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Groupoid

In mathematics, a groupoid is a category in which every morphism is an isomorphism, providing a generalization of the group structure to allow for multiple objects and partial compositions between them.^[1] Formally, it consists of a set of objects, a set of arrows (morphisms) equipped with source and target maps, a partial composition operation that is associative where defined, identity morphisms for each object, and inverses for every arrow, satisfying the usual axioms of categories with invertibility.^[1] This structure captures symmetries and equivalences in a broader sense than groups, where a group corresponds precisely to a groupoid with a single object.^[2] Groupoids arise naturally in various fields, such as algebraic topology via the fundamental groupoid of a space, which encodes homotopy classes of paths between points, and in differential geometry through Lie groupoids, which generalize Lie group actions on manifolds.^[3] For instance, the pair groupoid on a set B has objects as elements of B and arrows as ordered pairs (x, y), with composition (x, y) \circ (y, z) = (x, z), illustrating how groupoids model transitive relations or equivalences.^[3] Historically, the concept was introduced by Heinrich Brandt in 1926 to study quadratic forms and ideals, evolving through contributions from Charles Ehresmann in the 1950s to applications in foliation theory and higher category theory.^[2] While the category-theoretic definition dominates modern usage, alternative notions exist, such as the algebraic groupoid—a set with a binary operation satisfying only closure (no associativity required)—or the more general structure over a base set with partial multiplication, identities, and inverses, often used in semigroup theory.^[3] These variants highlight the flexibility of the term, with over 3,000 nonisomorphic algebraic groupoids on three elements, underscoring their combinatorial richness.^[3]

Definitions

Algebraic definition

In the algebraic approach, a groupoid is defined as a structure generalizing groups to allow for multiple objects and partial compositions between arrows connecting them. Formally, a groupoid G consists of two sets: G^0, the set of objects (also called vertices), and G^1, the set of arrows (or morphisms). There are source and target maps s, t: G^1 \to G^0, an identity section e: G^0 \to G^1 assigning to each object its identity arrow, and an inversion map i: G^1 \to G^1 assigning to each arrow its inverse. Composition is given by a partial binary operation m: G^1 \times_{G^0} G^1 \to G^1, where the domain of m is the fiber product G^1 \times_{G^0} G^1 = \{ (a,b) \in G^1 \times G^1 \mid t(a) = s(b) \}, ensuring arrows are composable only when the target of the first matches the source of the second.^[4] The structure satisfies group-like axioms restricted to composable arrows. Specifically, associativity holds whenever defined:

m(m(a,b),c) = m(a,m(b,c))

for all a,b,c \in G^1 such that t(b) = s(c) and t(a) = s(b). The identities act as left and right units:

m(a, e(t(a))) = a = m(e(s(a)), a)

for all a \in G^1. The inverses satisfy:

m(a, i(a)) = e(t(a)), \quad m(i(a), a) = e(s(a))

and moreover s(i(a)) = t(a), t(i(a)) = s(a), ensuring every arrow is invertible. These axioms ensure that the composable arrows between any pair of objects form a group.^[4] The algebraic notion of groupoid was introduced by Heinrich Brandt in 1926, motivated by the study of quadratic forms and ideals. It was later applied by B. Jónsson and A. Tarski in 1947 to direct decompositions of finite algebraic systems.^[4] A basic example is the discrete (or trivial) groupoid on a set X, where G^0 = X and G^1 = X with arrows consisting solely of the identity maps e(x) for each x \in X; here, composition is only possible for an arrow with itself, yielding the identity, and inverses are the identities themselves.^[4]

Category-theoretic definition

In category theory, a groupoid is defined as a category \mathbf{C} in which every morphism is an isomorphism, meaning that for every morphism f: A \to B in \mathbf{C}, there exists an inverse morphism g: B \to A such that g \circ f = \mathrm{id}_A and f \circ g = \mathrm{id}_B.^[4] This structure generalizes the notion of a group by allowing multiple objects, where the morphisms between objects capture invertible transformations akin to group elements acting on themselves.^[5] The components of a groupoid \mathbf{C} mirror those of a general category: it consists of a class of objects \mathrm{Ob}(\mathbf{C}), a collection of morphisms forming Hom-sets \mathbf{C}(A, B) for each pair of objects A, B, a composition operation \circ: \mathbf{C}(B, C) \times \mathbf{C}(A, B) \to \mathbf{C}(A, C) that is associative, and identity morphisms \mathrm{id}_A: A \to A for each object A satisfying the usual unit and compatibility axioms. Due to the invertibility of all morphisms, each Hom-set \mathbf{C}(A, B) forms a group under composition, with the identity \mathrm{id}_A (or \mathrm{id}_B when appropriate) as the unit element and inverses providing the group operation's reversibility.^[4] This categorical perspective emphasizes the role of groupoids in broader frameworks, such as functors between groupoids preserving composition and isomorphisms, facilitating generalizations in algebraic topology and higher category theory.^[6] Groupoids can also be defined internally within a fixed ambient category \mathbf{E} equipped with pullbacks, allowing for constructions in settings like topological spaces or schemes. An internal groupoid in \mathbf{E} comprises an object of objects X_0 \in \mathbf{E}, an object of arrows X_1 \in \mathbf{E}, source and target projections d_0, d_1: X_1 \to X_0, identity section s: X_0 \to X_1, and composition morphism m: X_1 \times_{X_0} X_1 \to X_1 (where the fiber product is over d_1 and d_0), satisfying the standard axioms of associativity, units, and domain/codomain compatibility via commutative diagrams in \mathbf{E}. Additionally, an inversion morphism i: X_1 \to X_1 provides inverses, ensuring every arrow is invertible in a manner compatible with the internal structure.^[7] This internal formulation is particularly useful for studying groupoids in non-set-theoretic categories, such as étale groupoids in algebraic geometry.^[8] A special case arises when a groupoid has a single object, reducing to a group: the endomorphism monoid at that object becomes a group under composition, with all elements invertible. Groupoids are often denoted with G_0 for the object of objects and G_1 for the object of arrows, with d_0, d_1: G_1 \to G_0 as the source and target maps, respectively, highlighting the arrow-object duality central to the theory.^[4] This notation underscores the category-theoretic emphasis on arrows as primary structure, distinguishing groupoids from purely algebraic presentations.^[5]

Equivalence of definitions

The algebraic and category-theoretic definitions of a groupoid are equivalent, meaning that any structure satisfying one set of axioms can be uniquely reconstructed as satisfying the other.^[4] This equivalence arises because both perspectives describe a collection of objects equipped with invertible arrows between them, where composition is defined only when compatible (via source and target).^[9] The category of groupoids in the algebraic sense is equivalent to the category of groupoids in the categorical sense via the forgetful functor that views a category as a directed graph with partial binary composition.^[10] To sketch the proof, start from a category-theoretic groupoid \mathcal{G}: its set of objects forms G^0, the set of morphisms forms G^1, the source and target maps are the domain and codomain functions, the identities are the identity morphisms, and the inverses are guaranteed by the isomorphism condition, yielding an algebraic groupoid. Conversely, given an algebraic groupoid G, construct a category with objects G^0, morphisms G^1, partial composition defined by the groupoid operation wherever the target of the first matches the source of the second, identity elements from G^0, and inverses from the groupoid structure; invertibility then ensures every morphism is an isomorphism, recovering the category-theoretic definition.^[4] This correspondence is bijective and preserves the algebraic and categorical axioms, including associativity and unit properties.^[9] A key consequence is that every algebraic groupoid arises from a category in which all morphisms are isomorphisms, and vice versa; furthermore, this equivalence induces a correspondence between morphisms in the two categories, namely the functors between category-theoretic groupoids that respect the algebraic structure.^[10] One subtlety in this equivalence concerns thin groupoids, where there is at most one morphism between any pair of objects; such structures correspond precisely to equivalence relations on the object set G^0, with morphisms representing equality within equivalence classes.^[4]

Basic structures

Vertex groups and isotropy groups

In a groupoid \mathcal{G} with object set \mathcal{G}^0, the vertex group at an object x \in \mathcal{G}^0, denoted \mathcal{G}(x) or \Aut_{\mathcal{G}}(x), consists of all arrows f: x \to x (endomorphisms of x) equipped with the restriction of the groupoid composition, forming a group whose identity is the identity arrow at x and whose inverses are the groupoid inverses.^[4] The structure of \mathcal{G}(x) captures the local symmetries or automorphisms available at x, determining the extent of invertibility in the endomorphism monoid at that object.^[4] The term isotropy group is often used as a synonym for the vertex group, particularly in contexts involving group actions or geometric realizations, where it emphasizes the stabilizers of points under the associated transformations.^[11] For instance, in such settings, the isotropy group at x comprises the transformations that fix x while acting on a neighborhood, aligning with the algebraic definition of endomorphisms in the groupoid.^[11] This terminology highlights the role of vertex groups in preserving local structure under symmetries. The loop groupoid (or isotropy subgroupoid) of \mathcal{G} is the sub-groupoid formed by all identity arrows on \mathcal{G}^0 together with the union of the vertex groups \mathcal{G}(x) for x \in \mathcal{G}^0; composition is possible only within each \mathcal{G}(x), yielding a disjoint union of groups over the objects.^[12] This structure isolates the "loop" components of \mathcal{G}, focusing exclusively on endomorphisms and providing a normal subgroupoid that encodes the full local group-theoretic data.^[12] The cardinality and structure of the vertex groups \mathcal{G}(x) govern key properties of local invertibility in \mathcal{G}; for example, if every vertex group is trivial (isomorphic to the trivial group, containing only the identity arrow), then \mathcal{G} reduces to a disjoint union of equivalence relations on its objects, with at most one arrow between any pair of objects in the same component.^[4] A representative example arises in the transformation groupoid G \ltimes X arising from a group G acting on a set X, where objects are elements of X and arrows are pairs (g, x): x \to g \cdot x for g \in G; here, the vertex group at x \in X is precisely the stabilizer subgroup \{ g \in G \mid g \cdot x = x \}, which inherits the group structure from G.^[13]

Orbits and transitive groupoids

In a groupoid G = (G^0, G^1, s, t, \circ, \mathrm{id}), the orbit of an object x \in G^0 is defined as the set \{ y \in G^0 \mid \exists f \in G^1 \text{ such that } s(f) = x \text{ and } t(f) = y \}, consisting of all objects reachable from x via a morphism.^[14] This relation induces an equivalence relation \sim on G^0, where y \sim z if and only if there exists a morphism f: y \to z, partitioning the objects into equivalence classes that form the orbits.^[14] The orbit space, denoted G^0 / \sim, is the quotient set of these equivalence classes, which corresponds to the set of connected components \pi_0(G).^[14] A groupoid is transitive if it has exactly one orbit, meaning that for every pair of objects x, y \in G^0, the hom-set G(x, y) is nonempty, so there exists at least one morphism from x to y; this is equivalent to the groupoid being connected in the sense of path components.^[4] In a transitive groupoid, all vertex groups (automorphism groups at objects) are isomorphic.^[4] A principal groupoid is one that is equivalent to a transitive groupoid with trivial vertex groups, where the automorphism group at each object consists solely of the identity morphism.^[14] Such groupoids arise, for example, from the action of a group G on a set X that is both free and transitive, yielding a transformation groupoid X \rtimes G with exactly one morphism between any pair of objects in the orbit.^[4] Any groupoid decomposes as a disjoint union of its transitive components, where the components are the maximal transitive subgroupoids, and the set of these components is \pi_0(G).^[4] This decomposition respects the structure of the orbits, with each component corresponding to a single equivalence class under \sim.^[14]

Subgroupoids and quotient groupoids

A subgroupoid of a groupoid G is a subcollection H \subseteq G consisting of subsets H^0 \subseteq G^0 of objects and H^1 \subseteq G^1 of arrows such that H is closed under the source and target maps, composition (where defined), inverses, and identities, thereby forming a groupoid in its own right.^[4] A subgroupoid H is called wide if it includes all objects of G, that is, H^0 = G^0, while possibly having fewer arrows.^[15] In contrast, a full subgroupoid H includes all arrows of G between any pair of its objects, meaning that for every x, y \in H^0, the hom-sets satisfy H(x, y) = G(x, y).^[4] A subgroupoid H of G is normal if it is closed under conjugation by elements of G, specifically, for all g \in G^1 and h \in H^1 with compatible source and target, g h g^{-1} \in H^1.^[15] This condition generalizes the notion of a normal subgroup in group theory and ensures that the conjugation action of G preserves H. Normal subgroupoids are often wide, though the definition does not strictly require it; however, in many constructions, such as quotients, wide normality is assumed to align objects properly.^[4] Given a normal subgroupoid H of G, the quotient groupoid G/H is constructed with objects given by the orbits of G^0 under the action induced by H, denoted G^0 / \sim_H, where two objects x, y \in G^0 are equivalent if there exists h \in H^1 with source h = x and target h = y.^[4] The arrows of G/H are the equivalence classes of arrows in G^1 modulo the H-action, where f \sim_H g if f and g connect the same orbit representatives and differ by left or right multiplication by elements of H. Composition in the quotient is defined by = [f g] whenever f and g are composable in G, with [ \cdot ] denoting the equivalence class; this operation is well-defined due to the normality of H.^[16] The quotient construction yields a groupoid morphism \pi: G \to G/H that is universal among morphisms factoring through H, projecting objects to their orbits and arrows to their classes.^[15] Wide and full subgroupoids play key roles in decompositions; for instance, the maximal transitive subgroupoids of G are full and wide within their components.^[4] Quotient groupoids capture symmetries modulo subgroups, analogous to orbit spaces in group actions, and are essential for studying invariants like homology of groupoids.^[15]

Examples

Groupoids from equivalence relations

Given an equivalence relation \sim on a set X, a groupoid can be constructed with objects the elements of X and morphisms the ordered pairs (x, y) such that x \sim y. The source map s: (x, y) \mapsto x and target map t: (x, y) \mapsto y are the natural projections, the identity morphism at each x \in X is (x, x), composition is defined by (x, y) \circ (y, z) = (x, z) whenever y \sim z, and the inverse of (x, y) is (y, x). This construction satisfies the axioms of a groupoid, with associativity holding whenever compositions are defined and units and inverses behaving as required.^[4] Such a groupoid is thin, meaning that between any two objects there is at most one morphism, as the equivalence relation provides a unique witness (or none) for x \sim y. The isotropy groups (or vertex groups), consisting of automorphisms at each object x, are trivial and contain only the identity (x, x). The connected components of the groupoid are precisely the equivalence classes of \sim, and within each component the groupoid is transitive, with a unique morphism connecting any pair of objects in the same class.^[17]^[4] The total number of morphisms equals \sum_i |C_i|^2, where the sum is over the equivalence classes C_i of \sim and |C_i| denotes the cardinality of C_i; this follows because the morphisms within each class C_i form the complete set C_i \times C_i. These thin groupoids from equivalence relations provide the simplest non-trivial examples beyond discrete groupoids (where \sim is the equality relation).^[4] A natural generalization is the pair groupoid on X, also called the coarse or haul groupoid, obtained by taking the universal equivalence relation where every pair (x, y) \in X \times X is a morphism (i.e., all elements are equivalent). Here, composition is always (x, y) \circ (y, z) = (x, z), yielding a single transitive thin component unless X is empty. This structure captures the full relational framework without partitioning into classes.^[18]^[4]

Fundamental groupoids

The fundamental groupoid of a topological space X, denoted \Pi_1(X) or \pi_1(X), is a groupoid whose objects are the points of X and whose arrows from x to y are the homotopy classes of paths in X from x to y, where a path is a continuous map \gamma: [0,1] \to X with \gamma(0) = x and \gamma(1) = y, and homotopies are required to be relative to the endpoints.^[19] Composition of arrows is given by concatenation of paths: if \alpha is a path from x to z and \beta from z to y, then \beta \cdot \alpha is the path from x to y obtained by traversing \alpha followed by \beta, up to homotopy relative to endpoints; identities are constant paths, and inverses are reparametrizations of paths traversed in reverse.^[19] This construction makes \Pi_1(X) a functor from the category of topological spaces (with continuous maps) to the category of groupoids, preserving path-connected components as orbits.^[19] A key property is that the vertex group (or isotropy group) at any point x \in X is precisely the fundamental group \pi_1(X, x), consisting of homotopy classes of loops based at x.^[19] If X is path-connected, then \Pi_1(X) is transitive (a single orbit) and fully faithful in the sense that the inclusion of the fundamental group at a basepoint captures the local homotopy structure, though the full groupoid encodes global path information between points.^[19] More formally, the set of arrows from x to y can be denoted \rel_{x,y} = \{ homotopy classes of paths from x to y relative to endpoints\}, forming a principal homogeneous space under the action of \pi_1(X, x) on the left and \pi_1(X, y) on the right when x and y are in the same path component.^[19] The concept of the fundamental groupoid emerged in topology as the edge-path groupoid in Kurt Reidemeister's 1932 work on knot theory and combinatorial topology, providing a multi-object framework for path homotopies beyond the single-basepoint fundamental group introduced by Poincaré in 1895.^[19] It was further developed in the mid-20th century, with Charles Ehresmann incorporating groupoids into differential geometry and fibered structures around 1950, and Ronald Brown extending it in 1967 to multiple basepoints \Pi_1(X, A) for a subspace A \subset X, enabling a more flexible Seifert–van Kampen theorem for computing homotopy types.^[19] This generalization positions the fundamental groupoid as a bridge between groups and higher categorical structures in homotopy theory. A representative example is the circle S^1, where the fundamental groupoid \Pi_1(S^1) has objects the points of S^1, a single orbit since S^1 is path-connected, and vertex groups at each point isomorphic to \mathbb{Z}, generated by the homotopy class of the standard loop winding once around the circle.^[19] The arrows between distinct points x, y \in S^1 are in bijection with \mathbb{Z}, corresponding to winding numbers of paths from x to y. This structure illustrates how the groupoid captures both local loop information (via \pi_1) and global connectivity via paths.

Groupoids from group actions

A group acting on a set gives rise to a groupoid known as the transformation groupoid or action groupoid. Given a group G acting on a set X, the transformation groupoid X // G has objects the elements of X, and morphisms from x to y in X given by the elements g \in G such that g \cdot x = y; if no such g exists, the hom-set is empty. Equivalently, the morphisms can be represented as pairs (g, x) with source x and target g \cdot x, and composition defined by (h, y) \circ (g, x) = (h g, x) whenever y = g \cdot x. The identity morphism at x is (e, x), where e is the identity in G. This construction captures the symmetries induced by the group action, with each orbit under the action forming a transitive sub-groupoid.^[20]^[21] In the transformation groupoid X // G, the vertex group (or isotropy group) at an object x \in X is the stabilizer subgroup G_x = \{g \in G \mid g \cdot x = x\}, which acts as the group of automorphisms of x. The orbit of x, denoted G \cdot x = \{g \cdot x \mid g \in G\}, corresponds to the set of objects reachable from x via morphisms, and the groupoid restricted to this orbit is transitive, meaning any two objects in the orbit are connected by a morphism. The size of the orbit is given by the index of the stabilizer: |G \cdot x| = |G / G_x|, which equals the number of distinct targets reachable from x. The total number of morphisms emanating from x is |G|, since each g \in G defines a unique arrow (g, x) to g \cdot x, and for each y \in G \cdot x, the hom-set \mathrm{Hom}(x, y) has size |G_x|.^[20]^[5]^[22] A concrete example arises when G is the symmetric group S_n acting by permutations on the finite set X = \{1, 2, \dots, n\}; the resulting permutation groupoid has objects the points in X and morphisms the permutations mapping one point to another, modeling all possible rearrangements while preserving the set structure. In a geometric context, consider a finite group G acting on an algebraic variety V; the quotient stack [V / G] is represented by the action groupoid V // G, where objects are points in V and morphisms account for G-equivariant identifications, allowing the stack to capture ramified or singular quotients that ordinary varieties cannot. This construction is rigid if and only if the action is free, meaning stabilizers are trivial.^[5]^[23] The inertia groupoid of the action groupoid X // G is the subgroupoid consisting of those morphisms (g, x) where g \cdot x = x, i.e., the fixed points under group elements; its objects are pairs (x, g) with g \in G_x, and morphisms are elements of the centralizer of g in G_x. This subgroupoid encodes the "loops" or fixed-point data at each object, playing a key role in equivariant cohomology and orbifold invariants by summing over conjugacy classes of stabilizers. For instance, in the permutation groupoid example, the inertia captures permutations that fix specific points, reflecting cycle structures.^[24]^[25]

Stacky and geometric groupoids

In geometric contexts, groupoids provide a framework for modeling spaces with singularities, such as orbifolds and stacks, by capturing both the underlying topology and the local symmetries or automorphisms at singular points. Étale groupoids, in particular, are used to represent these objects, where the source and target maps are local diffeomorphisms, allowing for a smooth structure that accommodates quotient phenomena like group actions on manifolds.^[26] An étale groupoid is a smooth groupoid G \rightrightarrows X in which the source map s: G_1 \to X (and similarly the target map t: G_1 \to X) is a local diffeomorphism, ensuring that the structure is compatible with the differential geometry of the base space X. In the algebraic setting, this corresponds to the maps being étale, meaning local isomorphisms in the scheme sense. For orbifolds, which are spaces locally modeled by quotients of manifolds by finite group actions, the associated orbifold groupoid is an effective étale groupoid where the maps (s, t): G_1 \to X \times X are proper submersions.^[27]^[26]^[25] A concrete example arises in the construction of an orbifold groupoid from a manifold with a finite group action. The objects are points in the manifold, while arrows include not only identity maps but also deck transformations arising from the group action, capturing the local symmetries and singularities at fixed points. This groupoid presentation uniquely determines the orbifold structure via its category of equivariant sheaves.^[27]^[27] Stacky groupoids extend this to present Deligne-Mumford stacks, which generalize orbifolds to allow infinite stabilizers in a controlled way. A stacky groupoid is a groupoid object in a suitable category (such as manifolds or schemes) that presents a stack, with source and target maps as surjective projections and multiplication defined up to isomorphism. For instance, the quotient stack [X/G] for a manifold X and Lie group G acting on it is presented by the action groupoid G \ltimes X \rightrightarrows X, where arrows encode both points in X and group elements acting on them, modeling the stack's descent data.^[28]^[28] A groupoid is effective if, for each object x \in X, the induced map from the isotropy group G_x to the diffeomorphism group of a neighborhood U_x is injective, ensuring that stabilizers act faithfully without kernel. Equivalently, the inertia groupoid (comprising arrows with identical source and target) consists only of identity arrows except at points with trivial isotropy; in this case, the groupoid corresponds to a smooth manifold without singularities.^[25]^[24] The foundational work on orbifold groupoids as effective étale structures was developed by Moerdijk and Pronk in 1997, building on earlier notions of V-manifolds and providing a sheaf-theoretic characterization. These constructions have been applied in geometric quantization, where symplectic or étale groupoids model generalized symmetries on phase spaces, enabling quantization that commutes with reduction for Hamiltonian actions via continuous fields of Hilbert spaces.^[27]^[29]

Relation to groups

From groups to groupoids

A group G can be viewed as a groupoid with a single object, often denoted by *, where the set of arrows is the underlying set of G, and the composition of arrows corresponds to the group multiplication in G. The identity arrow is the unit element of G, and every arrow has an inverse given by the group inverse. This construction establishes an equivalence between the category of groups and the category of one-object groupoids, as the forgetful functor from one-object groupoids to groups is an equivalence of categories.^[30]^[31] The delooping of a group G, denoted BG, is the one-object groupoid with a single object * and automorphism group G. This groupoid models the classifying space of G in the sense that its geometric realization yields a space whose homotopy type captures the principal G-bundles up to isomorphism. Specifically, the fundamental groupoid of BG is equivalent to the one-object groupoid associated to G. For the classifying space BG, the zeroth homotopy group is \pi_0(BG) = \{*\}, reflecting its connectedness, and the first homotopy group at the basepoint is \pi_1(BG, *) = G.^[31]^[32] Groups embed into the broader category of groupoids through various constructions, such as action groupoids. For instance, the group G acts on itself by left multiplication, yielding a groupoid whose objects are the elements of G and whose arrows are pairs (g, x) \in G \times G, representing the map sending x to g x. Composition in this groupoid is defined by (g', y) \circ (g, x) = (g' g, x) when y = g x, with identities (e, x) for the unit e \in G. This action groupoid, often denoted G \ltimes G, is transitive and has trivial isotropy groups at each object.^[33] A trivial embedding arises by regarding the underlying set of G as a discrete groupoid, where the objects are the elements of G and the only arrows are the identity morphisms at each object. This construction yields a groupoid equivalent to the discrete category on the set G, with no non-trivial isomorphisms.^[31]

Groupoids as generalizations of groups

A groupoid generalizes the concept of a group by allowing a collection of objects alongside invertible arrows between them, where composition is defined only for compatible pairs of arrows (those where the target of the first matches the source of the second). In contrast, a group consists of a single object with a total multiplication operation on all elements, every element invertible, and an identity element satisfying associativity. Thus, any group may be viewed as a groupoid with one object, where the arrows are the group elements and composition is the group multiplication; conversely, the automorphism groupoid of a single object recovers the group structure. This perspective, originating with Heinrich Brandt's algebraic definition in 1926, extends group theory to structures with multiple components and partial operations while preserving essential properties like invertibility and associativity where defined.^[4] The core properties of groupoids maintain the reversible and associative nature of groups but relax the totality axiom. Every arrow g in a groupoid has a unique inverse g^{-1} such that g \circ g^{-1} = g^{-1} \circ g = \mathrm{id}_{\mathrm{target}(g)}, and composition g \circ h is associative whenever it is defined (i.e., \mathrm{target}(h) = \mathrm{source}(g)). To achieve a total operation akin to groups, one can construct the totalization of a groupoid, often via the holomorph or an associated bundle that adjoins formal compositions for non-composable pairs, effectively embedding the partial structure into a larger group-like entity. This construction highlights how groupoids capture "variable symmetries" across objects, generalizing the fixed symmetry of a single group's action on itself.^[4]^[34] A illustrative example is the pair groupoid on a set X, where the objects are the elements of X and the arrows are all ordered pairs (x, y) \in X \times X, with source \mathrm{source}(x, y) = y, target \mathrm{target}(x, y) = x, identity at x given by (x, x), inverse of (x, y) as (y, x), and composition (x, y) \circ (y, z) = (x, z). This structure exemplifies partial compositions between multiple objects, with trivial automorphisms at each object, allowing direct "transportations" between any pair of elements. Such generalizations prove advantageous in modeling symmetries with domain-dependent composability, as seen in foliation theory, where holonomy groupoids encode leafwise symmetries on manifolds, capturing infinitesimal transformations that vary along different leaves without assuming global uniformity.^[3]^[4]^[35]

Weak equivalences and homotopy

In the context of groupoid homotopy theory, a weak equivalence is a functor F: \mathcal{G} \to \mathcal{H} between groupoids that is fully faithful and essentially surjective on objects. Fully faithful means that for every pair of objects x, y in \mathcal{G}, the induced map on hom-sets \mathcal{G}(x, y) \to \mathcal{H}(F(x), F(y)) is a bijection, while essentially surjective means that every object in \mathcal{H} is isomorphic to the image of some object under F. Such functors induce isomorphisms on the associated homotopy categories, where one localizes at the weak equivalences to obtain a category up to homotopy. This notion equips the category of groupoids with a model category structure, where weak equivalences serve as the morphisms that preserve homotopical information, fibrations are isofibrations (functors that lift isomorphisms), and cofibrations are monomorphisms. Homotopy between functors of groupoids is captured by natural transformations, which provide a 2-categorical structure on the category of groupoids. Specifically, two functors F, G: \mathcal{C} \to \mathcal{D} between groupoids are homotopic if there exists a natural transformation \eta: F \Rightarrow G, consisting of components \eta_x: F(x) \to G(x) that are compatible with the morphisms in \mathcal{C}. Since all morphisms in groupoids are isomorphisms, such natural transformations are invertible, aligning the 2-category of groupoids (with functors as 1-morphisms and natural transformations as 2-morphisms) closely with homotopy notions in higher category theory. This framework inverts weak equivalences to form the homotopy 2-category of groupoids, where homotopies mediate between equivalent representations of the same homotopical data. The basic homotopy invariants of a groupoid \mathcal{G} are given by its fundamental "homotopy groups": the zeroth homotopy set \pi_0(\mathcal{G}) is the set of isomorphism classes of objects in \mathcal{G}, capturing the connected components, while for a fixed object x \in \mathcal{G}, the first homotopy group \pi_1(\mathcal{G}, x) is the vertex group (or automorphism group) \mathcal{G}(x, x), which is a group under composition. These coincide with the homotopy groups of the classifying space of \mathcal{G}, up to the basepoint corresponding to x. A key tool for realizing the homotopy type of a groupoid is the nerve construction, which associates to a groupoid \mathcal{G} (with object set \mathcal{G}_0 and morphism set \mathcal{G}_1) a simplicial set N\mathcal{G} whose n-simplices are chains of n composable morphisms g_1: x_0 \to x_1, g_2: x_1 \to x_2, ..., g_n: x_{n-1} \to x_n in \mathcal{G}.^[36] Face maps compose adjacent morphisms or drop endpoints, while degeneracy maps insert identity morphisms.^[36] The geometric realization |N\mathcal{G}| serves as the classifying space of \mathcal{G}, a topological space whose homotopy type encodes the structure of \mathcal{G}; in particular, \pi_0(|N\mathcal{G}|) \cong \pi_0(\mathcal{G}) and \pi_1(|N\mathcal{G}|, ) \cong \pi_1(\mathcal{G}, x) for x \in \mathcal{G}_0.^[36] This construction models the weak homotopy type of groupoids as Kan complexes, with higher horns filling uniquely due to invertibility of morphisms.^[36] The homotopy-theoretic perspective on groupoids was pioneered by Graeme Segal in his 1968 work, where he extended the nerve construction from groups to general categories, showing that the classifying space captures cohomology theories and spectral sequences associated to the homotopy type. Segal's approach established groupoids as models for 1-types in homotopy theory, bridging algebraic and topological structures.

Category of groupoids

Composition and functors

In category theory, the category Grpd (also denoted Gpd) has groupoids as objects and functors between groupoids as morphisms.^[31] Composition in Grpd is defined by the usual composition of functors: for functors F: \mathcal{G} \to \mathcal{H} and G: \mathcal{H} \to \mathcal{K}, the composite G \circ F: \mathcal{G} \to \mathcal{K} maps objects and arrows via (G \circ F)_0 = G_0 \circ F_0 and (G \circ F)_1 = G_1 \circ F_1, preserving the groupoid structure as required.^[31] A functor F: \mathcal{G} \to \mathcal{H} between groupoids \mathcal{G} and \mathcal{H} consists of maps F_0: \mathcal{G}^0 \to \mathcal{H}^0 on objects and F_1: \mathcal{G}^1 \to \mathcal{H}^1 on arrows such that it preserves sources and targets: \mathrm{t} \circ F_1 = F_0 \circ \mathrm{t} and \mathrm{s} \circ F_1 = F_0 \circ \mathrm{s}, where \mathrm{s}, \mathrm{t} denote source and target; preserves composition: F_1 \circ m_{\mathcal{G}} = m_{\mathcal{H}} \circ (F_1 \times_{F_0} F_1), with m the multiplication and \times_{F_0} the pullback over F_0; and preserves identities: F_1(\eta_{\mathcal{G}}) = \eta_{\mathcal{H}} \circ F_0, where \eta denotes the unit.^[31] Such functors are strict in the sense that they preserve the groupoid operations exactly as defined, without weakening or homotopy.^[31] An isomorphism of groupoids is a functor F: \mathcal{G} \to \mathcal{H} that admits an inverse functor G: \mathcal{H} \to \mathcal{G} satisfying G \circ F = \mathrm{[id](/page/id)}_{\mathcal{G}} and F \circ G = \mathrm{[id](/page/id)}_{\mathcal{H}}, which implies that F is bijective on both objects and arrows.^[31] A functor F: \mathcal{G} \to \mathcal{H} is an equivalence of groupoids if it is fully faithful—meaning the induced map \mathrm{Hom}_{\mathcal{G}}(x,y) \to \mathrm{Hom}_{\mathcal{H}}(F x, F y) is bijective for all objects x,y in \mathcal{G}—and essentially surjective, i.e., every object in \mathcal{H} is isomorphic to the image under F of some object in \mathcal{G}.^[31] This notion aligns with the general definition of categorical equivalence, where such functors induce isomorphisms on homotopy categories or classify structures up to weak isomorphism.^[31] A representative example is the forgetful functor U: \mathbf{Grpd} \to \mathbf{Cat} that sends a groupoid to its underlying category, forgetting the property that all morphisms are isomorphisms while preserving the category structure.^[31]

Natural transformations and 2-categories

A natural transformation \eta: F \Rightarrow G between two functors F, G: \mathcal{C} \to \mathcal{D} consists of a family of morphisms \{\eta_x: F(x) \to G(x)\}_{x \in \mathrm{Ob}(\mathcal{C})} in \mathcal{D}, one for each object x of \mathcal{C}, such that for every morphism f: x \to y in \mathcal{C}, the following diagram commutes:

\begin{CD} F(x) @>{\eta_x}>> G(x) \\ @V{F(f)}VV @VV{G(f)}V \\ F(y) @>>{\eta_y}> G(y) \end{CD}

This naturality condition ensures that \eta respects the structure of the functors.^[37] In the context of groupoids, where \mathcal{D} is a groupoid and thus all morphisms in \mathcal{D} are isomorphisms, each component \eta_x is automatically an isomorphism, making every natural transformation between groupoid functors a vertical isomorphism.^[38] This property distinguishes the 2-categorical structure of groupoids from that of general categories.^[4] The category of groupoids, denoted \mathbf{Grpd}, extends naturally to a strict 2-category, where the 0-cells are groupoids, the 1-cells are functors between groupoids, and the 2-cells are natural transformations between such functors.^[39] Composition in \mathbf{Grpd} proceeds in two ways: vertical composition of 2-cells (natural transformations) is performed pointwise in the target groupoid, while horizontal composition combines a functor with a natural transformation via whiskering. For functors L: \mathcal{A} \to \mathcal{B}, F, G: \mathcal{B} \to \mathcal{C}, and a natural transformation \eta: F \Rightarrow G, the left whiskering L * \eta: F \circ L \Rightarrow G \circ L has components (L * \eta)_x = \eta_{L(x)}, and the right whiskering \eta * R: R \circ F \Rightarrow R \circ G for R: \mathcal{C} \to \mathcal{D} has components (\eta * R)_y = R(\eta_y). These operations ensure associativity and unitality hold strictly in \mathbf{Grpd}.^[39] A natural transformation \eta: F \Rightarrow G in \mathbf{Grpd} is invertible if and only if each component \eta_x is an isomorphism in the target groupoid; however, since all morphisms in groupoids are isomorphisms, every natural transformation is invertible, reinforcing the 2-categorical coherence.^[38] This strict 2-categorical structure allows \mathbf{Grpd} to model higher-dimensional categorical phenomena effectively, with horizontal and vertical compositions aligning precisely without coherence isomorphisms.^[39]

Relation to categories and simplicial sets

Groupoids are closely related to the broader framework of categories, where the category of groupoids, denoted Grpd, embeds as a full subcategory of the category of small categories, Cat. Specifically, Grpd consists precisely of those objects in Cat for which every morphism is an isomorphism, with morphisms in Grpd being the functors between such categories and 2-morphisms being the natural isomorphisms. This embedding is full and faithful, preserving the structure of invertible morphisms inherent to groupoids.^[40] Equivalently, Grpd is the full subcategory of Cat on all groupoids, highlighting its role as a reflective and coreflective subcategory. For any small category C, the core core(C)—defined as the subcategory retaining all objects of C but restricting morphisms to only the isomorphisms—yields a groupoid, serving as the right adjoint to the inclusion functor Grpd → Cat. This coreflectivity underscores how groupoids capture the invertible aspects of arbitrary categories. Complementarily, Grpd is reflective in Cat via the groupoid completion functor, which is the left adjoint to the inclusion and formally inverts all morphisms in a given category C to produce a groupoid where every arrow becomes an isomorphism.^[41] A key connection to simplicial sets arises through the nerve functor N: Grpd → sSet, which assigns to each groupoid G a simplicial set encoding its structure. The n-simplices N(G)_n form the set of all n-tuples of composable morphisms in G, i.e.,

N(G)_n = \coprod_{(x_0, \dots, x_n) \in (\Ob(G))^{n+1}} \prod_{i=1}^n G(x_{i-1}, x_i),

where the face maps compose adjacent morphisms or omit vertices, and degeneracy maps insert identity morphisms. The geometric realization |NG| of this nerve is the classifying space of G, a topological space whose homotopy type reflects the weak equivalence class of G. This construction extends naturally to the nerve functor on Cat, but restricts particularly well-behaved on Grpd.^[36] The nerve functor N preserves weak equivalences in Grpd, where these are the equivalences of categories (functors that are essentially surjective on objects and fully faithful on hom-sets). Moreover, if f: G → H is a weak equivalence of groupoids, then the induced map |Nf|: |NG| → |NH| is a homotopy equivalence of topological spaces, ensuring that the classifying spaces capture the homotopy theory of groupoids up to equivalence. In contrast, while the nerve on Cat detects homotopy types via geometric realization, the coreflective structure via groupoid completion allows Grpd to serve as a homotopy-theoretic completion of Cat within sSet.^[36]^[42] These relations have deep 2-categorical underpinnings, as explored in early work on 2-categories where groupoids feature prominently as (2,1)-categories with invertible 2-morphisms. Ross Street's 1972 development of the formal theory of monads in 2-categories laid foundational aspects for understanding such embeddings and functors in higher-dimensional settings.

Applications

In algebraic topology

In algebraic topology, groupoids provide a framework for encoding pathwise information in topological spaces, extending the role of the fundamental group by capturing homotopy classes of paths between arbitrary points rather than just loops at a fixed basepoint. The fundamental groupoid \Pi_1(X) of a topological space X has as objects the points of X, with morphisms given by homotopy classes of paths connecting those points, where homotopies preserve endpoints.^[31] This structure detects higher connectivity properties than the fundamental group \pi_1(X); for instance, \Pi_1(X) is weakly equivalent to the discrete groupoid on the path components \pi_0(X) if and only if every path component of X is simply connected (i.e., \pi_1(X,x) = 0 for all x \in X), revealing global simply connectedness (vanishing \pi_1) beyond local loop information.^[43] A key application arises in covering space theory, where for a covering map p: Y \to X, the induced functor p_*: \Pi_1(Y) \to \Pi_1(X) is a covering morphism of groupoids, meaning it satisfies a unique path-lifting property relative to the deck transformation groupoid acting on paths in Y.^[44] The covering groupoid thus models the monodromy of paths in X via lifts to Y, with objects points in X and morphisms equivalence classes of lifted paths in Y modulo deck transformations, providing a groupoidal analogue to the classical Galois correspondence between covers and subgroups of \pi_1(X).^[45] If p is a homotopy equivalence, then \Pi_1(X) \simeq \Pi_1(Y) as groupoids, preserving the full path homotopy type of the space.^[31] Groupoids also feature prominently in Postnikov towers, which decompose the homotopy type of a space X into stages approximating its higher homotopy groups. The k-th stage of the tower is modeled by a k-groupoid, a higher categorical structure where 1-morphisms are paths, 2-morphisms are homotopies between paths, and so on up to k-dimensional cells, allowing the tower to encode the action of lower homotopy groups on higher ones via groupoid fibrations.^[43] This approach, developed through higher-dimensional groupoid models, facilitates computations of homotopy invariants by successively killing higher homotopy groups while tracking the resulting groupoidal data. In foliation theory, the holonomy groupoid of a foliated manifold encodes the transverse geometry by associating to each leafwise path the induced germ of holonomy transformation on transverse sections, where the morphism space consists of equivalence classes of leafwise paths identified if they induce the same transverse holonomy. This structure computes transverse invariants, such as holonomy groups that are images of leafwise fundamental groups in transverse automorphisms. The leafwise fundamental groupoid, consisting of homotopy classes of paths within leaves (also called the homotopy groupoid in some contexts), refines this by not quotienting by transverse equivalence, enabling the study of both intrinsic leaf topology and the foliation's transverse structure through its action on leaf paths. Groupoids further apply to stratified spaces, where proper Lie groupoids induce a canonical stratification on orbit spaces, modeling the homotopy type of singular strata via groupoid cohomology and path spaces.^[46] For instance, in the orbit space of a proper groupoid action, the strata correspond to connected components of isotropy groupoids, allowing computation of homotopy invariants that detect the space's decomposition into smooth and singular layers.^[47]

In algebraic geometry

In algebraic geometry, groupoids serve as a fundamental tool for constructing quotient stacks that capture geometric quotients while accounting for stabilizer data. For a group scheme G acting on a scheme X, the quotient stack [X/G] is presented by the action groupoid G \ltimes X, whose objects are points of X and whose arrows are pairs (x, g) with source x and target g \cdot x. This presentation ensures that the stack parametrizes G-torsors over test schemes, providing a fine moduli space for equivariant objects even when the coarse quotient X/G exhibits singularities due to non-free actions.^[48]^[49] Čech groupoids further illustrate the role of groupoids in descent theory and cohomology computations. Given an open cover \{U_i\}_{i \in I} of a scheme X, the associated Čech groupoid has objects the disjoint union \coprod_{i \in I} U_i and arrows consisting of elements of \coprod_{i,j \in I} U_i \times_X U_j, with source and target maps given by the two projections. The nerve of this groupoid computes the Čech cohomology of sheaves on X, modeling descent data for gluing local objects into global ones. In particular, the first Čech cohomology group \check{H}^1(X, G) classifies isomorphism classes of G-torsors over X. G-gerbes—stacky groupoids that are locally nonempty and locally connected, banded by a group related to the abelianization of G—are instead classified by the second cohomology group \check{H}^2(X, \hat{G}).^[50]^[51] Deligne-Mumford stacks, which parametrize families of objects with finite stabilizers, admit presentations by étale groupoids. Specifically, such a stack \mathcal{X} is isomorphic to [R \rightrightarrows U], where U \to \mathcal{X} is an étale surjective morphism from a scheme U, and R = U \times_{\mathcal{X}} U forms an étale groupoid scheme over U with quasi-compact relative diagonal. This structure ensures that \mathcal{X} has a representable diagonal and is algebraic, allowing resolution of singularities via étale atlases. A prominent example is the moduli stack \overline{\mathcal{M}}_{g,n} of stable curves of genus g with n marked points, presented as the stack whose objects are stable n-pointed curves and whose arrows are isomorphisms preserving marked points; this Deligne-Mumford stack has dimension $3g - 3 + n and coarse moduli space \overline{M}_{g,n}.^[49]^[52] Recent developments in derived algebraic geometry, as formalized by Toën and Vezzosi, extend these notions by incorporating \infty-groupoids to handle homotopical phenomena in stacks. Derived Artin stacks are defined as geometric realizations of smooth \infty-groupoids in derived schemes, enabling the study of derived moduli problems such as deformations of singular varieties or perfect complexes. For instance, the derived stack of perfect complexes on a scheme incorporates higher homotopy groups via \infty-gerbes, resolving obstructions in classical stack presentations and providing a framework for motivic and homotopical invariants.^[53]

In higher category theory

In higher category theory, groupoids generalize to n-groupoids through a process of categorification, where the structure is built recursively: an n-groupoid consists of objects, whose 1-morphisms form an (n-1)-groupoid, and higher-dimensional morphisms up to dimension n are all required to be invertible, with composition satisfying appropriate coherence conditions.^[54] This recursive definition captures weak equivalences and higher-dimensional invertibility, extending the ordinary groupoid (n=1) to model more complex categorical structures.^[55] The notion culminates in ∞-groupoids, which are weak n-groupoids for all finite n, providing algebraic models for homotopy types of topological spaces.^[56] Kan complexes in the category of simplicial sets serve as a primary model for ∞-groupoids, where the simplicial structure encodes higher homotopies, and the homotopy hypothesis equates the homotopy category of spaces with the ∞-category of ∞-groupoids up to weak equivalence.^[56] For a simplicial set K, its 1-truncation \tau_{\leq 1} K is the groupoid whose objects are the path components \pi_0(K) and whose automorphism groups at each object x are given by the fundamental groups \pi_1(K, x), effectively collapsing higher homotopy information while preserving the 1-categorical structure.^[56] In the framework of ∞-categories developed by Lurie, groupoids appear as 1-truncated objects, where the ∞-category of ∞-groupoids is the maximal ∞-groupoid within the ∞-topos of spaces, and truncations provide a way to extract lower-dimensional invariants.^[56] John Baez has applied higher groupoids in higher gauge theory, where 2-groupoids model 2-connections on principal 2-bundles, generalizing Yang-Mills theory to incorporate string-like extended objects and their parallel transport.^[57] A canonical example is the fundamental ∞-groupoid of a topological space X, denoted \Pi_\infty(X), whose objects are points of X, 1-morphisms are paths, 2-morphisms are homotopies between paths, and higher cells represent higher homotopies, fully capturing the homotopy type of X.^[54] More recent models for ∞-groupoids include Rezk's complete Segal spaces, introduced in the early 2000s as bisimplicial sets satisfying Segal conditions and completeness, which provide a combinatorial framework equivalent to Kan complexes for presenting homotopy types and enable explicit constructions in homotopy theory.^[58] These models facilitate the study of ∞-categories enriched over ∞-groupoids, bridging algebraic and geometric perspectives on higher structures.

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Base-free interpretation of the fundamental group via groupoids
Dec 12, 2015 · This means that the fundamental groupoid contains all information about the fundamental group, as well as the space itself. It turns out that ...
[46]
covering spaces and the fundamental groupoid
Apr 22, 2012 · A covering map of spaces induces a covering morphism of fundamental groupoids. The category of covering morphisms of a groupoid G is equivalent ...Equivalence Between Covering Groupoids and Covering SpacesHow do you construct the lifted topology of a groupoid cover?More results from math.stackexchange.com
[47]
Geometry of orbit spaces of proper Lie groupoids - NASA/ADS
First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung.
[48]
Orbispaces as differentiable stratified spaces
Nov 1, 2017 · We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid.
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Lemma 94.15.2 (04Y5)—The Stacks project
### Summary of Quotient Stacks [X/G] via Action Groupoids in Algebraic Geometry
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[PDF] Deligne–Mumford stacks
These are known now as Deligne–Mumford stacks. They are all isomorphic to stacks of the form. [R ⇉ U], where R ⇉ U is an groupoid scheme with étale structure ...
[51]
Čech groupoid in nLab
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Section 101.28 (06QB): Gerbes—The Stacks project
We say an algebraic stack \mathcal{X} is a gerbe if there exists a morphism \mathcal{X} \to X where X is an algebraic space which turns \mathcal{X} into a gerbe ...
[53]
Section 99.15 (0D4Y): The stack of curves—The Stacks project
In this section we prove the stack of curves is algebraic. For a further discussion of moduli of curves, we refer the reader to Moduli of Curves, Section 109.1.
[54]
[PDF] Derived algebraic geometry
Derived algebraic geometry is an extension of algebraic geometry whose main purpose is to propose a setting to treat geometrically special situations ...
[55]
[PDF] An Introduction to n-Categories John C. Baez Department of ...
May 13, 1997 · We discuss applications of weak n-categories to various subjects including homotopy theory and topological quantum field theory, and review the ...
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[PDF] LECTURES ON n-CATEGORIES AND COHOMOLOGY
n-Groupoids are the same as homotopy n-types. Here 'n-groupoids' means weak n-groupoids, in which everything is invertible up to higher-level morphisms, in the ...
[57]
[PDF] Higher topos theory / Jacob Lurie
One goal of algebraic topology is to study the topology of X by means of algebraic invariants, such as the singular cohomology groups Hn(X; G) of X with co-.
[58]
[hep-th/0412325] Higher Gauge Theory: 2-Connections on 2-Bundles
Dec 30, 2004 · We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold.
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[PDF] Charles Rezk - A MODEL FOR THE HOMOTOPY THEORY OF ...
Complete Segal spaces arise naturally in situations where one can do homotopy theory. Any category gives rise to a complete Segal space by means of a ...