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Principal 2-bundles and quotient 2-stacks

Elena Caviglia

Abstract

We generalize principal bundles and quotient stacks to the two-categorical context of bisites. We introduce a notion of principal 2-bundle that makes sense for a 2-category with finite flexible limits, endowed with a bitopology. We then use principal 2-bundles to explicitly construct quotient-pre-2-stacks, which are the analogues of quotient stacks one dimension higher. In order to perform this construction, we prove that principal 2-bundles are closed under iso-comma objects and we restrict ourselves to (2,1)-categories. Finally, we prove that, if the bisite is subcanonical and the underlying (2,1)-category satisfies some mild conditions, quotient pre-2-stacks are 2-stacks.

Principal 2-bundles and quotient 2-stacks

Abstract

We generalize principal bundles and quotient stacks to the two-categorical context of bisites. We introduce a notion of principal 2-bundle that makes sense for a 2-category with finite flexible limits, endowed with a bitopology. We then use principal 2-bundles to explicitly construct quotient-pre-2-stacks, which are the analogues of quotient stacks one dimension higher. In order to perform this construction, we prove that principal 2-bundles are closed under iso-comma objects and we restrict ourselves to (2,1)-categories. Finally, we prove that, if the bisite is subcanonical and the underlying (2,1)-category satisfies some mild conditions, quotient pre-2-stacks are 2-stacks.
Paper Structure (1 section, 1 theorem, 2 equations)

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

Table of Contents

  1. Principal 2-bundles

Key Result

Theorem 1

Let $\mathpzc{K}\xspace$ be a bicocomplete $(2,1)$-category with finite flexible limits and such that iso-comma objects preserve bicolimits. Let then $\tau$ be a subcanonical bitopology on $\mathpzc{K}\xspace$ and $\mathcal{X}\xspace,\mathcal{G}\xspace\in \mathpzc{K}\xspace$ with $\mathcal{G}\xspace

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2: iso-comma objects
  • Definition 1.3