In relational doctrines with quotients, the extensional quotient completion is characterized by the existence of a projective cover, and this transfers to doctrines of algebras for quotient-preserving monads.
Barr, Exact categories
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Projective covers, doctrines of algebras and the relational quotient completion
In relational doctrines with quotients, the extensional quotient completion is characterized by the existence of a projective cover, and this transfers to doctrines of algebras for quotient-preserving monads.