Skip to main content

Cover Relations on Categories


A cover relation on a category ℂ is a binary relation ⊏ on the class of morphisms of ℂ, which is defined only for those pairs of morphisms which have the same codomain, and which has the following two properties: (i) if fg and h is composable with f, then hfhg, (ii) if fg and f is composable with e then feg. We study cover relations arising from a special type of factorization systems, and cover relations arising from a special type of monoidal structures.

This is a preview of subscription content, access via your institution.


  1. Bourn, D.: Mal’cev categories and fibration of pointed objects. Appl. Categ. Structures 4, 307–327 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  2. Bourn, D.: Intrinsic centrality and associated classifying properties. J. Algebra 256, 126–145 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  3. Bourn, D., Janelidze, G.: Centralizars in action accessible categories. Les Cahiers du Laboratoire de Mathematiques Pures et Appliquees Joseph Liouville 344, 2007 (preprint)

  4. Dikranjan, D., Tholen, W.: Categorical structure of closure operators, Mathematics and Its Applications, vol. 346. Kluwer (1995)

  5. Freyd, P.J.: Algebra-valued functors in general and tensor products in particular. Colloq. Math. 14, 89–106 (1966)

    MATH  MathSciNet  Google Scholar 

  6. Freyd, P.J., Kelly, G.M.: Categories of continuous functors I. J. Pure Appl. Algebra 2, 169–191 (1972)

    MATH  Article  MathSciNet  Google Scholar 

  7. Janelidze, Z.: Closedness properties of internal relations V: Linear Mal’tsev conditions. Algebra Universalis 58, 105–117 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  8. Lawvere, F.W.: Functorial semantics of algebraic theories, PhD Thesis, Columbia University (1963)

  9. Linton, F.E.J.: Autonomous equational categories. J. Math. Mech. 15, 637–642 (1966)

    MATH  MathSciNet  Google Scholar 

  10. Mac Lane, S.: Categories for the working mathematician. Graduate Texts in Mathematics, vol. 5. Springer-Verlag (1971)

  11. Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic: a first introduction to topos theory, Springer (1992)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Zurab Janelidze.

Additional information

Supported by Claude Leon Foundation, INTAS (06-1000017-8609) and Georgian National Science Foundation (GNSF/ST06/3-004).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Janelidze, Z. Cover Relations on Categories. Appl Categor Struct 17, 351–371 (2009).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Cover relation
  • Factorization system
  • Monoidal category

Mathematics Subject Classifications (2000)

  • 18A32
  • 18D10