In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, group objects are certain generalizations of groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 which are built on more complicated structures than sets. A typical example of a group object is a topological group

Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

, a group whose underlying set is a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 such that the group operations are continuous.

Definition

Formally, we start with a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

). A group object in C is an object G of C together with morphisms

• m : G × G → G (thought of as the "group multiplication")
• e : 1 → G (thought of as the "inclusion of the identity element")
• inv: G → G (thought of as the "inversion operation")

such that the following properties (modeled on the group axioms) are satisfied

• m is associative, i.e. m(m × id_G) = m (id_G × m) as morphisms G × G × G → G; here we identify G × (G × G) in a canonical manner with (G × G) × G.
• e is a two-sided unit of m, i.e. m (id_G × e) = p₁, where p₁ : G × 1 → G is the canonical projection, and m (e × id_G) = p₂, where p₂ : 1 × G → G is the canonical projection
• inv is a two-sided inverse for m, i.e. if d : G → G × G is the diagonal map, and e_G : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (id_G × inv) d = e_G and m (inv × id_G) d = e_G.

Another way to state the above is to say G is a group object in a category C if for every object X in C, there is a group structure on the morphisms hom(X, G) from X to G such that the association of X to hom(X, G) is a covariant functor (from C to the category of groups).

Examples

• A group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
   can be viewed as a group object in the category of sets
  Set theory
  Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
  
  . The map m is the group operation, the map e (whose domain is a singleton) picks out the identity element of the group, and the map inv assigns to every group element its inverse. e_G : G → G is the map that sends every element of G to the identity element.
• A topological group
  Topological group
  In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
  
   is a group object in the category of topological spaces
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
   with continuous functions.
• A Lie group
  Lie group
  In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
  
   is a group object in the category of smooth manifolds
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
   with smooth maps.
• A Lie supergroup is a group object in the category of supermanifold
  Supermanifold
  In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...
  
  s.
• An algebraic group
  Algebraic group
  In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
  
   is a group object in the category of algebraic varieties
  Algebraic variety
  In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
  
  . In modern algebraic geometry
  Algebraic geometry
  Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
  
  , one considers the more general group scheme
  Group scheme
  In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
  
  s, group objects in the category of scheme
  Scheme (mathematics)
  In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
  
  s.
• A localic group is a group object in the category of locales.
• The group objects in the category of groups (or monoid
  Monoid
  In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
  
  s) are essentially the Abelian group
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
  s. The reason for this is that, if inv is assumed to be a homomorphism, then G must be abelian. More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then (A,m,e,inv) is a group object in the category of groups (or monoids). Conversely, if (A,m,e,inv) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group. See also Eckmann-Hilton argument
  Eckmann-Hilton argument
  In mathematics, the Eckmann–Hilton argument is an argument about two monoid structures on a set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting monoid demonstrated to be commutative...
  
  .
• Given a category C with finite coproduct
  Coproduct
  In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
  
  s, a cogroup object is an object G of C together with a "comultiplication" m: G → G G, a "coidentity" e: G → 0, and a "coinversion" inv: G → G, which satisfy the dual
  Dual (category theory)
  In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
  
   versions of the axioms for group objects. Here 0 is the initial object
  Initial object
  In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
  
   of C. Cogroup objects occur naturally in algebraic topology
  Algebraic topology
  Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
  
  .

Group theory generalized

Much of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

 can be formulated in the context of the more general group objects. The notions of group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

, subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

, normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

 and the isomorphism theorem

Isomorphism theorem

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...

s are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, normally cannot be generalized to group objects in a straightforward manner.

See also

• Hopf algebra
  Hopf algebra
  In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
  
  s can be seen as a generalization of group objects to monoidal categories
  Monoidal category
  In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
  
  .