Equivalences in Bicategories

Document Type : Research Paper


Department of Mathematics, Universit'e Choua"ib Doukkali, El Jadida, Morocco.



In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.


[1] Abbad, O., Categorical classifications of extensions, Ph.D. Thesis (in preparation).
[2] Abbad, O. and Vitale, E.M.,  Faithful calculus of fractions, Cah. Topol. Géom. Différ. Catég. 54(3) (2013), 221-239.
[3] Bénabou, J., “Introduction to Bicategories”, in: Reports of the Midwest Category Seminar, Lecture Notes in Math. 47, Springer, Berlin 1967, 1-77.
[4] Borceux, F., “Handbook of Categorical Algebra 1”, Cambridge University Press, 1994.
[5] Bunge M. and Paré, R., Stacks and equivalence of indexed categories, Cah. Topol. Géom. Différ. Catég. 20(4) (1979), 373-399.
[6] Baez, John C., Higher-dimensional algebra II: 2-Hilbert Spaces, ArXiv:qalg/9609018v2, (1998).
[7] Everaert, T., Kieboom, R.W., and Van der Linden, T., Model structures for homotopy of internal categories, Theory Appl. Categ. 15(3), (2005), 66-94.
[8] Leinster, T., Basic bicategories, ArXiv:math/9810017v1, (1998).
[9] Mac Lane, S., “Categories for theWorking Mathematician”, Graduate Texts in Mathematics, Springer Verlag, New York, 2nd Edition, 1998.
[10] Pronk, D., Etendues and stacks as bicategories of fractions, Compos. Math. 102 (1996), 243-303.