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Coassociative coalgebras

2003; Elsevier BV; Linguagem: Inglês

10.1016/s1570-7954(03)80072-4

1570-7954

Walter Michaelis,

Homotopy and Cohomology in Algebraic Topology

This chapter provides an overview of the theory of coassociative coalgebras. The counterpart of a unitary algebra is a unitary coalgebra. The counterpart of an associative algebra is an associative coalgebra. The counterpart of Lie algebra is Lie coalgebra. The underlying vector space, V, of algebra is denoted by A; the underlying vector space, V, of a coalgebra is denoted by C; the underlying vector space of a Lie algebra is denoted by L; and the underlying vector space of a Lie coalgebra is denoted by M. The detailed definition of an associative coalgebra is provided, the associativity of the multiplication of algebra in an element-free way by requiring the commutativity of a certain diagram is also expressed. Algebra is defined by taking the defining diagrams for a coalgebra and reversing arrows. The formal, technical device that is added to display the precise way in which algebras and coalgebras are dual to one another is that of a monoidal category.