AAAA} inc \circ p \simeq id \,.

For the first this is immediate, since we even have equality

p \circ inc \;=\; id \,.

For the second we observe that choosing

\eta(y) \coloneqq f_{x_i,y}

yields a naturality square by the above construction:

\array{ x_i &\overset{f_{x_i,y}}{\longrightarrow}& y \\ {}^{ \mathllap{ f_{x_i,z} \circ f \circ f_{x_i,y}^{-1} } }\downarrow && \downarrow^{\mathrlap{f}} \\ x_i &\underset{f_{x_i,z}}{\longrightarrow}& z } \,.

Proposition

(weak homotopy equivalence is equivalence of groupoids)

Let $F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ be a homomorphism of groupoids.

Assuming the axiom of choice then the following are equivalent:

$F$ is an equivalence of groupoids (def. 6);
$F$ is a weak homotopy equivalence in that
1. it induces an bijection of sets of connected components (def. 7);
  
  $\pi_0(F) \;\colon\; \pi_0(\mathcal{G}_1) \overset{\simeq}{\longrightarrow} \pi_0(\mathcal{G}_0) \,,$
for each object $x \in \mathcal{G}_1$ it induces an isomorphism of automorphism groups (def. 8):

$F_{x,x} \;\colon\; Aut_{\mathcal{G}_1}(x) \overset{\simeq}{\longrightarrow} Aut_{\mathcal{G}_2}(F_0(x)) \,.$

Proof

In one direction, if $F$ has an inverse up to natural isomorphism, then this induces by definition a bijection on connected components, and it induces isomorphism on homotopy groups by lemma 3.

In the other direction, choose equivalences to skeleta as in prop. 4 to get a commuting diagram in the 1-category of groupoids as follows:

\array{ \mathcal{G}_1 &\underoverset{\simeq}{inc_1}{\longleftarrow}& \underset{i \in \pi_0(\mathcal{G}_1)}{\sqcup} Aut_{\mathcal{G}_1}(x_i) \\ {}^{\mathllap{F}}\downarrow && \downarrow^{\mathrlap{\tilde F }} \\ \mathcal{G}_2 &\underoverset{inc_2}{\simeq}{\longleftarrow}& \underset{i \in \pi_0(\mathcal{G}_1)}{\sqcup} Aut_{\mathcal{G}_2}(F_0(x_i)) } \,.

Here $inc_1$ and $inc_2$ are equivalences of groupoids by prop. 4. Moreover, by assumption that $F$ is a weak homotopy equivalence $\tilde F$ is the union of of deloopings of isomorphisms of groups, and hence has a strict inverse, in particular a homotopy inverse, hence is in particular an euivalence of groupoids.

In conclusion, when regarded as a diagram in the homotopy category $Ho(Grpd)$ (def. 5), the top, bottom and right moprhism of the above diagram are isomorphisms. It follows that also $f$ is an isomorphism in $Ho(Grpd)$ . But this means exactly that it is a homotopy equivalence of groupoids, by def. 6.

Proposition

(groupoid representations are products of group representations)

Assuming the axiom of choice then the following holds:

Let $\mathcal{G}$ be a groupoid. Then its category of groupoid representations is equivalent to the product category indexed by the set of connected components $\pi_0(\mathcal{G})$ (def. 7) of group representations of the automorphism group $G_i \coloneqq Aut_{\mathcal{G}}(x_i)$ (def. 8) for $x_i$ any object in the $i$ th connected component:

Rep(\mathcal{G}) \;\simeq\; \underset{i \in \pi_0(\mathcal{G})}{\prod} Rep(G_i) \,.

Proof

Let $\mathcal{C}$ be the category that the representation is on. Then by definition

Rep(\mathcal{G}) = Hom( \mathcal{G} , \mathcal{C} ) \,.

Consider the injection functor of the skeleton (from lemma 4)

inc \;\colon\; \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i \overset{}{\longrightarrow} \mathcal{G} \,.

By lemma 1 the pre-composition with this constitutes a functor

inc^\ast \;\colon\; Hom( \mathcal{G}, \mathcal{C} ) \longrightarrow Hom( \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i, \mathcal{C} )

and by combining lemma 4 with lemma 2 this is an equivalence of categories. Finally, by example 9 the category on the right is the product of group representation categories as claimed.

\subsection{As 2-coskeletal Kan complexes}

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

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

\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

\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.

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.

\section{Related concepts}

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	contractible-if-inhabited	(-1)-groupoid/truth value	(0,1)-sheaf/ideal	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/2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/3-stack	h-3-groupoid
h-level $n+2$	$n$ -truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/n-stack	h- $n$ -groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/∞-stack	h- $\infty$ -groupoid

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

\section{References}

Apparently the earliest occurence of the concept of groupoids (see also Brandt groupoid):

Heinrich Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Mathematische Annalen 96 1 (1927) 360-366 [[doi:10.1007/BF01209171](https://doi.org/10.1007%2FBF01209171)]

Further early discussion:

George W. Mackey, §11 of: Ergodic theory and virtual groups, Mathematische Annalen 166 (1966) 187–207 [[doi:10.1007/BF01361167](https://doi.org/10.1007/BF01361167)]
Philip Higgins, Presentations of Groupoids, with Applications to Groups, Proc. Camb. Phil. Soc., 60 (1964) 7-20 [[doi;10.1017/S0305004100037397](https://doi.org/10.1017/S0305004100037397)]
Nicolas Bourbaki, Topologie Algébrique, Chapitres 1 à 4, Springer (1998, 2016) [[doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8]
Philip J. Higgins, Categories and Groupoids, Mathematical Studies 32, van Nostrand New York (1971), Reprints in Theory and Applications of Categories 7 (2005) 1-195 [[tac:tr7] (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html), pdf]

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 43 7 (1996) [[pdf](http://www.ams.org/notices/199607/weinstein.pdf), pdf]

Further exposition:

Ronnie Brown, From groups to groupoids: A brief survey, at Groupoids in Mathematics, Bulletin of the London Mathematical Society 19 2 (1987) 113-134 [[doi:10.1112/blms/19.2.113](https://doi.org/10.1112/blms/19.2.113), pdf]
Introduction to Topology -- 2, Groupoids
Alberto Santini, Topological groupoids (2011) [[pdf](http://web.math.ku.dk/~moller/students/alberto.pdf), pdf]

(about groupoids in topology, notably fundamental groupoids – not about topological groupoids)

With an eye towards homotopy theory:

Birgit Richter, Section 1.2 of: From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)

nLab groupoid (changes)

Context

Algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Homotopy theory

Type theory

Category theory

Concepts

Universal constructions

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Extensions

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