Abstract
Concordant-dissonant and monotone-light factorisation systems on categories, ways to construct them, and conditions for them to coincide, as well as their examples are studied in this article. These factorisation systems are constructed from a reflection induced from a ground adjunction and a specified prefactorisation system. Furthermore, we give additional conditions, under which the monotone-light and the concordant-dissonant factorisations coincide for sub-reflections of the induced reflection. The adjunctions given by right Kan extensions, from the category of presheaves on sets, turn out to be very well-behaved examples, provided they satisfy the cogenerating set condition, which allows to describe the four classes of morphisms in the reflective and concordant-dissonant (= monotone-light) factorisations. It is also noticed that the faithfulness of the composite of the left-adjoint with the Yoneda embedding can be seen as a generalisation of the cogenerating set condition. Using this generalisation it is possible to present a convenient simplified version of the sufficient conditions above for the case of an adjunction from the category of presheaves on sets into a cocomplete category, satisfying the faithfulness of the abovementioned composite. Then, the same is done for induced sub-reflections from categories of models of (limit) sketches; in particular this explains why the monotone-light factorisation for categories via preordered sets is just the restriction of the same factorisation for simplicial sets via ordered simplicial complexes.
Similar content being viewed by others
References
Carboni, A., Janelidze, G., Kelly, G.M., Paré, R.: On localization and stabilization for factorization systems. Appl. Categ. Struct. 5, 1–58 (1997)
Cassidy, C., Hébert, M., Kelly, G.M.: Reflective subcategories, localizations and factorization systems. J. Aust. Math. Soc. 38A, 287–329 (1985)
Eilenberg, S.: Sur les transformations continues d’espaces métriques compacts. Fundam. Math. 22, 292–296 (1934)
Freyd, P.J., Kelly, G.M.: Categories of continous functors, I. J. Pure Appl. Algebra 2, 169–191 (1972)
Janelidze, G., Sobral, M., Tholen, W.: Beyond barr exactness: effective descent morphisms. In: Categorical Foundations. Special Topics in Order, Topology, Algebra and Sheaf Theory. Cambridge University Press (2004)
Janelidze, G., Tholen, W.: Functorial factorization, well-pointedness and separability. J. Pure Appl. Algebra 142, 99–130 (1999)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer (1998)
Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer (1992)
Whyburn, G.T.: Non-alternating transformations. Am. J. Math. 56, 294–302 (1934)
Xarez, J.J.: The monotone-light factorization for categories via preorders. In: Galois Theory, Hopf Algebras and Semiabelian Categories, pp. 533–541, Fields Inst. Commun., vol. 43. Amer. Math. Soc., Providence (2004)
Xarez, J.J.: A Galois theory with stable units for simplicial sets. Theory Appl. Categ. 15, 178–193 (2006)
Xarez, J.J.: Well-behaved epireflections for Kan extensions. Appl. Categ. Struct. 18, 219–230 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xarez, J.J. Concordant and Monotone Morphisms. Appl Categor Struct 21, 393–415 (2013). https://doi.org/10.1007/s10485-011-9270-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-011-9270-3
Keywords
- Kan extensions
- Cogenerating set
- Epireflection
- Stable units
- Prefactorisation
- Concordant-dissonant factorisation
- Monotone-light factorisation
- Dense subcategory
- Regular category
- Presheaves
- Models of sketches
- Descent theory
- Galois theory
- Simplicial set