The elementary construction of formal anafunctors

Document Type : Research Paper


School of Mathematical Sciences, The University of Adelaide, Adelaide, Australia


This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.


[1] Bartels, T., Higher gauge theory I: 2-Bundles, Ph.D. thesis, University of California Riverside, 2006, arXiv:math.CT/0410328.
[2] Garner, R. and Bourke, J., Two-dimensional regularity and exactness, J. Pure Appl. Algebra 218 (2014), 1346-1371, arXiv:1304.5275, doi:10.1016/j.jpaa.2013.11.021.
[3] Johnson, N. and Yau, D., "2-Dimensional Categories", Oxford University Press, in press. Draft version available at arXiv:2002.06055.
[4] Karagila, A., Embedding orders into cardinals with DC_k, Fund. Math. 226 (2014), 143-156, arXiv:1212.4396, doi:10.4064/fm226-2-4.
[5] Land, M., Nikolaus, T., and SzumiƂo, K., Localization of cofibration categories and groupoid C*-algebras, Algebr. Geom. Topol. 17(5) (2017), 3007-3020, arXiv:1609.03805, doi:10.2140/agt.2017.17.3007.
[6] Mac Lane, S., "Categories for the working mathematician", Springer-Verlag, 1971, doi:10.1007/978-1-4757-4721-8.
[7] Makkai, M., Avoiding the axiom of choice in general category theory, J. Pure Appl. Algebra 108 (1996), 109-173, doi:10.1016/0022-4049(95)00029-1.
[8] Pronk, D., Etendues and stacks as bicategories of fractions, Compositio Math. 102(3) (1996), 243-303,
[9] Pronk, D. and Warren, M., Bicategorical fibration structures and stacks, Theory Appl. Categ. 29(29) (2014), 836-873, arXiv:1303.0340.
[10] Roberts, D.M., Internal categories, anafunctors and localisation, Theory Appl. Categ. 26(29) (2012), 788-829, arXiv:1101.2363.
[11] Roberts, D.M., The weak choice principle WISC may fail in the category of sets, Studia Logica 103 (2015), 1005-1017, arXiv:1311.3074, doi:10.1007/s11225-015-9603- 6.
[12] Roberts, D.M., On certain 2-categories admitting localisation by bicategories of fractions, Appl. Categ. Structures 24(4) (2016), 373-384, arXiv:1402.7108, doi:10.1007/s10485-015-9400-4.
[13] Simpson, C., Explaining Gabriel-Zisman localization to the computer, J. Automat. Reason. 36(3) (2006), 259-285, arXiv:math/0506471, doi:10.1007/s10817-006-9038- x.
[14] Street, R., Two-dimensional sheaf theory, J. Pure Appl. Algebra 23(3) (1982), 251- 270, doi:10.1016/0022-4049(82)90101 3.
[15] van den Berg, B. and Moerdijk, I., The axiom of multiple choice and models for constructive set theory, J. Math. Log. 14(1) (2014), arXiv:1204.4045, doi:10.1142/50219061314500056.