Abstract initiality

Lutz Schröder; Horst Herrlich

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 575-583
  • ISSN: 0010-2628

Abstract

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We study morphisms that are initial w.r.t. all functors in a given conglomerate. Several results and counterexamples are obtained concerning the relation of such properties to different notions of subobject. E.g., strong monomorphisms are initial w.r.t. all faithful adjoint functors, but not necessarily w.r.t. all faithful monomorphism-preserving functors; morphisms that are initial w.r.t. all faithful monomorphism-preserving functors are monomorphisms, but need not be extremal; and (under weak additional conditions) a morphism is initial w.r.t. all faithful functors that map extremal monomorphisms to monomorphisms iff it is an extremal monomorphism.

How to cite

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Schröder, Lutz, and Herrlich, Horst. "Abstract initiality." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 575-583. <http://eudml.org/doc/248620>.

@article{Schröder2000,
abstract = {We study morphisms that are initial w.r.t. all functors in a given conglomerate. Several results and counterexamples are obtained concerning the relation of such properties to different notions of subobject. E.g., strong monomorphisms are initial w.r.t. all faithful adjoint functors, but not necessarily w.r.t. all faithful monomorphism-preserving functors; morphisms that are initial w.r.t. all faithful monomorphism-preserving functors are monomorphisms, but need not be extremal; and (under weak additional conditions) a morphism is initial w.r.t. all faithful functors that map extremal monomorphisms to monomorphisms iff it is an extremal monomorphism.},
author = {Schröder, Lutz, Herrlich, Horst},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {initial morphism; (extremal) monomorphism; faithful functor; semicategory; initial morphism; monomorphism; faithful functor; semicategory},
language = {eng},
number = {3},
pages = {575-583},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Abstract initiality},
url = {http://eudml.org/doc/248620},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Schröder, Lutz
AU - Herrlich, Horst
TI - Abstract initiality
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 575
EP - 583
AB - We study morphisms that are initial w.r.t. all functors in a given conglomerate. Several results and counterexamples are obtained concerning the relation of such properties to different notions of subobject. E.g., strong monomorphisms are initial w.r.t. all faithful adjoint functors, but not necessarily w.r.t. all faithful monomorphism-preserving functors; morphisms that are initial w.r.t. all faithful monomorphism-preserving functors are monomorphisms, but need not be extremal; and (under weak additional conditions) a morphism is initial w.r.t. all faithful functors that map extremal monomorphisms to monomorphisms iff it is an extremal monomorphism.
LA - eng
KW - initial morphism; (extremal) monomorphism; faithful functor; semicategory; initial morphism; monomorphism; faithful functor; semicategory
UR - http://eudml.org/doc/248620
ER -

References

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  1. Adámek J., Herrlich H., Strecker G.E., Abstract and Concrete Categories, Wiley Interscience New York (1990). (1990) MR1051419
  2. Bénabou J., Fibered categories and the foundations of category theory, J. Symb. Logic 50 (1985), 10-37. (1985) MR0780520
  3. Borceux F., Handbook of Categorical Algebra 2, Cambridge University Press (1994). (1994) Zbl0843.18001MR1313497
  4. Grothendieck A., Catégories fibrées et descente, Revêtements étales et groupe fondamental, Séminaire de Géometrie Algébrique du Bois-Marie 1960/61 (SGA 1), Exposé VI, 3rd ed. Institut des Hautes Etudes Scientifiques Paris 1963 reprint Springer Lect. Notes Math. 224 1971 145-194. (1971) MR0354651
  5. Herrlich H., Strecker G.E., Category Theory, 2nd ed. Heldermann Berlin (1979). (1979) Zbl0437.18001MR0571016
  6. Hong S.S., Categories in which every monosource is initial, Kyungpook Math. J. 15 (1975), 133-139. (1975) MR0369466
  7. Klop J.W., Term rewriting systems, Handbook of Logic in Computer Science, vol. 2 (S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, eds.), Oxford University Press, 1992, pp.1-116. Zbl1030.68053MR1381696
  8. Schröder L., Composition graphs and free extensions of categories, German PhD Thesis, University of Bremen Logos Verlag Berlin (1999). (1999) 
  9. Schröder L., Traces of epimorphism classes, J. Pure Appl. Algebra, submitted. 
  10. Schröder L., Herrlich H., Free adjunction of morphisms, Appl. Cat. Struct., to appear. MR1799731
  11. Schröder L., Herrlich H., Free factorizations, Appl. Cat. Struct., to appear. MR1866871

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