HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
and
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
in several equivalent ways. A groupoid can be seen as a: * ''
Group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
'' with a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
replacing the
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
; * ''
Category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
'' in which every
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
is invertible. A category of this sort can be viewed as augmented with a
unary operation In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to ''binary operations'', which use two operands. An example is any function , where is a set; the function is a unary operation ...
on the morphisms, called ''inverse'' by analogy with
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. A groupoid where there is only one object is a usual group. In the presence of
dependent typing In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers lik ...
, a category in general can be viewed as a typed
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that . Special cases include: * ''
Setoid In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof theory and in type-theoretic foun ...
s'': sets that come with an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
, * ''
G-set In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under func ...
s'': sets equipped with an
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of a group . Groupoids are often used to reason about
geometrical Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
objects such as
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s. introduced groupoids implicitly via
Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let '' ...
s.


Definitions


Algebraic

A groupoid can be viewed as an algebraic structure consisting of a set with a binary
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
. Precisely, it is a non-empty set G with a
unary operation In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to ''binary operations'', which use two operands. An example is any function , where is a set; the function is a unary operation ...
, and a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
. Here * is not a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
because it is not necessarily defined for all pairs of elements of . The precise conditions under which * is defined are not articulated here and vary by situation. The operations \ast and −1 have the following axiomatic properties: For all , , and c in , # ''
Associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
'': If a*b and b*c are defined, then (a * b) * c and a * (b * c) are defined and are equal. Conversely, if one of (a * b) * c or a * (b * c) is defined, then they are both defined (and they are equal to each other), and a*b and b * c are also defined. # ''
Inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the ...
'': a^ * a and a* are always defined. # ''
Identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
'': If a * b is defined, then , and . (The previous two axioms already show that these expressions are defined and unambiguous.) Two easy and convenient properties follow from these axioms: * , * If a * b is defined, then .


Category-theoretic

A groupoid is a
small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
in which every
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
, i.e., invertible. More explicitly, a groupoid G is a set G_0 of ''objects'' with * for each pair of objects ''x'' and ''y'', a (possibly empty) set ''G''(''x'',''y'') of ''morphisms'' (or ''arrows'') from ''x'' to ''y''; we write ''f'' : ''x'' → ''y'' to indicate that ''f'' is an element of ''G''(''x'',''y''); * for every object ''x'', a designated element \mathrm_x of ''G''(''x'', ''x''); * for each triple of objects ''x'', ''y'', and ''z'', a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
; * for each pair of objects ''x'', ''y'', a function \mathrm: G(x, y) \rightarrow G(y, x): f \mapsto f^ satisfying, for any ''f'' : ''x'' → ''y'', ''g'' : ''y'' → ''z'', and ''h'' : ''z'' → ''w'': ** and ; ** ; ** f f^ = \mathrm_y and . If ''f'' is an element of ''G''(''x'',''y''), then ''x'' is called the source of ''f'', written ''s''(''f''), and ''y'' is called the target of ''f'', written ''t''(''f''). A groupoid ''G'' is sometimes denoted as , where G_1 is the set of all morphisms, and the two arrows G_1 \to G_0 represent the source and the target. More generally, one can consider a
groupoid object In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Defini ...
in an arbitrary category admitting finite fiber products.


Comparing the definitions

The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let ''G'' be the
disjoint union In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of all of the sets ''G''(''x'',''y'') (i.e. the sets of morphisms from ''x'' to ''y''). Then \mathrm and \mathrm become partial operations on ''G'', and \mathrm will in fact be defined everywhere. We define ∗ to be \mathrm and −1 to be , which gives a groupoid in the algebraic sense. Explicit reference to ''G''0 (and hence to ) can be dropped. Conversely, given a groupoid ''G'' in the algebraic sense, define an equivalence relation \sim on its elements by a \sim b iff ''a'' ∗ ''a''−1 = ''b'' ∗ ''b''−1. Let ''G''0 be the set of equivalence classes of , i.e. . Denote ''a'' ∗ ''a''−1 by 1_x if a\in G with . Now define G(x, y) as the set of all elements ''f'' such that 1_x*f*1_y exists. Given f \in G(x,y) and , their composite is defined as . To see that this is well defined, observe that since (1_x*f)*1_y and 1_y*(g*1_z) exist, so does . The identity morphism on ''x'' is then , and the category-theoretic inverse of ''f'' is ''f''−1. Sets in the definitions above may be replaced with
class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...
es, as is generally the case in category theory.


Vertex groups and orbits

Given a groupoid ''G'', the vertex groups or isotropy groups or object groups in ''G'' are the subsets of the form ''G''(''x'',''x''), where ''x'' is any object of ''G''. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group. The orbit of a groupoid ''G'' at a point x \in X is given by the set s(t^(x)) \subseteq X containing every point that can be joined to x by a morphism in G. If two points x and y are in the same orbits, their vertex groups G(x) and G(y) are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
: if f is any morphism from x to , then the isomorphism is given by the mapping . Orbits form a partition of the set X, and a groupoid is called transitive if it has only one orbit (equivalently, if it is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...
for counterexamples).


Subgroupoids and morphisms

A subgroupoid of G \rightrightarrows X is a
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
H \rightrightarrows Y that is itself a groupoid. It is called wide or full if it is
wide WIDE or Wide may refer to: * Wide (cricket), a type of illegal delivery to a batter *Wide and narrow data Wide and narrow (sometimes un-stacked and stacked, or wide and tall) are terms used to describe two different presentations for tabular data ...
or full as a subcategory, i.e., respectively, if X = Y or G(x,y)=H(x,y) for every . A groupoid morphism is simply a functor between two (category-theoretic) groupoids. Particular kinds of morphisms of groupoids are of interest. A morphism p: E \to B of groupoids is called a
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all ma ...
if for each object x of E and each morphism b of B starting at p(x) there is a morphism e of E starting at x such that . A fibration is called a covering morphism or covering of groupoids if further such an e is unique. The covering morphisms of groupoids are especially useful because they can be used to model
covering map In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms ...
s of spaces. It is also true that the category of covering morphisms of a given groupoid B is equivalent to the category of actions of the groupoid B on sets.


Examples


Fundamental groupoid

Given a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, let G_0 be the set . The morphisms from the point p to the point q are
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desir ...
s from p to , with two paths being equivalent if they are
homotopic In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is
associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
. This groupoid is called the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
of , denoted \pi_1(X) (or sometimes, ). The usual fundamental group \pi_1(X,x) is then the vertex group for the point . The orbits of the fundamental groupoid \pi_1(X) are the path-connected components of . Accordingly, the fundamental groupoid of a
path-connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equiva ...
as categories (see the section
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...
for the general theory). An important extension of this idea is to consider the fundamental groupoid \pi_1(X,A) where A\subset X is a chosen set of "base points". Here \pi_1(X,A) is a (full) subgroupoid of , where one considers only paths whose endpoints belong to . The set A may be chosen according to the geometry of the situation at hand.


Equivalence relation

If X is a
setoid In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof theory and in type-theoretic foun ...
, i.e. a set with an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
, then a groupoid "representing" this equivalence relation can be formed as follows: * The objects of the groupoid are the elements of ; *For any two elements x and y in , there is a single morphism from x to y (denote by ) if and only if ; *The composition of (z,y) and (y,x) is . The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples: * If every element of X is in relation with every other element of , we obtain the pair groupoid of , which has the entire X \times X as set of arrows, and which is transitive. * If every element of X is only in relation with itself, one obtains the unit groupoid, which has X as set of arrows, , and which is completely intransitive (every singleton \ is an orbit).


Examples

* If f: X_0 \to Y is a smooth
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
submersion Submersion may refer to: *Being or going underwater, as via submarine, underwater diving, or scuba diving *Submersion (coastal management), the sustainable cyclic portion of foreshore erosion *Submersion (mathematics) * Submersion (Stargate Atlanti ...
of
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topology ...
, then X_0\times_YX_0 \subset X_0\times X_0 is an equivalence relation since Y has a topology isomorphic to the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
of X_0 under the surjective map of topological spaces. If we write, X_1 = X_0\times_YX_0 then we get a groupoid X_1 \rightrightarrows X_0, which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds. * If we relax the reflexivity requirement and consider ''partial equivalence relations'', then it becomes possible to consider
semidecidable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called ''PER models''. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the
effective topos In mathematics, the effective topos introduced by captures the mathematical idea of effectivity within the category theoretical framework. Definition Preliminaries Kleene realizability The topos is based on the partial combinatory algebra give ...
introduced by
Martin Hyland (John) Martin Elliott Hyland is professor of mathematical logic at the University of Cambridge and a fellow of King's College, Cambridge. His interests include mathematical logic, category theory, and theoretical computer science. Education Hy ...
.


ÄŒech groupoid

A ÄŒech groupoidp. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover \mathcal = \_ of some manifold . Its objects are given by the disjoint union \mathcal_0 = \coprod U_i , and its arrows are the intersections \mathcal_1 = \coprod U_ . The source and target maps are then given by the induced maps
\begin s = \phi_j: U_ \to U_j\\ t = \phi_i: U_ \to U_i \end
and the inclusion map
\varepsilon: U_i \to U_
giving the structure of a groupoid. In fact, this can be further extended by setting
\mathcal_n = \mathcal_1\times_ \cdots \times_\mathcal_1
as the n-iterated fiber product where the \mathcal_n represents n-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since
\begin U_ & \to & U_ \\ \downarrow & & \downarrow \\ U_ & \to & U_ \end
is a cartesian diagram where the maps to U_i are the target maps. This construction can be seen as a model for some
∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standa ...
s. Also, another artifact of this construction is k-cocycles
sigma Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator ...
\in \check^k(\mathcal,\underline)
for some constant
sheaf of abelian groups In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ...
can be represented as a function
\sigma:\coprod U_ \to A
giving an explicit representation of cohomology classes.


Group action

If the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
G acts on the set , then we can form the
action groupoid In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action :X \times G \to X, we get the groupoid \mathcal (= a category whose morphisms are all invertible) where ...
(or transformation groupoid) representing this
group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
as follows: * The objects are the elements of ; * For any two elements x and y in , the
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s from x to y correspond to the elements g of G such that ; *
Composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
of morphisms interprets the
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
of . More explicitly, the ''action groupoid'' is a small category with \mathrm(C)=X and \mathrm(C)=G\times X and with source and target maps s(g,x) = x and . It is often denoted G \ltimes X (or X\rtimes G for a right action). Multiplication (or composition) in the groupoid is then , which is defined provided . For x in , the vertex group consists of those (g,x) with , which is just the isotropy subgroup at x for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
of the group action, and the groupoid is transitive if and only if the group action is transitive. Another way to describe G-sets is the
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
, where \mathrm is the groupoid (category) with one element and
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the group . Indeed, every functor F of this category defines a set X=F(\mathrm) and for every g in G (i.e. for every morphism in ) induces a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
F_g : . The categorical structure of the functor F assures us that F defines a G-action on the set . The (unique)
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
F : \mathrm \to \mathrm is the Cayley representation of . In fact, this functor is isomorphic to \mathrm(\mathrm,-) and so sends \mathrm(\mathrm) to the set \mathrm(\mathrm,\mathrm) which is by definition the "set" G and the morphism g of \mathrm (i.e. the element g of ) to the permutation F_g of the set . We deduce from the
Yoneda embedding In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
that the group G is isomorphic to the group , a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of the group of
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s of .


Finite set

Consider the group action of \mathbb/2 on the finite set X = \ where 1 acts by taking each number to its negative, so -2 \mapsto 2 and . The quotient groupoid /G/math> is the set of equivalence classes from this group action , and /math> has a group action of \mathbb/2 on it.


Quotient variety

Any finite group G that maps to GL(n) gives a group action on the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
\mathbb^n (since this is the group of automorphisms). Then, a quotient groupoid can be of the form , which has one point with stabilizer G at the origin. Examples like these form the basis for the theory of
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...
s. Another commonly studied family of orbifolds are
weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''. Prope ...
s \mathbb(n_1,\ldots, n_k) and subspaces of them, such as Calabi–Yau orbifolds.


Inertia groupoid

The inertia groupoid of a groupoid is roughly a groupoid of loops in the given groupoid.


Fiber product of groupoids

Given a diagram of groupoids with groupoid morphisms : \begin & & X \\ & & \downarrow \\ Y &\rightarrow & Z \end where f:X\to Z and , we can form the groupoid X\times_ZY whose objects are triples , where , , and \phi: f(x) \to g(y) in . Morphisms can be defined as a pair of morphisms (\alpha,\beta) where \alpha: x \to x' and \beta: y \to y' such that for triples , there is a commutative diagram in Z of , g(\beta):g(y) \to g(y') and the .


Homological algebra

A two term complex : C_1 ~\overset~ C_0 of objects in a
concrete Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
Abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...
can be used to form a groupoid. It has as objects the set C_0 and as arrows the set ; the source morphism is just the projection onto C_0 while the target morphism is the addition of projection onto C_1 composed with d and projection onto . That is, given , we have : t(c_1 + c_0) = d(c_1) + c_0. Of course, if the abelian category is the category of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on a scheme, then this construction can be used to form a
presheaf In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ...
of groupoids.


Puzzles

While puzzles such as the Rubik's Cube can be modeled using group theory (see
Rubik's Cube group The Rubik's Cube group (G, \cdot ) represents the mathematical structure of the Rubik's Cube mechanical puzzle. Each element of the set G corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this ...
), certain puzzles are better modeled as groupoids. The transformations of the
fifteen puzzle The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and more) is a sliding puzzle. It has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 tile positions wide, with one unoccupied pos ...
form a groupoid (not a group, as not all moves can be composed). This groupoid acts on configurations.


Mathieu groupoid

The
Mathieu groupoid In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced by and studied in detail by . Construction The projective plane of order 3 has 13 point ...
is a groupoid introduced by
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...
acting on 13 points such that the elements fixing a point form a copy of the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
M12.


Relation to groups

If a groupoid has only one object, then the set of its morphisms forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
generalize to groupoids, with the notion of
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
replacing that of
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
. Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) . By transitivity, there will only be one
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
under the action. Note that the isomorphism just mentioned is not unique, and there is no
natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object , a
group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the ...
h from G(x_0) to , and for each x other than , a morphism in G from x_0 to . If a groupoid is not transitive, then it is isomorphic to a
disjoint union In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of groupoids of the above type, also called its connected components (possibly with different groups G and sets X for each connected component). In category-theoretic terms, each connected component of a groupoid is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equiva ...
(but not
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the ''multiplicity'' of ...
of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets , but only the groups . For example, *The fundamental groupoid of X is equivalent to the collection of the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
s of each
path-connected component In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties th ...
of , but an isomorphism requires specifying the set of points in each component; *The set X with the equivalence relation \sim is equivalent (as a groupoid) to one copy of the
trivial group In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usu ...
for each
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
, but an isomorphism requires specifying what each equivalence class is; *The set X equipped with an
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of the group G is equivalent (as a groupoid) to one copy of G for each
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
of the action, but an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
requires specifying what set each orbit is. The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not
natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each G(x) in terms of a single group, and this choice can be arbitrary. In the example from
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, one would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component. As a more illuminating example, the classification of groupoids with one
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s with one endomorphism is nontrivial. Morphisms of groupoids come in more kinds than those of groups: we have, for example,
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all ma ...
s, covering morphisms,
universal morphism In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
s, and quotient morphisms. Thus a subgroup H of a group G yields an action of G on the set of
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of H in G and hence a covering morphism p from, say, K to , where K is a groupoid with vertex groups isomorphic to . In this way, presentations of the group G can be "lifted" to presentations of the groupoid , and this is a useful way of obtaining information about presentations of the subgroup . For further information, see the books by Higgins and by Brown in the References.


Category of groupoids

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, and is denoted by Grpd. The category Grpd is, like the category of small categories,
Cartesian closed In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...
: for any groupoids H,K we can construct a groupoid \operatorname(H,K) whose objects are the morphisms H \to K and whose arrows are the natural equivalences of morphisms. Thus if H,K are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids G,H,K there is a natural bijection : \operatorname(G \times H, K) \cong \operatorname(G, \operatorname(H,K)). This result is of interest even if all the groupoids G,H,K are just groups. Another important property of Grpd is that it is both
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
and
cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in ...
.


Relation to

Cat The cat (''Felis catus''), also referred to as the domestic cat or house cat, is a small domesticated carnivorous mammal. It is the only domesticated species of the family Felidae. Advances in archaeology and genetics have shown that the ...

The inclusion i : \mathbf \to \mathbf has both a left and a right
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
: : \hom_(C ^ G) \cong \hom_(C, i(G)) : \hom_(i(G), C) \cong \hom_(G, \mathrm(C)) Here, C ^/math> denotes the
localization of a category In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in gene ...
that inverts every morphism, and \mathrm(C) denotes the subcategory of all isomorphisms.


Relation to sSet

The
nerve functor In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The simplicial set, geometric realization of this simplicial set is a top ...
N : \mathbf \to \mathbf embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a
Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
. The nerve has a left adjoint : \hom_(\pi_1(X), G) \cong \hom_(X, N(G)) Here, \pi_1(X) denotes the fundamental groupoid of the simplicial set .


Groupoids in Grpd

There is an additional structure which can be derived from groupoids internal to the category of groupoids, double-groupoids. Because Grpd is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids \mathcal_1,\mathcal_0 with functors
s,t: \mathcal_1 \to \mathcal_0
and an embedding given by an identity functor
i:\mathcal_0 \to\mathcal_1
One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares
\begin \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow & \bullet \end and \begin \bullet & \xrightarrow & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end
with a the same morphism, they can be vertically conjoined giving a diagram
\begin \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end
which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.


Groupoids with geometric structures

When studying geometrical objects, the arising groupoids often carry a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, turning them into topological groupoids, or even some
differentiable structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for d ...
, turning them into
Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are sm ...
s. These last objects can be also studied in terms of their associated
Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...
s, in analogy to the relation between
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s. Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in
Poisson geometry In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hami ...
one has the notion of a symplectic groupoid, which is a Lie groupoid endowed with a compatible
symplectic form In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping \omega : V \times V \to F that is ; Bilinear: ...
. Similarly, one can have groupoids with a compatible
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, or complex structure, etc.


See also

*
∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standa ...
*
2-group In mathematics, particularly category theory, a is a groupoid with a way to multiply objects and morphisms, making it resemble a group. They are part of a larger hierarchy of . They were introduced by Hoàng Xuân Sính in the late 1960s unde ...
*
Homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory ap ...
* Inverse category * Groupoid algebra (not to be confused with algebraic groupoid) * R-algebroid


Notes


References

* * Brown, Ronald, 1987,
From groups to groupoids: a brief survey
, ''Bull. London Math. Soc.'' 19: 113–34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references. * —, 2006.

' Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application. * —

Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
and in group
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. Many references. * * * F. Borceux, G. Janelidze, 2001,
Galois theories.
' Cambridge Univ. Press. Shows how generalisations of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
lead to Galois groupoids. * Cannas da Silva, A., and A. Weinstein,
Geometric Models for Noncommutative Algebras.
' Especially Part VI. * Golubitsky, M., Ian Stewart, 2006,
Nonlinear dynamics of networks: the groupoid formalism
, ''Bull. Amer. Math. Soc.'' 43: 305–64 * * Higgins, P. J., "The fundamental groupoid of a
graph of groups In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, ...
", J. London Math. Soc. (2) 13 (1976) 145–149. * Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an
orbit space In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under fun ...
", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115–122. *Higgins, P. J., 1971. ''Categories and groupoids''. Van Nostrand Notes in Mathematics. Republished in ''Reprints in Theory and Applications of Categories'', No. 7 (2005) pp. 1–195
freely downloadable
Substantial introduction to
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g.
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
. * Mackenzie, K. C. H., 2005.
General theory of Lie groupoids and Lie algebroids
'. Cambridge Univ. Press. * Weinstein, Alan,
Groupoids: unifying internal and external symmetry – A tour through some examples
. Also available i
Postscript
Notices of the AMS, July 1996, pp. 744–752. * Weinstein, Alan,
The Geometry of Momentum
(2002) * R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In ''Algebraic and geometric combinatorics'', volume 423 of ''Contemp. Math''., 305–324. Amer. Math. Soc., Providence, RI (2006) * * {{nlab, id=core, title=core Algebraic structures Category theory Homotopy theory