Abstract Hazratpour

The notion of (op)fibration in the 2-category of toposes and geometric morphisms have close connections with topological properties. For example, every local homeomorphism is an opfibration. This connection is in line with the conception of toposes as generalized spaces. To study fibrations of toposes, Johnstone [1]  has defined fibrations internal to 2-categories.  Johnstone's definition requires neither strictness of 2-categories nor the existence of the structure of strict pullbacks and comma objects in them. Indeed, this definition is particularly well-suited for the class of 2-categories such as 2-category of bounded toposes in which we do not expect diagrams of 1-cells to commute strictly. Moreover, Johnstone also established the equivalence of his definition with the representable definition of internal fibrations. However, this definition is rather complicated and difficult to work with in practice.

First, I will give a simpler and syntactic definition of fibration in the Vickers's 2-category of contexts for arithmetic universes [2,3]. This syntactic formulation is based on Chevalley’s internal characterization of fibrations obtained as a theorem in [4].  I shall then describe how our finitary syntactic definition of (op)fibrations will give rise to (op)fibrations of toposes in the sense of Johnstone.

 [1] Johnstone, P. (1993), ‘Fibrations and partial products in a 2-category’, AppliedCategorical Structures Vol.1, 141–179

 [2] Vickers, S. (2017), ‘Arithmetic universes and classifying toposes’. Cahiers de topologie et géométrie différentielle catégorique 58 no. 4 (2017), pp. 213-248

 [3] Vickers, S. (2016), ‘Sketches for arithmetic universes’.URL: https://arxiv.org/abs/1608.01559

 [4] Street, R. (1974), ‘Fibrations and yoneda’s lemma in a 2-category’, Lecture Notes in Math., Springer, Berlin Vol.420, 104–133.

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