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axiom of stack completions

Axiom of stack completion

Axiom of stack completion

Contents

Definition

An internal category 𝔸\mathbb{A} in an elementary topos \mathcal{E} is called an (intrinsic) stack if the indexed category that it represents, (X)(X,𝔸)Cat(X\in \mathcal{E}) \mapsto \mathcal{E}(X,\mathbb{A}) \in Cat, is a stack for the regular topology of \mathcal{E}. An internal functor is called a weak equivalence if it is internally essentially surjective and fully faithful; this is strictly weaker than being an internal equivalence if the axiom of choice does not hold in \mathcal{E}. However, if f:𝔸𝔹f:\mathbb{A} \to \mathbb{B} is a weak equivalence and \mathbb{C} is a stack, then the induced functor Cat()(𝔹,)Cat()(𝔸,)\mathrm{Cat}(\mathcal{E})(\mathbb{B},\mathbb {C})\to \mathrm{Cat}(\mathcal{E})(\mathbb{A},\mathbb{C}) is an equivalence.

We say that \mathcal{E} satisfies the axiom of stack completions (ASC) if every internal category admits a weak equivalence to an internal category that is a stack (see Bunge-Hermida).

Examples

References

  • Bunge and Hermida, “Pseudomonadicity and 2-stack completions”

  • Joyal and Tierney, “Strong stacks and classifying spaces”

Created on May 23, 2021 at 11:51:29. See the history of this page for a list of all contributions to it.