nLab
groupoid

This article is about the notion prevalent in category theory and homotopy theory, also known as a Brandt groupoid (Brandt 27). For the notion involving a globally defined binary operation, see magma.

Context

Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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

xfy x \overset{f}{\longrightarrow} y

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

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

such that this composition is associative and such that for each object xx there is identity transformation xid Xxx \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 xfyx \overset{f}{\longrightarrow} y there is a transformation the other way around yf 1xy \overset{f^{-1}}{\longrightarrow} x such that

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

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

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

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

If one chooses a point xXx \in X, then the corresponding group at that point is the fundamental group π 1(X,x)Aut Π 1(X)(x)\pi_1(X,x) \coloneqq Aut_{\Pi_1(X)}(x) of XX 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:XYf \;\colon\; X \to Y is a continuous function between topological spaces, then postcomposition of paths with this function induces a groupoid homomorphism f *:Π 1(X)Π 1(Y)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 Grp (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 genralization 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:XYf,g \;\colon\; X \longrightarrow Y are two continuous functions between topological spaces, and if η:fg\eta \;\colon\; f \Rightarrow g is a homotopy from ff to gg, then the homotopy relative boundary classes of the paths η(x,):[0,1]Y\eta(x,-) \;\colon\; [0,1] \to Y constitute a natural transformation between f *,g *:Π 1(X)Π y(Y)f_*, g_\ast \;\colon\; \Pi_1(X) \to \Pi_y(Y) in that for all paths x 1[γ]x 2x_1 \overset{[\gamma]}{\longrightarrow} x_2 in XX we have the “conjugation relation”

[η(x 1,)][fγ]=[gγ][η(x 2,)]AAAAi.e.AAAAf(x 1) [η(x 1,)] g(x 1) [fγ] [fγ] f(x 1) [η(x 2,)] g(x 2). [\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{[f \circ \gamma]}} \\ f(x_1) &\underset{[\eta(x_2,-)]}{\longrightarrow}& g(x_2) } \,.

One may take care of the existence of these conjugation actions/natutaral transformation in two way:

  1. 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(Grp)Ho(Grp) of Grp (def. 4 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)Ho(Top 1 CW)Ho(Top CW). 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.

  2. 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 Grp (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)Ho(Grpd) or of the (2,1)-category Grpd, because conjugation action is is not part of the definition of GrpGrp. Instead in homotopy theory it is pointed one-object groupoids which are equivalent to groups

GrpGrpd */. 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 accont, the difference between groupoids and groups is rather mild, after all: Every groupoid is equvalent (isomorphic in Ho(Grpd)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.

Definition

Groupoids

Definition

(groupoid – dependently typed definition)

A small groupoid 𝒢\mathcal{G} is

  1. a set XX, to be called the set of objects;

  2. for all pairs of objects (x,y)X×X(x,y) \in X \times X a set Hom(x,y)Hom(x,y), to be called the set of morphisms with domain or source xx and codomain or target yy;

  3. for all triples of objects (x,y,z)X×X×X(x,y,z) \in X \times X \times X a function

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

    to be called composition

  4. for all objects xXx \in X an element

    id xHom(x,x) id_x \in Hom(x,x)

    to be called the identity morphism on xx;

  5. for all pairs x,yHom(x,y)x,y \in Hom(x,y) of obects a function

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

    to be called the inverse-assigning function

such that

  1. (associativity) for all quadruples of objects x 1,x 2,x 3,x 4Xx_1, x_2, x_3, x_4 \in X and all triples of morphisms fHom(x 1,x 2)f \in Hom(x_1,x_2), gHom(x 2,x 3)g \in Hom(x_2,x_3) and hHom(x 3,x 4)h \in Hom(x_3,x_4) an equality

    h(gf)=(hg)f h \circ (g \circ f) \;=\; (h \circ g) \circ f
  2. (unitality) for all pairs of objects x,yXx,y \in X and all moprhisms fHom(x,y)f \in Hom(x,y) equalities

    id yf=fAAAAfid x=f id_y \circ f = f \phantom{AAAA} f \circ id_x = f
  3. (invertibility) for all pairs of objects x,yXx,y \in X and every morphism fHom(x,y)f \in Hom(x,y) equalities

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

If 𝒢 1,𝒢 2\mathcal{G}_1, \mathcal{G}_2 are two groupoids, then a homomorphism or functor between them, denoted

F:𝒢 1𝒢 2 F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

is

  1. a function F 0:X 1X 2F_0 \;\colon\; X_1 \longrightarrow X_2 between the respective sets of objects;

  2. for each pair x,yX 1x,y \in X_1 of objects a function

    F x,y:Hom 𝒢 1(x,y)Hom 𝒢 2(F 0(x),F 0(y)) 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

  1. (respect for composition) for all triples x,y,zX 1x,y,z \in X_1 and all fHom(x,y)f \in Hom(x,y) and gHom(y,z)g \in Hom(y,z) an equality

    F y,z(g) 2F x,y(f)=F x,z(g 1f) F_{y,z}(g) \circ_2 F_{x,y}(f) \;=\; F_{x,z}(g\circ_1 f)
  2. (respect for identities) for all xXx \in X an equality

    F x,x(id x)=id F 0(x). F_{x,x}(id_x) = id_{F_0(x)} \,.

For 𝒢 1,𝒢 2\mathcal{G}_1, \mathcal{G}_2 two groupoids, and for F,G:𝒢 1𝒢 2F,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)

η:FG \eta \;\colon\; F \Rightarrow G

is

  • for each object xX 1x \in X_1 of 𝒢 1\mathcal{G}_1 a morphism η xHom 𝒢 2(F(x),G(y))\eta_{x} \in Hom_{\mathcal{G}_2}(F(x), G(y))

such that

  • for all x,yX 1x,y \in X_1 and fHom 𝒢 1(x,y)f \in Hom_{\mathcal{G}_1}(x,y) an equality

    η y 2F(f)=G(f)η xAAAAAAF(x) η x G(x) F(f) G(f) F(y) η y G(y) \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 𝒢 1,𝒢 2\mathcal{G}_1, \mathcal{G}_2 two groupoids and F,G,H:𝒢 1𝒢 2F, G, H \colon \mathcal{G}_1 \longrightarrow \mathcal{G}_2 three functors between them and η 1:FRihtarrowG\eta_1 \;\colon\; F \Rihtarrow G and η 2:GH\eta_2 \;\colon\; G \Rightarrow H conjugation actions/natural isomorphisms between these, there is the composite

η 2:η 1:FH \eta_2 \colon \eta_1 \;\colon\; F \Rightarrow H

with components the composite of the components

(η 2η 1)(x)η 2(x)η 1(x). (\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(𝒢 1,𝒢 2) 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 categories)

A small groupoid (def. 1) is equivalently a small category in which all morphisms are isomorphisms.

While therefore groupoid theory may be regarded as a special case of 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 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 𝒢\mathcal{G} is

  1. a set G 0G_0 called its set of objects;

  2. a set G 1G_1 called its set of morphisms or set of arrows

  3. a pair of functions

    s,t:G 1G 0 s, t \;\colon\; G_1 \rightrightarrows G_0

    called source and target or domain and codomain

    (an element gG 1g\in G_1 is denoted by s(g)gt(g)s(g) \overset{g}{\longrightarrow} t(g));

  4. a multiplication or composition map m:G 1× s,G 0,tG 1G 1m \colon G_1 \times_{s, G_0, t} G_1 \to G_1, usually denoted as ghg h for m(g,h)m(g,h), which satisfies

    1. s(gh)=s(h)s(g h)=s(h), t(gh)=t(g)t(g h)=t(g), and

    2. associativity: (gh)k=g(hk)(g h)k=g(h k),

    3. identity section: e:G 0G 1e: G_0\to G_1, such that e(t(g))g=g=ge(s(g))e(t(g))g=g=ge(s(g)) (in particular, se=tes\circ e= t \circ e),

    4. inverse, i:G 1G 1i: G_1 \to G_1, also denoted by i(g)=g 1i(g)=g^{-1}, such that for all gGg\in G,

      g 1g=e(s(g)),gg 1=e(t(g)). g^{-1} g = e(s(g)), \quad g g^{-1} = e(t(g)).

Remark

(Notation and Terminology)

If x,yx,y are objects (also called vertices) of the groupoid GG then the set of morphisms (also called arrows) from xx to yy is written G(x,y)G(x,y), or other notations for hom-sets. The set G(x,x)G(x,x) (which is a group under composition) is also written G(x)G(x) and called the vertex group of GG at xx. It is also written Aut G(x)Aut_G(x) and called the automorphism group of xx in GG, or written π 1(G,x)\pi_1(G,x) and called the fundamental group of GG at xx (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:xya:x \to y and b:yzb: y \to z as ab:xza b:x \to z, although a more traditional notation would be bab a. The two conventions can be distinguished by writing a;ba; b or bab\circ a (which is the most traditional notation for categories). See composition for further discussion.

A groupoid GG is connected, or transitive?, if G(x,y)G(x,y) is nonempty for all x,yOb(G)x,y \in Ob(G); it is called inhabited?, or nonempty, if it has at least one object. A maximal inhabited connected subgroupoid? of GG is called a component of GG, and GG is then the disjoint union (the coproduct in Grpd\Grpd) of its connected components. The set of components of GG is written π 0(G)\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.

Categories of groupoids

Remark

(1-category of groupoids)

From def. 1 we see that there is a category whose

But since this 1-category does not reflect the existence of homotopies/natural isomorphisms between homomorphsims/functors of groupoids (def. 1) 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 𝒢 1\mathcal{G}_1, 𝒢 2\mathcal{G}_2, 𝒢 3\mathcal{G}_3, 𝒢 4\mathcal{G}_4 be groupoid and let

𝒢 1F 1𝒢 2ηAAF 2AAAAF 2AA𝒢 3F 3𝒢 3 \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}_3

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

𝒢 1AAAAF 2η F 1F 3F 2F 1F 3F 2F 1AAAA𝒢 2 \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_2 \cdot \eta_ \cdot F_1 }}} \phantom{AAAA}\mathcal{G}_2

between the respective composites, with components given by

(F 2ηF 1)(x)F 2(η(F 1(x))). (F_2 \cdot \eta \cdot F_1)(x) \;\coloneqq\; F_2(\eta(F_1(x))) \,.

This operation constitutes a groupoid homomorphism/functor

F 3()F 1:Hom Grpd(𝒢 2,𝒢 3)Hom Grp(𝒢 1,𝒢 4). 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

𝒢 2F η 1 G η 2 H𝒢 3 \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(η 2η 1F 1=(F 3η 2F 1)(F 3η 1F 1). 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 xx any object of 𝒢 1\mathcal{G}_1 we find

(F 3(η 2η 1F 1)(x) F 2((η 2η 1)(F 1(X))) F 3(η 2(F 1(x))η 1(F 1(x))) =F 2(η 2(F 1(x)))F 2(η 1(F 1(X))) =((F 3η 2F 1)(F 3η 1F 1))(x). \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 (vbertical) composition of homotopies, except the third equality, which is the functoriality of F 2F_2.

Lemma

(horizontal composition of homotopies)

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

𝒢 1η 1AAF 1AAAAF 1AA𝒢 2η 2AAF 2AAAAF 2AA𝒢 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}} {\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

𝒢 1η 2η 1F 2F 1F 2F 1𝒢 3 \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

𝒢 1η 1AAF 1AAAAF 1AA𝒢 2AAF 2AA𝒢 3 𝒢 1AAF 1AA𝒢 2η 2AAF 2AAAAF 2AA𝒢 3 \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

𝒢 1AAF 1AA𝒢 2η 2AAF 2AAAAF 2AA𝒢 3 𝒢 1η 1AAF 1AAAAF 1AA𝒢 2AAF 2AA𝒢 3 \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

η 2η 1(η 2F 1)(F 2η 1) \eta_2 \cdot \eta_1 \;\coloneqq\; (\eta_2 \cdot F'_1) \circ (F_2 \cdot \eta_1)

while in the second case we get

η 2η 1(F 2η 1)(η 2F 1). \eta_2 \cdot \eta_1 \;\coloneqq\; ( F'_2 \cdot \eta_1 ) \circ (\eta_2 \cdot F_1) \,.

These two definitions coincide.

Proof

For xx an object of 𝒢 1\mathcal{G}_1, then we need that the following square diagram commutes in 𝒢 3\mathcal{G}_3

F 2(F 1(x)) (F 2η 1)(x) F 2(F 1(x)) (η 2F 1)(x) (η 2F 1)(x) F 2(F 1(x)) (F 2η 1)(x) F 2(F 1(y))AAAA=AAAAF 2(F 1(x)) F 2(η 1(x)) F 2(F 1(x)) η 2(F 1(x)) η 2(F 1(x)) F 2(F 1(x)) F 2(η 1(x)) F 2(F 1(y)). \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 η 2\eta_2 applied to the morphism η 1(x)\eta_1(x) in 𝒢 2\mathcal{G}_2.

Proposition

(horizontal composition with homotopy is natural transformation)

Consider groupoids, homomorphisms and homotopies of the form

𝒢 1η 1F 1F 1𝒢 2AAAAAAA𝒢 3η 3F 3F 3𝒢 4. \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)

(η 3()η 1):(F 3()F 1)(F 3()F 1):Hom Grpd(𝒢 2,𝒢 2)Hom Grpd(𝒢 1,𝒢 4). ( \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}_2) \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)Ho(Grpd) whose

This is usually denoted Ho(Grpd)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, hece a strict 2-category with hom-groupoids.

Equivalences of groupoids

Definition

(equivalence of groupoids)

Given two groupoids 𝒢 1\mathcal{G}_1 and 𝒢 2\mathcal{G}_2, then a homomorphism

F:𝒢 1𝒢 2 F\;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

is an equivalence it it is an isomorphism in the homotopy category Ho(Grpd)Ho(Grpd) (def. 4), hence if there exists a homomorphism the other way around

G:𝒢 2𝒢 1 G \;\colon\; \mathcal{G}_2 \longrightarrow \mathcal{G}_1

and a homotopy/natural transformations of the form

GFid 𝒢 1AAAAFGid 𝒢 2. 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 𝒢\mathcal{G} with set of objects XX, then the relation “there exists a morphism from xx to yy”, i.e.

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

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

π 0(𝒢) \pi_0(\mathcal{G})

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

Definition

(automorphism groups)

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

Definition

(weak homotopy equivalence of groupoids)

Let 𝒢 1\mathcal{G}_1 and 𝒢 2\mathcal{G}_2 be groupoids. Then a morphism (functor)

F:𝒢 1𝒢 2 F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2

is called a weak homotopy equivalence if

  1. it induces a bijection on connected components (def. 6):

    π 0(F):π 0(𝒢 1)π 0(𝒢 2) \pi_0(F) \;\colon\; \pi_0(\mathcal{G}_1) \overset{\simeq}{\longrightarrow} \pi_0(\mathcal{G}_2)
  2. for each object xx of 𝒢 1\mathcal{G}_1 the morphism

    F x,x:Aut 𝒢 1(x)Aut 𝒢 2(F 0(X)) 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. 7)

Groupoid representations

Definition

(groupoid representation)

Let 𝒢\mathcal{G} be a groupoid. Then:

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

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

to the groupoid core of the category Vect of vector spaces (example 7). Hence this is

  1. For each object xx of 𝒢\mathcal{G} a vector space V xV_x;

  2. for each morphism xfyx \overset{f}{\longrightarrow} y of 𝒢\mathcal{G} a linear map ρ(f):V xV y\rho(f) \;\colon\; V_x \to V_y

such that

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

    ρ(g)ρ(f)=ρ(gf) \rho(g) \circ \rho(f) = \rho(g \circ f)
  2. (respect for identities) for each object xx of the groupoid we have an equality

    ρ(id x)=id V x. \rho(id_x) = id_{V_x} \,.

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

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

to the groupoid core of Set. Hence this is

  1. For each object xx of 𝒢\mathcal{G} a set S xS_x;

  2. for each morphism xfyx \overset{f}{\longrightarrow} y of 𝒢\mathcal{G} a function ρ(f):S xS y\rho(f) \;\colon\; S_x \to S_y

such that composition and identities are respected, as above.

For ρ 1\rho_1 and ρ 2\rho_2 two such representations, then a homomorphism of representations

ϕ:ρ 1ρ 2 \phi \;\colon\; \rho_1 \longrightarrow \rho_2

is a natural transformation between these functors, hence is

  • for each object xx of the groupoid a (linear) function

    (V 1) xϕ(x)(V 2) x (V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x
  • such that for all morphisms xfyx \overset{f}{\longrightarrow} y we have

    ϕ(y)ρ 1(f)=ρ 2(x)ϕ(x)AAAAAA(V 1) x ϕ(x) (V 2) x ρ 1(f) ϕ 2(f) (V 1) y ϕ(y) (V 2) y \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 𝒢\mathcal{G} and homomorphisms between them constitute a category, called the representation category Rep Grpd(𝒢)Rep_{Grpd}(\mathcal{G}).

Examples

Example

(fundamental groupoid)

Let XX be a topological space. For x,yXx,y \in X two points, write P x,yXP_{x,y}X for the set of paths in XX from xx to yy. Consider the equivalence relationhomotopy relative boundary” on this set and write

Hom(x,y)(P x,yX)/ 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 XX of points in the topological space. This is the called the fundamental groupoid Π 1(X)\Pi_1(X) of XX.

This construction extends to a functor

Π 1:TopGrp \Pi_1 \;\colon\; Top \longrightarrow Grp

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

Π 1:Ho(Top)Ho(Grpd) \Pi_1 \;\colon\; Ho(Top) \longrightarrow Ho(Grpd)

on homotopy categories.

Example

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

If XX and YY are topological spaces and f:XYf \;\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:Π 1(X)Π 1(Y) F_f \;\colon\; \Pi_1(X) \longrightarrow \Pi_1(Y)

given on objects by the underlying function of ff

(F f) 0f (F_f)_0 \coloneqq f

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

(F f) x,y:(x[γ]y)(f(x)[fγ]f(y)). (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

Π 1:TopGrpd 1 \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:XYf,g \;\colon\; X \longrightarrow Y are two continuous function and

η:fg \eta \;\colon\; f \Rightarrow g

is a left homotopy between them, hence a continuous function

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

such that η(,0)=f\eta(-,0) = f and η(,1)=g\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

Π 1:Ho(Top)Ho(Grpd). \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 hY X \simeq_h Y

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

Π 1(X)Π 1(Y). \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 XX any set, there is the discrete groupoid Disc(X)Disc(X), whose set of objects is XX and whose only morphisms are identity morphisms.

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

Example

(delooping of a group)

Let GG be a group. Then there is a groupoid, denoted BGB G, with a single object pp, with morphisms

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

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

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

For G 1,G 2G_1, G_2 two groups, then there is a natural bijection between group homomorphisms ϕ:G 1G 2\phi \colon G_1 \to G_2 and groupoid homomorphisms GG 1B G2G G_1 \to B_ G_2: the latter are all of the form BϕB \phi, with (Bϕ) 0(B \phi)_0 uniquely fixed and (Bϕ) p,p=ϕ(B \phi)_{p,p} = \phi.

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

B():GrpGrpd 1 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

Bϕ 1,Bϕ 2:BGBH B \phi_1, B \phi_2 \;\colon\; B G \longrightarrow B H

By definition, such a homotopy (natural transformation) η:Bϕ 1Bϕ 2\eta \;\colon\; B \phi_1 \Rightarrow B \phi_2 is a choice of a single elemet η pH\eta_p \in H such that for all gGg \in G we have

ϕ 2(g)=hϕ 1(g)h 1AAAAAAAAAp h p ϕ 1(g) ϕ 2(g) p h p \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

ϕ 2=Ad hϕ 1. \phi_2 = Ad_h \circ \phi_1 \,.

Therefore notably the induced functor

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

to the homotopy category of groupoids is not fully faithful.

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

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

to the category of pointed objects of Grpd. Since groupoid homomorphisms BG 1BG 2B 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():GrpHo(Grpd */) B(-) \;\colon\; Grp \hookrightarrow Ho(Grpd^{\ast/})

then also the homotopies are required to preserve the absepoint, 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 {𝒢 i} iI\{\mathcal{G}_i\}_{i \in I} be a set of groupoids. Then their disjoint union (coproduct) is the groupoid

iI𝒢 i \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 𝒢 i\mathcal{G}_i if both objects are form this groupoid, and is empty otherwise.

Example

(disjoint union of delooping groupoids)

Let {G i} iI\{G_i\}_{i \in I} be a set of groups. Then there is a groupoid iIBG i\underset{i \in I}{\sqcup} B G_i which is the disjoint union groupoid (example 5) of the delooping groupoids BG iB G_i (example 4).

Its set of objects is the index set II, and

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

(groupoid core of a category)

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

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

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

For instance

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

    For 𝒞\mathcal{C} FinSet then the skeleton of this groupoid (prop. 4) is the disjoint union of deloopings (example 6) of all the symmetric groups:

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

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

    Core(FinVect)nGL(n). Core(FinVect) \simeq \underset{n \in \mathbb{N}}{\sqcup} GL(n) \,.
Example

(groupoid representation of delooping groupoid is group representation)

If BGB G is the delooping groupoid of a group GG (example 4), then a groupoid representation of BGB G according to def. 8 is equivalently a group representation of the group GG:

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

\,

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

  1. From any action of a group HH on a set XX we obtain an action groupoid or “weak quotientX//HX/ \!\! /H. This is also written XHX \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={*}X=\{*\} this gives the groupoid BH\mathbf{B}H, above.

  2. A setoid, that is a set XX equipped with an equivalence relation EE, becomes a groupoid with the multiplication (x,y);(y,z)=(x,z)(x,y);(y,z) = (x,z) for all (x,y),(y,z)E(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/EX/E.

  3. In particular, if EE is the universal relation X×XX \times X, then we get the square groupoid? X 2X^2, also called the trivial groupoid? on XX. Despite the latter name, there is an important special case, namely the groupoid I={0,1} 2I= \{0,1\}^2. This groupoid has non-identity elements ι:01,ι 1:10\iota:0 \to 1, \iota^{-1}: 1 \to 0, and can be regarded as a groupoid model of the unit interval [0,1][0,1] in topology.

  4. More generally, if we choose some subset SS of the points of a space XX, then we have a full subgroupoid of π 1(X)\pi_1(X) containing only those points in SS, denoted π 1(X,S)\pi_1(X,S). This can result in much more manageable groupoids; for instance Π 1([0,1],{0,1})\Pi_1([0,1],\{0,1\}) is the groupoid II considered above, while Π 1([0,1])\Pi_1([0,1]) has uncountably many objects (but is equivalent to II).

  5. If Γ\Gamma is a directed graph or quiver, then the free groupoid F(Γ)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 Γ\Gamma has one vertex xx and one loop u:xxu:x \to x. The free groupoid on Γ\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 Γ\Gamma; it is the point. As a groupoid, it is not the same as the free groupoid on Γ\Gamma, although it is the same as the free setoid on the free groupoid on Γ\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 Γ\Gamma and the point, which as a graph does not have uu.)

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

  7. 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.

Properties

Equivalences of groupoids

Lemma

(automorphism group depends on basepoint only up to conjugation)

For 𝒢\mathcal{G} a groupoid, let xx and yy be two objects in the same connected component (def. 6). Then there is a group isomorphism

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

between their automorphism groups (def. 7).

Proof

By assumption, there exists some morphism from xx to yy

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

The operation of conjugation with this morphism

Aut 𝒢(x) Ad f Aut 𝒢(y) g f 1gf \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} iI\{G_i\}_{i \in I} and {H j} jJ\{H_j\}_{j \in J} be sets of groups and consider a homomorphism (functor)

F:sqcupiIG ijJH j 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 4).

Then the following are equivalent:

  1. FF is an equivalence of groupoids (def. 5);

  2. FF is a weak homotopy equivalence (def. 3).

Proof

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

In the other direction, assume that FF is an equivalence of groupoids, and let GG 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

α:GFidAAAAβ:FGid \alpha \;\colon\; G \circ F \Rightarrow id \phantom{AAAA} \beta \;\colon\; F \circ G \Rightarrow id

are in each separate delooping groupoid BH jB H_j of the form

* α * G F 0(i),F 0(i)(F i,i(f)) id * α *AAAAAAAAAAAAAAAAAAAAA* β * F G 0(j),G 0(j)(G j,j(f)) id * β * \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,iF_{i,i} and F j,jF_{j,j} are group isomorphisms.

Proposition

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

Assuming the axiom of choice, then:

For 𝒢\mathcal{G} any groupoid, then there exists a set {G i} iI\{G_i\}_{i \in I} of groups and an equivalence of groupoids (def. 5)

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

between 𝒢\mathcal{G} and a disjoint union of delooping groupoids (example 6). This is called a skeleton of 𝒢\mathcal{G}.

Concretely, this exists for I=π 0(𝒢)I = \pi_0(\mathcal{G}) the set of connected components of 𝒢\mathcal{G} (def. 6) and for G iAut 𝒢(x)G_i \coloneqq Aut_{\mathcal{G}}(x) the automorphism group (def. 7) of any object xx in the given connected component.

Proof

Using the axiom of choice we may find a set {x i} iπ 0(𝒢)\{x_i\}_{i \in \pi_0(\mathcal{G})} of objects of 𝒢\mathcal{G}, with x ix_i being in the connected component iπ 0(𝒢)i \in \pi_0(\mathcal{G}).

This choice induces a functor

inc:iπ 0(𝒢)Aut 𝒢(x i)𝒢 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 iπ 0(𝒢)i \in \pi_0(\mathcal{G}) and for each object yy in that connected component a morphism

x if x i,yy. x_i \overset{f_{x_i,y}}{\longrightarrow} y \,.

Using this we obtain a functor the other way around

p:𝒢iπ 0(𝒢)Aut 𝒢(x i) 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 yy, zz of 𝒢\mathcal{G} is given by conjugation with the morphisms choosen above:

Hom 𝒢(y,z) p y,z Aut 𝒢(x i) y y f x i,y x i f f z z f x i,z 1 x i. \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

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

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

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

For the second we observe that choosing

η(y)f x i,y \eta(y) \coloneqq f_{x_i,y}

yields a naturality square by the above construction:

x i f x i,y y f x i,zff x i,y 1 f x i f x i,z z. \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:𝒢 1𝒢 2F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2 be a homomorphism of groupoids.

Assuming the axiom of choice then the following are equivalent:

  1. FF is an equivalence of groupoids (def. 5);

  2. FF is a weak homotopy equivalence in that

    1. it induces an bijection of sets of connected components (def. 6);

      π 0(F):π 0(𝒢 1)π 0(𝒢 0), \pi_0(F) \;\colon\; \pi_0(\mathcal{G}_1) \overset{\simeq}{\longrightarrow} \pi_0(\mathcal{G}_0) \,,
  3. for each object x𝒢 1x \in \mathcal{G}_1 it induces an isomorphis of automorphism groups (def. 7):

    F x,x:Aut 𝒢 1(x)Aut 𝒢 2(F 0(x)). F_{x,x} \;\colon\; Aut_{\mathcal{G}_1}(x) \overset{\simeq}{\longrightarrow} Aut_{\mathcal{G}_2}(F_0(x)) \,.
Proof

In one direction, if FF 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:

𝒢 1 inc 1 iπ 0(𝒢 1)Aut 𝒢 1(x i) F F˜ 𝒢 2 inc 2 iπ 0(𝒢 1)Aut 𝒢 2(F 0(x i)). \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 1inc_1 and inc 2inc_2 are equivalences of groupoids by prop. 4. Moreover, by assumption that FF is a weak homotopy equivalence F˜\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)Ho(Grpd) (def. 4), the top, bottom and right moprhism of the above diagram are isomorphisms. It follows that also ff is an isomorphism in Ho(Grpd)Ho(Grpd). But this means exactly that it is a homotopy equivalence of groupoids, by def. 5.

Proposition

(groupoid representations are products of group representations)

Assuming the axiom of choice then the following holds:

Let 𝒢\mathcal{G} be a groupoid. Then its category of groupoid representations is equivalent to the product category indexed by the set of connected components π 0(𝒢)\pi_0(\mathcal{G}) (def. 6) of group representations of the automorphism group G iAut 𝒢(x i)G_i \coloneqq Aut_{\mathcal{G}}(x_i) (def. 7) for x ix_i any object in the iith connected component:

Rep(𝒢)iπ 0(𝒢)Rep(G i). Rep(\mathcal{G}) \;\simeq\; \underset{i \in \pi_0(\mathcal{G})}{\prod} Rep(G_i) \,.
Proof

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

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

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

inc:iπ 0(𝒢)BG i𝒢. 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 *:Hom(𝒢,𝒞)Hom(iπ 0(𝒢)BG i,𝒞) 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 8 the category on the right is the product of group representation categories as claimed.

As 2-coskeletal Kan complexes

Groupoids KK are equivalent to 1-hypergroupoids, which are in particular 2-coskeletal Kan complexes N(K)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)gf(f,g) \mapsto g \circ f in the grouopoid is encoded in the 2-simplices of the Kan complex

y f = g x gf z. \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

y g z h f gf x h(gf) wandy g z h f hg x (hg)f w \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

y z x w=y z x w. \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.

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)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

References

An early occurence of the concept is

  • H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Mathematische Annalen, (1927) 96 (1): 360–366, doi:10.1007/BF01209171

(see Brandt 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

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

Further exposition includes

Technical discussion is for instance in the following references.

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

  • Philip Higgins, Categories and Groupoids, van Nostrand, New York, 1971,Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195. (web)

  • Ronnie Brown, Topology and groupoids, Booksurge, 2006. (web)

  • 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 August 30, 2017 17:48:26 by Urs Schreiber (77.56.177.247)