Birkhoffs variety theorem for coalgebras (Q2759839)

For a set functor \(F:\text{Set}\to \text{Set}\), a coalgebra of type \(F\) is a pair \((A,\phi)\) where \(A\) is a set and \(\phi :A\to FA\) is a mapping (then a mapping \(f:A\to B\) is a homomorphism from a coalgebra \((A,\phi)\) into a coalgebra \((B,\psi)\) whenever \(\psi\circ f=Ff\circ\phi\)). A family of coalgebras of type \(F\) forms a covariety if it is closed under sums, homomorphic images and subcoalgebras. If \(F\) satisfies a special boundedness condition (any small functor satisfies it) then any covariety has co-free coalgebras. The coequations are defined in such a way that, under the boundedness condition for \(F\), an analogue of the dual Birkhoff theorem is satisfied.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].