We show how coalgebras can be presented by operations and equations. This is a special case of Linton’s approach to algebras over a general base category X, namely where X is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic.
A. Kurz and J. Rosický, “Modal Predicates and Coequations,” Electronic Notes in Theoretical Computer Science, vol. 65, no. 1, pp. 156–175, Oct. 2002. DOI: 10.1016/S1571-0661(04)80364-5
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