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Keywords:
Discrete Fibrations; 2-Fibrations; Monadicity; Lax Transformations
Discrete Fibrations; 2-Fibrations; Monadicity; Lax Transformations
Summary:
This paper is a study of 2-dimensional discrete fibrations. A definition is proposed as a specialization of 2-fibrations. It is shown that discrete 2-fibrations correspond via a category of elements construction to contravariant category-valued 2-functors. The second part of the paper is dedicated to a monadicity result. It is shown that 2-fibrations are algebras for a monad given by an action of Bénabou’s cylinders construction. This should be seen as categorifying the monad for which ordinary fibrations are algebras, namely, that given by an action of an ordinary arrow category. Monadicity of discrete 2-fibrations is recovered by restricting the cylinders monad. To support the correctness of this categorification, it is shown that cylinders are a cotensor in a 3-categorical structure of 2-categories, 2-functors, lax natural transformations and modifications. This is an example of a lax 3-category which is introduced here to describe this universality.
References:
[1] Artin, M., Grothendieck, A., Verdier, J. L.: Théorie des topos et cohomologie etale des schémas (SGA 4) tome 1. Lecture notes in mathematics, Springer-Verlag, Berlin, DOI:10.1007/BFb0081551 DOI 10.1007/BFb0081551
[2] Baković, Igor: Fibrations of bicategories. Preprint available at https://www2.irb.hr/korisnici/ibakovic/groth2fib.pdf
[3] Bénabou, Jean: Introduction to bicategories. Reports of the midwest category seminar i, pages 1-77, Lecture notes in mathematics 47
[4] Bénabou, Jean: Fibered categories and the foundations of naive category theory. The Journal of Symbolic Logic, Vol. 50, Iss. 1, 10-37 DOI 10.2307/2273784
[5] Bird, G. J.: Limits in 2-categories of locally-presentable categories. PhD thesis, University of Sydney
[6] Borceux, Francis: Handbook of categorical algebra 1: Basic category theory. Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, DOI:10.1017/CBO9780511525858 DOI 10.1017/CBO9780511525858
[7] Buckley, Mitchell: Fibred 2-categories and bicategories. Journal of Pure and Applied Algebra, Vol. 218, Iss. 6, 1034-1074, DOI:10.1016/j.jpaa.2013.11.002 DOI 10.1016/j.jpaa.2013.11.002 | MR 3153612
[8] Cruttwell, G. S. H., Lambert, M. J., Pronk, D. A., Szyld, M.: Double fibrations. Theory and Applications of Categories, Vol. 38, Iss. 35, 1326-1394 MR 4520578
[9] Descotte, M. E., Dubuc, E. J., Szyld, M.: Sigma colimits in 2-categories and flat pseudofunctors. Advances in Mathematics, Vol. 333, 266-313, DOI:10.1016/j.aim.2018.05.021 DOI 10.1016/j.aim.2018.05.021 | MR 3818078
[10] Diaconescu, Radu: Change of base for some toposes. PhD thesis, Dalhousie University
[11] Diaconescu, Radu: Change of base for toposes with generators. Journal of Pure and Applied Algebra, Vol. 6, Iss. 3, 191-218, DOI:10.1016/0022-4049(75)90015-8 DOI 10.1016/0022-4049(75)90015-8
[12] Gordon, R., Power, A. J., Street, Ross: Coherence for tricategories. American Mathematical Society, Volume 117, DOI:10.1090/memo/0558 DOI 10.1090/memo/0558
[13] Grandis, Marco, Paré, Robert: Limits in double categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 40, Iss. 3, 162-220
[14] Grandis, Marco, Paré, Robert: Intercategories. Theory and Applications of Categories, Vol. 30, Iss. 38, 1215-1255 MR 3402490
[15] Gray, John W.: Fibred and cofibred categories. Proceedings of the conference on categorical algebra, la jolla 1965, pages 21-83,
[16] Gray, John W.: Formal category theory: Adjointness for 2-categories. Lecture notes in mathematics, Springer-Verlag, Berlin, DOI:10.1007/BFb0061280 DOI 10.1007/BFb0061280
[17] Grothendieck, Alexander: Categories fibrees et descente. Revêtements étales et groupe fondamental: Séminaire de géométrie algébrique du bois marie 1960-61 (SGA 1), pages 145-194, Lecture notes in mathematics 224
[18] Hardie, K. A., Kamps, K. H., Kieboom, R. W.: A homotopy 2-groupoid of a hausdorff space. Applied Categorical Structures, Vol. 8, 209-234
[19] Hermida, Claudio: Some properties of fib as a fibred 2-category. Journal of Pure and Applied Algebra, Vol. 134, Iss. 1, 83-109, DOI:10.1016/S0022-4049(97)00129-1 DOI 10.1016/S0022-4049(97)00129-1
[20] Jacobs, Bart: Categorical logic and type theory. Studies in logic and the foundations of mathematics, Elsevier
[21] Johnstone, P. T.: Fibrations and partial products in a 2-category. Applied Categorical Structures, Vol. 1, 141-179 DOI 10.1007/BF00880041
[22] Johnstone, P. T.: Sketches of an elephant: A topos theory compendium, volume 1. Oxford logic guides, Clarendon Press, London MR 1953060
[23] Johnstone, P. T.: Topos theory. Dover MR 0470019
[24] Kelly, G. M.: Basic concepts of enriched category theory. London math. Soc. Lecture notes series, Cambridge University Press, Cambridge
[25] Kelly, G. M.: Elementary observations on 2-categorical limits. Bulletin of the Austrailian Mathematical Soceity, Vol. 39, 301-317 DOI 10.1017/S0004972700002781
[26] Kelly, G. M., Street, Ross: Review of the elements of 2-categories. Category seminar: Proceedings sydney category theory seminar 1972 /1973, pages 75-103, Lecture notes in mathematics 420
[27] Lack, Stephen: A 2-categories companion. Towards higher categories, pages 105-191, The IMA volumes in mathematics and its applications 152 MR 2664622
[28] Lambert, Michael: An elementary account of flat 2-functors. PhD thesis, Dalhousie University
[29] Lambert, Michael: Discrete double fibrations. Theory and Applications of Categories, Vol. 37, Iss. 22, 671-708 DOI 10.70930/tac/zvxpxw4p | MR 4276762
[30] Loregian, Fosco, Riehl, Emily: Categorical notions of fibration. Expositiones Mathematicae, Vol. 38, 496-514 DOI 10.1016/j.exmath.2019.02.004 | MR 4177953
[31] Mac Lane, Saunders: Category theory for the working mathematician. Graduate texts in mathematics, Springer-Verlag, Berlin, DOI:10.1007/978-1-4757-4721-8 DOI 10.1007/978-1-4757-4721-8
[32] Mac Lane, Saunders, Moerdijk, Ieke: Sheaves in geometry and logic: A first introduction to topos theory. Springer-Verlag, DOI:10.1007/978-1-4612-0927-0 DOI 10.1007/978-1-4612-0927-0
[33] Moerdijk, Ieke, Svennson, Jan-Alve: Algebraic classification of equivariant homotopy 2-types, I. Journal of Pure and Applied Algebra, Vol. 89, Iss. 1-2, 187-216 DOI 10.1016/0022-4049(93)90094-A
[34] Shulman, Michael: Framed bicategories and monoidal fibrations. Theory and Applications of Categories, Vol. 20, Iss. 18, 650-673 DOI 10.70930/tac/2m83wy59 | MR 2534210
[35] Shulman, Mike: The problem with lax functors. Accessed: 2019-12-03, https://golem.ph.utexas.edu/category/2009/12/the_problem_with_lax_functors.html
[36] Street, Ross: The formal theory of monads. Journal of Pure and Applied Algebra, Vol. 2, Iss. 2, 149-168, DOI:10.1016/0022-4049(72)90019-9 DOI 10.1016/0022-4049(72)90019-9
[37] Street, Ross: Fibrations and Yoneda’s lemma in a 2-category. Category seminar: Proceedings sydney category theory seminar 1972 /1973, pages 104-133, Lecture notes in mathematics 420
[38] Street, Ross: Fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 21, Iss. 2, 111-160
[39] Street, Ross: Conspectus of variable categories. Journal of Pure and Applied Algebra, Vol. 21, Iss. 3, 307-338 DOI 10.1016/0022-4049(81)90021-9
[40] Weber, Mark: Yoneda structures from 2-toposes. Applied Categorical Structures, Vol. 15, 259-323, DOI:10.1007/s10485-007-9079-2 DOI 10.1007/s10485-007-9079-2 | MR 2320763
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