Abstract
Exact squares in Cat are not necessarily absolute (i.e., preserved by any 2-functor Cat → Cat), or even preserved by any 2-functor given by exponentiation (−)ℐ : Cat → Cat: if a square is preserved by exponentiation it will be called a contractible exact square. We will characterize diagrammatically these contractible squares, and among them the contractible categories, and the so called fibering and cofibering squares, with especially the comma squares and the adjunction squares. As an application we conclude with a diagrammatical characterization of absolutely absolute Kan extensions and especially of absolutely final functors and of absolutely absolute colimits.
Article PDF
Similar content being viewed by others
References
Bénabou, J.: Introduction to bicategories. L.N.M., vol. 47, pp. 1–77. Springer, Berlin (1967)
Bénabou, J.: Les distributeurs. In: Rapport 33, Institut of Pure and Applied mathematics. Louvain-la-Neuve (1973)
Bénabou, J.: Distributors at work. Lecture Notes of a Course Given by Jean Bénabou in June 2000 at TU Darmstadt taken by Thomas Streicher (avalaible at www.mathematik.tu-darmstadt.de/~streicher/)
Bourn, D., Cordier, J.-M.: Distributeurs et théorie de la forme. Cahiers Top. Géo. Diff. XXI-2, 161–189 (1980)
Cordier, J.-M., Porter, T.: Shape Theory: Categorical Methods of Approximation, p. 207. Ellis Horwwood Limited, Chichester (1989)
Ehresmann, C.: Catégories doubles et catégories structurées. C.R.A.S. 256, 1198–1201 (1963)
Ehresmann, C.: Catégorie double des quintettes; applications covariantes. C.R.A.S. 256, 1891–1894. Paris (1963)
Gray, J.W.: Fibred and cofibred categories. In: Proceedings Conference on Categorical Algemeen at La Jolla, pp. 21–83. Springer (1966)
Gray, J.W.: The calculus of comma categories. Not. Am. Math. Soc. 14, 937 (1967)
Guitart, R.: Extensions de Kan Absolues. Mathematisches Forschunginstitut Oberwolfach, Tagunsbericht Kategorien. 42–44, (also available at: http://rene.guitart.pagesperso-orange.fr/) (1977)
Guitart, R.: Relations et carrés exacts. Ann. Sci. Math. Qué. juillet IV(2), 103–125 (1980)
Guitart, R.: Carrés exacts et carrés déductifs. Diagrammes 6, 17 (1981)
Guitart, R.: An anabelian definition of abelian homology. Cahiers Top. Géo. Diff. XLVIII-4, 261–269 (2007)
Guitart, R., Van den Bril, L.: Note sur la détermination des homologies par les carrés exacts. Diagrammes 6, 7 (1981)
Guitart, R., Van den Bril, L.: Calcul des satellites et présentations des bimodules á l’aide des carrés exacts, I et II. Cahiers Top. Géo. Diff. XXIV(4), 299–330, 333–369 (1983)
Harting, R.: Distributoren und Kan-Erweiterungen. Archiv Math. XXIX-4 (1977)
Isbell, J.: Epimorphisms and dominions. In: Proceedings Conference on Categorical Algebra, La Jolla, California, 1965. Springer, New-York (1966)
Kahn, B., Maltsiniotis, G.: Structures de dérivabilité. Adv. Math. 218, 1286–1318 (2008)
Kan, D.M.: Adjoint functors. Trans. Am. Math. Soc. 87, 294–329 (1958)
Kelly, G.M., Street, R.: Review of the elements of 2-categories. L.N.M., vol. 420, pp. 75–103. Springer (1974)
Lawvere, W.: Functorial Semantics of Algebraic Theories. Thesis, Columbia University (1963)
Maltsiniotis, G.: Carrés exacts homotopiques, et dérivateurs. Cahiers Top. Géo. Diff. Cat. LIII-1, 3–63 (2012)
Paré, R.: Absoluteness Properties in Category Theory. Thesis, McGill (1969)
Paré, R.: Absolute Coequalizers. S.L.N. 86, 132–145 (1969)
Paré, R.: On absolute colimits. J. Alg. 19, 80–95 (1971)
Riquet, J.: Deducibility and exactness. Logic Journal of IGPL, vol. 6, (2), pp. 341–347. Oxford University Press (1998)
Street, R.: Fibrations and Yoneda’s lemma in a 2-category. L.N.M., vol. 420, pp. 104–133. Springer (1974)
Street, R.: Cosmoi of internal categories. Transactions of the AMS 258(2), 271–318 (1980)
Walters, R.S., Walters, R.F.C.: The comprehensive factorization of a functor. Bull. Am. Math. Soc. 79, 936–941 (1973)
Van den Bril, L.: Carré exacts de Hilton dans des contextes non-abéliens. Ann. Sc. Math. Qué., juillet IV(2), 153–173 (1980)
Van den Bril, L.: Exactitude dans les Yoneda-structures. Cahiers Top. Géo. Diff. Cat. XXIII-2, 215–224 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
In homage to my friend George Janelidze, for his 60th birthday
Rights and permissions
About this article
Cite this article
Guitart, R. Contractible Exact Squares. Appl Categor Struct 22, 873–898 (2014). https://doi.org/10.1007/s10485-013-9353-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-013-9353-4