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It is proved that any category $\cal{K}$ which is equivalent to a simultaneously reflective and coreflective full subcategory of presheaves $[\cal{A}^{op},Set]$, is itself equivalent to the category of the form $[\cal{B}^{op},Set]$ and the inclusion is induced by a functor $\cal{A} \to \cal{B}$ which is surjective on objects. We obtain a characterization of such functors.
Moreover, the base category $Set$ can be replaced with any symmetric monoidal closed category $V$ which is complete and cocomplete, and then analogy of the above result holds if we replace categories by $V$-categories and functors by $V$-functors.
As a consequence we are able to derive well-known results on simultaneously reflective and coreflective categories of sets, Abelian groups, etc.
@article{TAC_2002_10_a15,
author = {Robert El Bashir and Jiri Velebil},
title = {Simultaneously {Reflective} {And} {Coreflective} {Subcategories} of {Presheaves}},
journal = {Theory and applications of categories},
pages = {410--423},
publisher = {mathdoc},
volume = {10},
year = {2002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2002_10_a15/}
}
TY - JOUR AU - Robert El Bashir AU - Jiri Velebil TI - Simultaneously Reflective And Coreflective Subcategories of Presheaves JO - Theory and applications of categories PY - 2002 SP - 410 EP - 423 VL - 10 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2002_10_a15/ LA - en ID - TAC_2002_10_a15 ER -
Robert El Bashir; Jiri Velebil. Simultaneously Reflective And Coreflective Subcategories of Presheaves. Theory and applications of categories, Tome 10 (2002), pp. 410-423. http://geodesic.mathdoc.fr/item/TAC_2002_10_a15/