antipode. The group algebra of a finite group gives an example. Category-theoretical definition In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object in a monoidal category consists of an object A of C together with four morphisms such that is a monoid object in C, is a comonoid object in C, the diagrams and commute (for simplicity the diagrams are given here in the case where the monoidal category C is strict) and are known as Frobenius conditions. More compactly, a Frobenius algebra in C is a so-called Frobenius monoidal functor A:1 → C, where 1 is the category consisting of one object and one arrow. A Frobenius algebra is called isometric or special if . Applications Frobenius algebras originally were studied as part of an investigation into the representation theory of finite groups, and have contributed to the study of number theory, algebraic geometry, and combinatorics. They have been used to study Hopf algebras, coding theory, and cohomology rings of compact oriented manifolds. Topological quantum field theories Recently, it has been seen that they play an important role in the algebraic treatment and axiomatic foundation of topological quantum field theory. A commutative Frobenius algebra determines uniquely (up to isomorphism) a (1+1)-dimensional TQFT. More precisely, the category of commutative Frobenius -algebras is equivalent to the category of symmetric strong monoidal functors from - (the category of 2-dimensional cobordisms between 1-dimensional manifolds) to (the category of vector spaces over ). The correspondence between TQFTs and Frobenius algebras is given as follows: 1-dimensional manifolds are disjoint unions of circles: a TQFT associates a vector space with a circle, and the tensor product of vector spaces with a disjoint union of circles, a TQFT associates (functorially) to each cobordism between manifolds a map between vector spaces, the map associated with a pair of pants (a cobordism between 1 circle and 2 circles) gives a product map or a coproduct map , depending on how the boundary components are grouped – which is commutative or cocommutative, and the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary. This relation between Frobenius algebras and (1+1)-dimensional TQFTs can be used to explain Khovanov's categorification of the Jones polynomial. Generalizations Frobenius extensions Let B be a subring sharing the identity element of a unital associative ring