A note on reflective subcategories defined by partial algebras (original) (raw)
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2015
Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflec-tions with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcate...
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Based on the monoid classifier ∆, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via inclusions and Kleene morphisms.
Paracategories I: internal parategories and saturated partial algebras
Eprint Arxiv Math 0303083, 2003
Based on the monoid classifier ∆, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via epimorphisms and (monic) Kleene morphisms and relate the latter to saturation.
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