Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.

Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).

Coalgebras occur naturally in a number of contexts (for example, universal enveloping algebras and group schemes).

There are also F-coalgebras, with important applications in computer science.

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

(Here ⊗ refers to the tensor product over K and id is the identity function.)

Equivalently, the following two diagrams commute:

In the first diagram we silently identify C ⊗ (C ⊗ C) with (C ⊗ C) ⊗ C; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces C, C ⊗ K and K ⊗ C are identified.

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