These are, in fact, the results obtained first in (Guitart, 1977), because to reach the notion of exact squares, we started from R. Paré (Paré, 1969) absolute limits, we went through the Beck-Chevalley condition (Chevalley, 1964-1965) and the zig-zag theorem of Isbell (Guitart, 1982), and we only belatedly observed the connection of these questions with the exact squares in abelian categories (in the sense of Lambek (Lambek, 1958), Mac Lane, Brinkman-Puppe, Hilton (Hilton, 1966)), which suggested 1·f [001V] and Remark 1·i [001Y] definitions of Section 1 [001A].
Bibliography
-
Guitart, R. (1977). Extensions de Kan absolues. Math. Forschunginstitut Oberwolfach, 42–44.Details
-
Paré, R. (1969). Absoluteness properties in Category Theory [PhD thesis]. McGill University.Details
-
Chevalley, C. (1964-1965). Séminaire sur la descente. Unpublished.Details
-
Guitart, R. (1982). Qu’est-ce que la logique dans une catégorie ? Cahiers De Topologie Et Géométrie Différentielle Catégoriques, 23(2), 115–148. http://eudml.org/doc/91293Details
-
Lambek, J. (1958). Goursats Theorem and the Zassenhaus Lemma. Canadian Journal of Mathematics, 10, 45–56. https://doi.org/10.4153/cjm-1958-005-6Details
-
Hilton, P. (1966). Correspondences and Exact Squares. In S. Eilenberg, D. K. Harrison, S. MacLane, & H. Röhrl (Eds.), Proceedings of the Conference on Categorical Algebra (pp. 254–271). Springer Berlin Heidelberg.Details