Abstract
Pronk’s theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness. This is the published version of arXiv:1402.7108.
Similar content being viewed by others
References
Abbad, O., Vitale, E.: Faithful calculus of fractions. Cahiers de Topologie et Géométrie Différentielle Catégoriques 54, 221–239 (2013)
Bartels, T.: Higher gauge theory I: 2-Bundles. Ph.D. thesis, University of California Riverside (2006). arXiv:math.CT/0410328
van den Berg, B., Moerdijk, I.: The axiom of multiple choice and models for constructive set theory. J. Math. Log. 14(1) (2014). arXiv:1204.4045
Bunge, M., Paré, R.: Stacks and equivalence of indexed categories. Cahiers Topologie Géom Différentielle 20(4), 373–399 (1979)
Karagila, A.: Embedding orders into cardinals with D C κ . Fundamenta Mathematicae 226, 143–156 (2014). arXiv:1212.4396
Leinster, T.: Basic bicategories (1998). arXiv:math.CT/9810017
Makkai, M.: Avoiding the axiom of choice in general category theory. J. Pure Appl. Algebra 108, 109–173 (1996)
Pronk, D.: Etendues and stacks as bicategories of fractions. Compos. Math. 102(3), 243–303 (1996)
Pronk, D., Warren, M.: Bicategorical fibration structures and stacks. Theory and Applications of Categories 29(29), 836–873 (2014)
Roberts, D.M.: Internal categories, anafunctors and localisation. Theory Appl. Categ. 26(29), 788–829 (2012). arXiv:1101.2363
Roberts, D.M.: The weak choice principle WISC may fail in the category of sets. Studia Logica (2015). doi:10.1007/s11,225-015-9603-6. arXiv:1311.3074
Shulman, M.: Exact completions and small sheaves. Theory Appl. Categ. 27, 97–173 (2012)
Street, R.: Two-dimensional sheaf theory. J. Pure Appl. Algebra 23(3), 251–270 (1982). doi:10.1016/0022-4049(82)90101-3
The Stacks Project Authors: Stacks Project. http://stacks.math.columbia.edu (2015)
Tommasini, M.: Some insights on bicategories of fractions - I (2014). arXiv:1410.3990
Vistoli, A.: Grothendieck topologies, fibred categories and descent theory. In: Fundamental algebraic geometry. Math. Surveys. Monogr. 123, 1–104. Amer. Math. Soc., Providence, RI (2005). arXiv:math/0412512. Available from, http://www.homepage.sns.it/vistoli/descent.pdf
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Roberts, D.M. On Certain 2-Categories Admitting Localisation by Bicategories of Fractions. Appl Categor Struct 24, 373–384 (2016). https://doi.org/10.1007/s10485-015-9400-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-015-9400-4