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Prereflections and reflections. (English) Zbl 0587.18002
A pair (T,\(\eta)\) is a pointed endofunctor of a category \({\mathcal K}\) if \(T: {\mathcal K}\to {\mathcal K}\) is an endofunctor and \(\eta\) : Id\({}_{{\mathcal K}}\to T\) is a natural transformation. If \(T\eta =\eta T\) then (T,\(\eta)\) is called well-pointed, and if for every pair of \({\mathcal K}\)-morphisms \(f: A\to B\), h: TA\(\to TB\) with \(h\circ \eta^ A=\eta^ B\circ f\) we have \(h=Tf\) then we say that (T,\(\eta)\) is a prereflection. If (T,\(\eta)\) is well-pointed and \(\eta\) T is an isomorphism then (T,\(\eta)\) is called a reflection. If a prereflection (T,\(\eta)\) is a reflection then the full subcategory of \({\mathcal K}\) formed by \({\mathcal K}\)-objects A such that \(\eta^ A\) is an isomorphism is a reflective subcategory of \({\mathcal K}\). The paper describes properties of reflections and prereflections, in particular several distinct characterizations of reflections and prereflections are given.
Reviewer: V.Koubek

MSC:
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18C99 Categories and theories
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