Hereditarity of closure operators and injectivity

Gabriele Castellini; Eraldo Giuli

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 149-157
  • ISSN: 0010-2628

Abstract

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A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let C be a regular closure operator induced by a subcategory 𝒜 . It is shown that, if every object of 𝒜 is a subobject of an 𝒜 -object which is injective with respect to a given class of monomorphisms, then the closure operator C is hereditary with respect to that class of monomorphisms.

How to cite

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Castellini, Gabriele, and Giuli, Eraldo. "Hereditarity of closure operators and injectivity." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 149-157. <http://eudml.org/doc/247417>.

@article{Castellini1992,
abstract = {A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let $C$ be a regular closure operator induced by a subcategory $\mathcal \{A\}$. It is shown that, if every object of $\mathcal \{A\}$ is a subobject of an $\mathcal \{A\}$-object which is injective with respect to a given class of monomorphisms, then the closure operator $C$ is hereditary with respect to that class of monomorphisms.},
author = {Castellini, Gabriele, Giuli, Eraldo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {closure operator; hereditary closure operator; injective object; factorization pair; monomorphisms; codomain functor; closure operator; hereditarity; factorization pair; injectivity; idempotent hulls},
language = {eng},
number = {1},
pages = {149-157},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Hereditarity of closure operators and injectivity},
url = {http://eudml.org/doc/247417},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Castellini, Gabriele
AU - Giuli, Eraldo
TI - Hereditarity of closure operators and injectivity
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 149
EP - 157
AB - A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let $C$ be a regular closure operator induced by a subcategory $\mathcal {A}$. It is shown that, if every object of $\mathcal {A}$ is a subobject of an $\mathcal {A}$-object which is injective with respect to a given class of monomorphisms, then the closure operator $C$ is hereditary with respect to that class of monomorphisms.
LA - eng
KW - closure operator; hereditary closure operator; injective object; factorization pair; monomorphisms; codomain functor; closure operator; hereditarity; factorization pair; injectivity; idempotent hulls
UR - http://eudml.org/doc/247417
ER -

References

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  1. Castellini G., Closure operators, monomorphisms and epimorphisms in categories of groups, Cahiers Topologie Geom. Differentielle Categoriques 27 (2) (1986), 151-167. (1986) Zbl0592.18001MR0850530
  2. Castellini G., Strecker G.E., Global closure operators vs. subcategories, Quaestiones Mathematicae 13 (1990), 417-424. (1990) Zbl0733.18001MR1084751
  3. Dikranjan D., Giuli E., Closure operators induced by topological epireflections, Coll. Math. Soc. J. Bolyai 41 (1983), 233-246. (1983) MR0863906
  4. Dikranjan D., Giuli E., Closure operators I, Topology Appl 27 (1987), 129-143. (1987) Zbl0634.54008MR0911687
  5. Dikranjan D., Giuli E., Factorizations, injectivity and compactness in categories of modules, Comm. Algebra 19 (1) (1991), 45-83. (1991) Zbl0726.16005MR1092551
  6. Dikranjan D., Giuli E., C -perfect morphisms and C -compactness, preprint. 
  7. Dikranjan D., Giuli E., Tholen W., Closure operators II, Proceedings of the Conference in Categorical Topology (Prague, 1988) World Scientific (1989), 297-335. MR1047909
  8. Giuli E., Bases of topological epireflections, Topology Appl. 11 (1980), 265-273. (1980) MR0585271
  9. Herrlich H., Strecker G.E., Category Theory, 2nd ed., Helderman Verlag, Berlin, 1979. Zbl1125.18300MR0571016
  10. Herrlich H., Salicrup G., Strecker G.E., Factorizations, denseness, separation, and relatively compact objects, Topology Appl. 27 (1987), 157-169. (1987) Zbl0629.18003MR0911689
  11. Koslowski J., Closure operators with prescribed properties, Category Theory and its Applications (Louvain-la-Neuve, 1987) Springer L.N.M. 1248 (1988), 208-220. Zbl0659.18005MR0975971
  12. Lambek J., Torsion theories, additive semantics and rings of quotients, Springer L.N.M. 177, 1971. Zbl0213.31601MR0284459
  13. Salbany S., Reflective subcategories and closure operators, Proceedings of the Conference in Categorical Topology (Mannheim, 1975), Springer L.N.M. 540 (1976), 548-565. Zbl0335.54003MR0451186
  14. Skula L., On a reflective subcategory of the category of all topological spaces, Trans. Amer. Mat. Soc. 142 (1969), 37-41. (1969) Zbl0185.50401MR0248718

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