03.03.13 · modern-geometry / lie

Groupoid as a small category with all morphisms invertible

shipped3 tiersLean: none

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. I; Brown — Topology and Groupoids Ch. 6–11

Intuition Beginner

A group records the symmetries of one object: every symmetry can be undone, and any two can be performed in sequence. But many situations carry symmetries that only make sense between specific points. A path in a space runs from one point to another; you cannot glue a second path onto it unless the second one starts where the first ends. A groupoid is the structure built for exactly this: a collection of reversible arrows, each running from a start point to an end point, that combine only when the end of one meets the start of the next.

You can read a groupoid two ways, and they say the same thing. The first way is to picture a small "diagram" of dots and arrows: between any two dots there may be several arrows, and every arrow has a reverse that cancels it. The second way is to list raw data: a set of points, a set of arrows, a rule that reads off where each arrow starts and ends, a partial rule for combining matching arrows, a do-nothing arrow at each point, and a reverse for each arrow.

Why bother? Because a single group is too rigid for symmetry that varies from point to point. Two points joined by a path, two fibres of a bundle, two states linked by an allowed move: each carries symmetry that lives between a pair of points, not at one. A groupoid packages all of these point-to-point symmetries into one object you can compute with.

Visual Beginner

Picture several dots on a page. Between some pairs of dots, draw arrows. Each arrow points from a start dot to an end dot, and every arrow comes with a partner pointing the opposite way that undoes it. At each dot sits a small loop: the do-nothing arrow that starts and ends there.

Two arrows join head-to-tail only when the end dot of the first is the start dot of the second. The result is a single arrow from the first start dot to the last end dot. The dots fall into clusters: two dots are in the same cluster when some arrow connects them. Inside one cluster every dot is reachable from every other; between clusters there are no arrows at all.

Worked example Beginner

Take three points, ${1, 2, 3}$ , and let the arrows be all ordered pairs: one arrow from $i$ to $j$ for every choice of $i$ and $j$ . Write the arrow from $i$ to $j$ as $(j, i)$ , read "from $i$ to $j$ ".

The start of $(j, i)$ is $i$ and the end is $j$ . To combine, the arrow $(k, j)$ after the arrow $(j, i)$ gives $(k, i)$ : start at $i$ , pass through $j$ , finish at $k$ . The combination $(k, j) \cdot (j, i) = (k, i)$ works only because the middle label matches.

Count the arrows that start and end at the same point: $(1, 1)$ , $(2, 2)$ , $(3, 3)$ . These are the do-nothing arrows, one per point. The reverse of $(2, 1)$ is $(1, 2)$ , and combining them gives $(1, 1)$ , the do-nothing arrow at $1$ . The arrows that both start and end at point $1$ are just $(1, 1)$ by itself, so the "symmetries sitting at point $1$ " form a one-element group.

What this tells us: even with three points, the arrows organize into start, end, combination, do-nothing arrows, and reverses that all fit together. This all-pairs example is the simplest groupoid that is not a group, and it is connected — every point reaches every other.

Check your understanding Beginner

Formal definition Intermediate+

There are two standard definitions of a groupoid, and they describe the same objects. The first is short; the second exposes the working data.

Definition (categorical). A groupoid is a small category in which every morphism is an isomorphism ^{[Mackenzie Ch. I]}. Small means the objects form a set $G_{0}$ and the morphisms form a set $G_{1}$ ; every morphism an isomorphism means each $g : x \to y$ has a two-sided inverse $g^{- 1} : y \to x$ with $g^{- 1} g = 1_{x}$ and $g g^{- 1} = 1_{y}$ .

Definition (algebraic). A groupoid consists of a set $G_{0}$ of objects, a set $G_{1}$ of arrows, two maps $α, β : G_{1} \to G_{0}$ (the source and target), a partially defined multiplication $m$ , a unit map $ε : G_{0} \to G_{1}$ , and an inversion $ι : G_{1} \to G_{1}$ , subject to the axioms below. Two arrows $g, h$ are composable when $α (g) = β (h)$ ; on the set of composable pairs $$ G_1 \times_{G_0} G_1 := {(g, h) \in G_1 \times G_1 : \alpha(g) = \beta(h)} $$ the multiplication produces $g h := m (g, h)$ . The axioms are:

(source/target of products) if $(g, h)$ is composable then $α (g h) = α (h)$ and $β (g h) = β (g)$ ;
(associativity) if $(g, h)$ and $(h, k)$ are composable then $(g h) k = g (hk)$ (both sides defined);
(units) writing $1_{x} := ε (x)$ , one has $α (1_{x}) = β (1_{x}) = x$ , and $1_{β (g)} g = g = g 1_{α (g)}$ for every arrow $g$ ;
(inverses) writing $g^{- 1} := ι (g)$ , one has $α (g^{- 1}) = β (g)$ , $β (g^{- 1}) = α (g)$ , and $g^{- 1} g = 1_{α (g)}$ , $g g^{- 1} = 1_{β (g)}$ .

Notation. Here $G_{0}$ is the object set, $G_{1}$ the arrow set, $α$ the source, $β$ the target, $1_{x}$ the unit arrow at $x$ , $g^{- 1}$ the inverse of $g$ , and $G_{1} \times_{G_{0}} G_{1}$ the set of composable pairs (the fibred product of $α$ and $β$ ). The composite $g h$ means "first $h$ , then $g$ ": this is the function-composition order, consistent with reading $g : x \to y$ as a morphism. A groupoid is written $G ⇉ G_{0}$ , the double arrow recording the pair $(α, β)$ .

For each pair $x, y \in G_{0}$ write $G (x, y) := α^{- 1} (x) \cap β^{- 1} (y)$ for the set of arrows from $x$ to $y$ . The set $G (x, x)$ is the vertex group (also isotropy or object group) at $x$ . Define a relation on $G_{0}$ by $x \sim y$ when $G (x, y) \neq = \emptyset$ ; this is the orbit relation, and its classes are the orbits. A groupoid is connected (or transitive) when it has a single orbit, equivalently when $G (x, y) \neq = \emptyset$ for all $x, y$ .

Counterexamples to common slips

An arbitrary small category is not a groupoid: the monoid $(N, +)$ as a one-object category has no inverse for the morphism $1$ . The defining extra demand is invertibility of every morphism.
The combination $g h$ requires $α (g) = β (h)$ ; demanding $α (g) = α (h)$ instead does not define an associative partial operation and is not the groupoid law.
A groupoid need not be connected. The disjoint union of two groups is a groupoid with two orbits and no arrow between them.

Key theorem with proof Intermediate+

Theorem (equivalence of the two definitions). The categorical and algebraic definitions of a groupoid describe the same structures: every small category with all morphisms invertible yields the algebraic data satisfying the axioms, and conversely every system $(G_{0}, G_{1}, α, β, m, ε, ι)$ satisfying the axioms is a small category in which every morphism is invertible. The two passages are mutually inverse.

Proof. From a small category $G$ with all morphisms invertible, set $G_{0} := Ob (G)$ and $G_{1} := ∐_{x, y} G (x, y)$ , the set of all morphisms ^{[Mackenzie Ch. I]}. For a morphism $g \in G (x, y)$ put $α (g) := x$ and $β (g) := y$ . Define $m (g, h) := g \circ h$ on composable pairs (where $α (g) = β (h)$ , so the codomain of $h$ equals the domain of $g$ ); define $ε (x) := id_{x}$ and $ι (g) := g^{- 1}$ , which exists by hypothesis. The category axioms give source/target of products, associativity, and the unit laws directly; the existence of inverses gives the inverse axiom. So the algebraic axioms hold.

Conversely, from algebraic data $(G_{0}, G_{1}, α, β, m, ε, ι)$ build a category $G$ with object set $G_{0}$ and hom-sets $G (x, y) := G (x, y) = α^{- 1} (x) \cap β^{- 1} (y)$ . Composition $G (y, z) \times G (x, y) \to G (x, z)$ is $m$ : if $g \in G (y, z)$ and $h \in G (x, y)$ then $α (g) = y = β (h)$ , so $(g, h)$ is composable, and the source/target axiom gives $α (g h) = α (h) = x$ , $β (g h) = β (g) = z$ , so $g h \in G (x, z)$ . Associativity of $m$ is the associativity axiom; the identity at $x$ is $1_{x} = ε (x) \in G (x, x)$ , and the unit axiom is the category unit law. Each $g \in G (x, y)$ has $g^{- 1} \in G (y, x)$ with $g^{- 1} g = 1_{x}$ and $g g^{- 1} = 1_{y}$ , so every morphism is an isomorphism.

These passages are mutually inverse. Starting from a category, extracting the algebraic data, and rebuilding the category returns the same hom-sets $G (x, y)$ , the same composition, the same identities, and the same inverses, because each was defined to match. Starting from algebraic data, building the category, and re-extracting returns $α, β$ (recorded by which hom-set an arrow lies in), $m$ (the composition), $ε$ (the identities), and $ι$ (the inverses) unchanged. Hence the two definitions are interchangeable. $□$

Bridge. This equivalence builds toward every later use of groupoids, because it lets one switch freely between the diagrammatic category picture and the concrete arrow-data picture; it appears again in the Lie groupoid 03.03.10, where the algebraic data is exactly what carries the smooth structure that the categorical phrasing cannot see, and in the fundamental groupoid 03.12.08, where paths-up-to-homotopy are the arrows of a connected groupoid. The categorical side makes functoriality and equivalence of groupoids immediate, while the algebraic side makes the source and target fibres, the vertex groups, and the orbit relation available as honest sets to compute with, and the two together are what let the structure theorems below be both stated cleanly and proved by hand.

Exercises Intermediate+

Exercise 4 (medium, proof).

Show that a groupoid in which there is at most one arrow between any two objects is the same data as an equivalence relation on $G_{0}$ .

Hint

Define $x R y$ when there is an arrow from $x$ to $y$ , and check reflexivity, symmetry, transitivity against units, inverses, composition.

Answer

Let $R \subseteq G_{0} \times G_{0}$ be ${(β (g), α (g)) : g \in G_{1}}$ . Units give $(x, x) \in R$ (reflexive); inverses give $(y, x) \in R \Rightarrow (x, y) \in R$ (symmetric); composition gives $(z, y), (y, x) \in R \Rightarrow (z, x) \in R$ (transitive). The at-most-one-arrow hypothesis makes $g \mapsto (β (g), α (g))$ a bijection $G_{1} \to R$ , with composition forced. Conversely an equivalence relation $R$ is a groupoid with arrow set $R$ , $α (y, x) = x$ , $β (y, x) = y$ , and the unique composition; reflexivity supplies units, symmetry supplies inverses, transitivity supplies products.

Lean formalization Intermediate+

lean_status: none — Mathlib has CategoryTheory.Groupoid (the categorical definition: a category with inv on every morphism and the two cancellation laws) and CategoryTheory.Groupoid.vertexGroup for $G (x, x)$ , but it does not package the equivalence with the algebraic source/target/partial-multiplication presentation, nor the connected-groupoid structure theorem. The statements below indicate the intended formal content; the equivalence statement is the part that has no current Mathlib home, which is why no Lean module is declared.

-- In Mathlib: the categorical definition (paraphrased).
-- class Groupoid (obj : Type u) extends Category obj where
--   inv : ∀ {X Y : obj}, (X ⟶ Y) → (Y ⟶ X)
--   inv_comp : ∀ {X Y} (f : X ⟶ Y), inv f ≫ f = 𝟙 Y
--   comp_inv : ∀ {X Y} (f : X ⟶ Y), f ≫ inv f = 𝟙 X

-- Target (not in Mathlib): the algebraic presentation and the equivalence.
structure AlgGroupoid where
  G0     : Type*
  G1     : Type*
  source : G1 → G0
  target : G1 → G0
  mul    : {p : G1 × G1 // source p.1 = target p.2} → G1
  unit   : G0 → G1
  inv    : G1 → G1
  -- source/target, associativity, unit, inverse axioms ...

-- The missing theorem: AlgGroupoid ≃ (Σ obj, Groupoid obj), mutually inverse.

The genuine obstacle is bookkeeping the partial multiplication: in the categorical version composition is total on each hom-set, while the algebraic version carries a single global arrow set with a partial product defined on the fibred product source p.1 = target p.2. Translating one to the other is the content of the equivalence theorem above.

Advanced results Master

Structure of a connected groupoid. Let $G ⇉ G_{0}$ be connected, fix $x_{0} \in G_{0}$ , and set $K := G (x_{0}, x_{0})$ . The bijection of Exercise 7 upgrades to an isomorphism of groupoids $$ G ;\cong; K \times \mathrm{Pair}(G_0), $$ where $Pair (G_{0}) = G_{0} \times G_{0}$ is the pair groupoid and the product on the right is the groupoid whose arrow set is $K \times G_{0} \times G_{0}$ with $α (k, y, x) = x$ , $β (k, y, x) = y$ , and $(k^{'}, z, y) (k, y, x) = (k^{'} k, z, x)$ . Equivalently $G$ is the constant groupoid on $K$ over $G_{0}$ in Higgins' sense, namely the product of the vertex group with the codiscrete (one-arrow-between-any-two-objects) groupoid on $G_{0}$ . As a category, $G$ is also equivalent to the one-object groupoid $K$ : the inclusion of $K$ as $G (x_{0}, x_{0})$ and the functor sending every object to $x_{0}$ and every arrow $g \in G (x, y)$ to $τ_{y}^{- 1} g τ_{x}$ are mutually quasi-inverse, with natural isomorphism supplied by the chosen arrows $τ_{y}$ . So a connected groupoid carries the same homotopy-theoretic content as a single group, while remembering its set of objects.

Decomposition into orbits. For a general groupoid, the orbit relation partitions $G_{0}$ into orbits ${O_{i}}_{i \in I}$ , and there are no arrows between distinct orbits, since an arrow $g$ has $α (g) \sim β (g)$ . Restricting $G$ to each orbit yields a connected subgroupoid $G ∣_{O_{i}}$ , and $$ G ;\cong; \coprod_{i \in I} G|{O_i} ;\cong; \coprod{i \in I} \big(K_i \times \mathrm{Pair}(O_i)\big), $$ where $K_{i}$ is the vertex group at any chosen base point of $O_{i}$ (well defined up to isomorphism by the conjugation result). Thus every groupoid is, up to isomorphism, a disjoint union of pieces each of the form (vertex group) $\times$ (pair groupoid), and up to equivalence of categories it is the disjoint union of its vertex groups, one per orbit. This is the groupoid form of the statement that a groupoid is determined up to equivalence by its set of orbits and the isomorphism class of vertex group on each.

The fundamental groupoid as a universal example. For a topological space $X$ , the fundamental groupoid $Π_{1} (X)$ has objects the points of $X$ and arrows the homotopy classes rel endpoints of paths 03.12.08. It is connected exactly when $X$ is path-connected, and then its vertex group at $x$ is $π_{1} (X, x)$ ; the orbit decomposition recovers the path components, and the conjugation isomorphism of vertex groups is the classical change-of-basepoint isomorphism of fundamental groups. The pair-groupoid factor records the choice of paths $τ_{y}$ between basepoints. Van Kampen's theorem is cleanest stated for $Π_{1}$ rather than $π_{1}$ , because the groupoid keeps several basepoints at once.

Functors, equivalence, and the skeleton. A morphism of groupoids is a functor: a pair of maps $G_{0} \to H_{0}$ , $G_{1} \to H_{1}$ respecting source, target, composition, and units (inverses are then automatic). An equivalence of groupoids is an equivalence of the underlying categories: a fully faithful, essentially surjective functor. The orbit decomposition shows that every groupoid is equivalent to its skeleton, the disjoint union of its vertex groups, one per orbit. Two groupoids are equivalent precisely when there is an orbit-set bijection matching isomorphic vertex groups. Equivalence, not isomorphism, is the right notion of sameness: it forgets the size of each orbit while keeping the symmetry $K_{i}$ and the partition into orbits.

Synthesis. A groupoid is two pictures of one structure: a small category in which every morphism is invertible, and a set of arrows over a set of objects with source, target, a partial associative product, units, and inverses. The equivalence theorem lets either picture be used wherever convenient, and the algebraic picture is what carries the analytic and smooth enrichments developed elsewhere. Three invariants organize the algebra: the vertex group $G (x, x)$ at each object, the orbit relation $x \sim y$ , and the conjugation isomorphism $ϕ_{a} (g) = a g a^{- 1}$ that identifies vertex groups along an orbit. From these, the structure theorems follow: a connected groupoid is isomorphic to (vertex group) $\times$ (pair groupoid) and equivalent as a category to its vertex group alone, while an arbitrary groupoid decomposes as a disjoint union of such connected pieces, one per orbit, hence is equivalent to its skeleton of vertex groups. The standard examples populate every position on this map: a group is the one-object connected groupoid; the pair groupoid of a set is the connected groupoid with one-element vertex group; an equivalence relation is the groupoid with at most one arrow between objects; the action groupoid $H ⋉ X$ has stabilizers as vertex groups; and the fundamental groupoid $Π_{1} (X)$ has $π_{1}$ as vertex groups and path-components as orbits. Across all of them the groupoid is the home for symmetry between pairs of objects rather than at one, with equivalence rather than isomorphism the matching notion of sameness.

Full proof set Master

Proposition (the vertex group is a group). In any groupoid $G$ , for each object $x$ the set $G (x, x)$ with the restricted multiplication is a group, with identity $1_{x}$ and inverses the groupoid inverses.

Proof. Every pair $g, h \in G (x, x)$ is composable, since $α (g) = x = β (h)$ , and $g h \in G (x, x)$ because $α (g h) = α (h) = x$ and $β (g h) = β (g) = x$ . So multiplication restricts to a total binary operation on $G (x, x)$ . It is associative by the groupoid associativity axiom. The unit $1_{x}$ lies in $G (x, x)$ and satisfies $1_{x} g = g = g 1_{x}$ for $g \in G (x, x)$ by the unit axiom. Each $g \in G (x, x)$ has $g^{- 1}$ with $α (g^{- 1}) = β (g) = x$ , $β (g^{- 1}) = α (g) = x$ , so $g^{- 1} \in G (x, x)$ , and $g^{- 1} g = 1_{x} = g g^{- 1}$ . Hence $G (x, x)$ is a group 01.02.01. $□$

Proposition (vertex groups along an orbit are isomorphic). If $a \in G (x, y)$ , then $ϕ_{a} : G (x, x) \to G (y, y)$ , $ϕ_{a} (g) = a g a^{- 1}$ , is a group isomorphism, and $ϕ_{a}$ depends on $a$ only up to an inner automorphism of $G (y, y)$ .

Proof. For $g \in G (x, x)$ the product $a g a^{- 1}$ is defined: $a^{- 1} \in G (y, x)$ , $g \in G (x, x)$ , $a \in G (x, y)$ , with sources and targets matching, and $a g a^{- 1} \in G (y, y)$ . It is a homomorphism: $ϕ_{a} (g) ϕ_{a} (g^{'}) = a g a^{- 1} a g^{'} a^{- 1} = a g g^{'} a^{- 1} = ϕ_{a} (g g^{'})$ , using $a^{- 1} a = 1_{x}$ . It is invertible with inverse $ϕ_{a^{- 1}}$ , because $a^{- 1} (a g a^{- 1}) a = g$ . For two choices $a, a^{'} \in G (x, y)$ , write $u := a^{'} a^{- 1} \in G (y, y)$ ; then $ϕ_{a^{'}} (g) = a^{'} g a^{' - 1} = u (a g a^{- 1}) u^{- 1} = Inn_{u} (ϕ_{a} (g))$ , so $ϕ_{a^{'}} = Inn_{u} \circ ϕ_{a}$ differs from $ϕ_{a}$ by the inner automorphism $Inn_{u}$ of $G (y, y)$ . $□$

Proposition (connected groupoid structure theorem). Let $G$ be connected, $x_{0} \in G_{0}$ , $K := G (x_{0}, x_{0})$ , and choose $τ_{y} \in G (x_{0}, y)$ with $τ_{x_{0}} = 1_{x_{0}}$ . Then $Θ : G \to K \times Pair (G_{0})$ , $Θ (g) = (τ_{β (g)}^{- 1} g τ_{α (g)}, β (g), α (g))$ , is an isomorphism of groupoids.

Proof. Write $x := α (g)$ , $y := β (g)$ , $k (g) := τ_{y}^{- 1} g τ_{x} \in K$ . The map $Θ$ has the inverse $(k, y, x) \mapsto τ_{y} k τ_{x}^{- 1}$ of Exercise 7, so it is a bijection on arrows; on objects it is the identity. It respects source and target by construction: the $Pair$ -component reads off $(β (g), α (g))$ . It preserves composition: for composable $g \in G (x, y)$ , $g^{'} \in G (y, z)$ , $$ k(g'g) = \tau_z^{-1}(g' g)\tau_x = (\tau_z^{-1} g' \tau_y)(\tau_y^{-1} g \tau_x) = k(g'),k(g), $$ inserting $τ_{y} τ_{y}^{- 1} = 1_{y}$ , while the pair components multiply as $(z, y) (y, x) = (z, x)$ . Hence $Θ (g^{'} g) = (k (g^{'}) k (g), z, x) = Θ (g^{'}) Θ (g)$ in $K \times Pair (G_{0})$ . It sends units to units: $k (1_{x}) = τ_{x}^{- 1} τ_{x} = 1_{x_{0}}$ , so $Θ (1_{x}) = (1_{x_{0}}, x, x)$ . A composition-and-unit-preserving bijection on objects and arrows is an isomorphism of groupoids. $□$

Proposition (a connected groupoid is equivalent to its vertex group). With $G$ , $x_{0}$ , $K$ as above, the inclusion $ι : K ↪ G$ (sending the single object to $x_{0}$ and $k$ to $k \in G (x_{0}, x_{0})$ ) is an equivalence of categories.

Proof. The functor $ι$ is fully faithful onto $G (x_{0}, x_{0}) = K$ and faithful in general because $K$ is a one-object groupoid mapping bijectively onto $G (x_{0}, x_{0})$ . Define $F : G \to K$ on objects by $F (y) = *$ (the single object) and on arrows by $F (g) = τ_{β (g)}^{- 1} g τ_{α (g)} = k (g)$ . The previous proposition shows $F$ is a functor (it is the $K$ -component of $Θ$ ). Then $F \circ ι = id_{K}$ since for $k \in K$ , $k (k) = τ_{x_{0}}^{- 1} k τ_{x_{0}} = k$ . And $ι \circ F$ is naturally isomorphic to $id_{G}$ : the family $η_{y} := τ_{y} \in G (x_{0}, y) = Hom_{G} ((ι F) (y), y)$ is a natural isomorphism, since for $g \in G (x, y)$ , $$ \eta_y \cdot (\iota F)(g) = \tau_y , k(g) = \tau_y \tau_y^{-1} g \tau_x = g,\tau_x = g \cdot \eta_x. $$ So $ι$ is fully faithful and essentially surjective (every object is isomorphic to $x_{0}$ by connectedness), hence an equivalence. $□$

Connections Master

A group 01.02.01 is exactly a groupoid with one object: the object set is a point, every pair of arrows is composable, the partial product becomes total, and the vertex-group construction returns the group itself. The structure theorem reduces to the tautology that a one-object connected groupoid is its own vertex group.
A Lie groupoid 03.03.10 is this algebraic structure carrying a compatible smooth structure: object and arrow sets become manifolds, source and target become surjective submersions, and the partial multiplication becomes smooth on the fibred-product manifold. The vertex group $G (x, x)$ becomes a Lie group, and the connected-groupoid structure theorem becomes the principal-bundle picture of a transitive Lie groupoid. This unit supplies the categorical and orbit-theoretic skeleton that the smooth theory enriches.
The fundamental groupoid 03.12.08 is the universal topological example: its vertex groups are the fundamental groups $π_{1} (X, x)$ , its orbits are path components, and the conjugation isomorphism of vertex groups is the change-of-basepoint isomorphism. The groupoid viewpoint is what makes the van Kampen theorem functorial across several basepoints at once.
A group action 03.03.02 yields the action groupoid $H ⋉ X$ , whose vertex groups are the stabilizers and whose orbits are the action orbits; transitivity of the groupoid is transitivity of the action. The groupoid repackages a possibly badly behaved quotient $X / H$ as a well-structured object, the algebraic precursor of the quotient stack.

Historical & philosophical context Master

The groupoid concept was isolated by Heinrich Brandt in 1927, in the arithmetic of quaternary quadratic forms and quaternion ideals, where the composition of ideal classes is defined only when the right module of one matches the left module of the next ^{[Brandt 1927]}. Brandt's "Gruppoid" is precisely a set with a partially defined, associative, invertible multiplication — the algebraic definition given above. The categorical reading came with the rise of category theory: a groupoid is a small category with all morphisms invertible, a description natural once Eilenberg and Mac Lane had categories in hand, and developed systematically by Charles Ehresmann from 1957 onward as part of his program on structured categories ^{[Ehresmann 1957]}.

The topological theory was carried by Ronald Brown, whose Topology and Groupoids (first edition 1968) argued that the fundamental groupoid, not the fundamental group, is the correct invariant for stating van Kampen's theorem and handling spaces with several basepoints ^{[Brown 2006]}. Philip Higgins' Categories and Groupoids (1971) gave the purely algebraic theory its standard form, including the structure of connected groupoids and the free-groupoid constructions ^{[Higgins 1971]}. The differential-geometric line, tracing back to Ehresmann's differentiable categories, was systematized by Kirill Mackenzie, whose monographs take the algebraic definition as the starting point for the smooth theory ^{[Mackenzie Ch. I]}.

Bibliography Master

@article{brandt1927,
  author    = {Brandt, Heinrich},
  title     = {\"Uber eine Verallgemeinerung des Gruppenbegriffes},
  journal   = {Mathematische Annalen},
  volume    = {96},
  pages     = {360--366},
  year      = {1927}
}

@incollection{ehresmann1957,
  author    = {Ehresmann, Charles},
  title     = {Gattungen von lokalen Strukturen},
  booktitle = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  volume    = {60},
  pages     = {49--77},
  year      = {1957}
}

@book{brown2006,
  author    = {Brown, Ronald},
  title     = {Topology and Groupoids},
  publisher = {BookSurge (3rd ed.; 1st ed. McGraw-Hill, 1968)},
  year      = {2006}
}

@book{higgins1971,
  author    = {Higgins, Philip J.},
  title     = {Categories and Groupoids},
  series    = {Van Nostrand Reinhold Mathematical Studies},
  volume    = {32},
  publisher = {Van Nostrand Reinhold},
  year      = {1971}
}

@book{mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@article{weinstein1996,
  author    = {Weinstein, Alan},
  title     = {Groupoids: unifying internal and external symmetry},
  journal   = {Notices of the American Mathematical Society},
  volume    = {43},
  number    = {7},
  pages     = {744--752},
  year      = {1996}
}

Groupoid — a small category with every morphism invertible, equivalently a set of arrows over a set of objects with partial composition, units, and inverses; its vertex groups, orbits, and conjugation isomorphisms decompose it up to equivalence into the disjoint union of its vertex groups, and it is the algebraic core that the Lie groupoid endows with smooth structure.

Prerequisites

01.02.01
03.03.01
03.03.10

Tier anchors

beginner: reversible arrows between points that compose only when endpoints match, with a do-nothing arrow at every point
intermediate: Brown — Topology and Groupoids Ch. 6–8; Higgins — Categories and Groupoids §1–§3
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. I; Brown — Topology and Groupoids Ch. 6–11

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. I, §1.1–1.4 (groupoids, the algebraic definition)
Brown — Topology and Groupoids · Ch. 6–11 (groupoids and the fundamental groupoid)
Higgins — Categories and Groupoids · §1–§5 (algebra of groupoids, connected groupoids)
Brandt — Über eine Verallgemeinerung des Gruppenbegriffes · Math. Ann. 96 (1927), the originating paper

Estimated time

beginner: 16m
intermediate: 45m
master: 75m