In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

Contents 1 Definition 2 Examples 3 Group theory generalized 4 See also 5 References

Definition

Formally, we start with a category C with finite products (i. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships e. C has a terminal object 1 and any two objects of C have a product). In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as 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.

Examples

• A group can be viewed as a group object in the category of sets. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, a singleton is a set with exactly one element e_G : G → G is the map that sends every element of G to the identity element.
• A topological group is a group object in the category of topological spaces with continuous functions. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function
• A Lie group is a group object in the category of smooth manifolds with smooth maps. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability
• A Lie supergroup is a group object in the category of supermanifolds. The concept of supergroup is a Generalization of that of group. In Physics and Mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from Supersymmetry.
• An algebraic group is a group object in the category of algebraic varieties. In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a group scheme is a Group object in the Category of schemes. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory.
• A localic group is a group object in the category of locales. In Mathematics, especially in Order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice.
• The group objects in the category of groups (or monoids) are essentially the Abelian groups. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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. In Mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an Argument about two Monoid
• Given a category C with finite coproducts, 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 versions of the axioms for group objects. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the Here 0 is the initial object of C. Cogroup objects occur naturally in 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

Group theory generalized

Much of group theory can be formulated in the context of the more general group objects. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural 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 straight-forward manner.

See also

• Hopf algebras can be seen as a generalization of group objects to monoidal categories. In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C

References

• Lang, Serge (2002), Algebra, vol. Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. 211 (3rd ed. ), Graduate Texts in Mathematics, Springer-Verlag .

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