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2-stacks over bisites

Elena Caviglia

Abstract

We generalize the concept of stack one dimension higher, introducing a notion of 2-stack suitable for a trihomomorphism from a 2-category equipped with a bitopology into the tricategory of bicategories. Moreover, we give a characterization of 2-stacks in terms of explicit conditions, that are easier to use in practice. These explicit conditions are effectiveness conditions for appropriate data of descent on objects, morphisms and 2-cells, generalizing the usual stacky gluing conditions one dimension higher. Furthermore, we prove some new results on bitopologies. The main one is that every object of a subcanonical bisite can be seen as the sigma-bicolimit of each covering bisieve over it. This generalizes one dimension higher a well-know result for subcanonical Grothendieck sites.

2-stacks over bisites

Abstract

We generalize the concept of stack one dimension higher, introducing a notion of 2-stack suitable for a trihomomorphism from a 2-category equipped with a bitopology into the tricategory of bicategories. Moreover, we give a characterization of 2-stacks in terms of explicit conditions, that are easier to use in practice. These explicit conditions are effectiveness conditions for appropriate data of descent on objects, morphisms and 2-cells, generalizing the usual stacky gluing conditions one dimension higher. Furthermore, we prove some new results on bitopologies. The main one is that every object of a subcanonical bisite can be seen as the sigma-bicolimit of each covering bisieve over it. This generalizes one dimension higher a well-know result for subcanonical Grothendieck sites.
Paper Structure (1 section, 2 theorems, 2 equations)

This paper contains 1 section, 2 theorems, 2 equations.

Table of Contents

  1. Preliminaries on tricategories

Key Result

Theorem 2

A trihomomorphism $F\colon \mathpzc{K}\xspace^{\operatorname{op}} \to \mathpzc{Bicat}\xspace$ is a 2-stack if and only if for every $C\in \mathpzc{K}\xspace$ and every covering bisieve $S\in \tau(C)$ the following conditions are satisfied:

Theorems & Definitions (4)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1