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A note on reflective subcategories defined by partial algebras. (English) Zbl 0556.18002

Given an endofunctor \(F: {\mathcal A}\to {\mathcal A}\) of a category \({\mathcal A}\), the category \({\mathcal A}(F)\) of algebras Fa\(\to a\) is considered. The main result states that if \({\mathcal A}(F)\) admits all free algebras (i.e. if the underlying functor \(U: {\mathcal A}(F)\to {\mathcal A}\) has a left adjoint) and coequalizers then each (generalized) partial algebra Fa\(\leftarrow x\to a\) has a free completion \(Fa^*\to a^*\). This is obtained as a corollary of a general theorem involving reflective subcategories of the comma category \(a\downarrow U\).
Reviewer: J.Reiterman

MSC:

18A25 Functor categories, comma categories
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
08C05 Categories of algebras
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