## 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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### References:

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