Coalgebra

The word “coalgebra” has two different but related meanings:

  1. In pure mathematics, a coalgebra is an algebraic structure dual to an associative algebra. It is an important ingredient in the concept of a Hopf algebra .
  2. In theoretical computer science, a “coalgebra” often means a coalgebra over an endofunctor , aka an F-coalgebra . Such coalgebras are used to model stateful computation.

This page is about both concepts, but where ambiguity is possible I restrict myself to the first usage. For the relationship between the two, see (Jacobs, 2016, Remark 1.0.1).

Literature

Coalgebra

See also Math.SE bibliography on Hopf algebras.

  • Michaelis, 2003: Coassociative coalgebras, in Handbook of Algebra, Vol. 3 (doi)
  • Street, 2007: Quantum groups: A path to current algebra (doi)

F-coalgebra

Introductions

  • Rutten, 2019: The method of coalgebra: exercises in coinduction (pdf)
  • Jacobs, 2016: Introduction to coalgebra: Towards mathematics of states and observation (doi)
  • Rutten, 2000: Universal coalgebra: a theory of systems (doi, tech report )
    • “Universal coalgebra” as dual to universal algebra
  • Corfield, 2011: Understanding the infinite II: Coalgebra (doi, nCat Cafe )
    • Corfield’s previous posts on nCat Cafe (1 ,2 ), citing several of the above

Applications

  • Pavlovic & Escardo, 1998: Calculus in coinductive form (doi, pdf)
  • Kozen & Ruozzi, 2009: Applications of metric coinduction (arxiv, pdf)