nLab axiom of stack completions

Axiom of stack completion

Contents

Definition

An internal category $\mathbb{A}$ in an elementary topos $\mathcal{E}$ is called an (intrinsic) stack if the indexed category that it represents, $(X\in \mathcal{E}) \mapsto \mathcal{E}(X,\mathbb{A}) \in Cat$, 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 $f:\mathbb{A} \to \mathbb{B}$ is a weak equivalence and $\mathbb{C}$ is a stack, then the induced functor $\mathrm{Cat}(\mathcal{E})(\mathbb{B},\mathbb {C})\to \mathrm{Cat}(\mathcal{E})(\mathbb{A},\mathbb{C})$ 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).

References

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

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

Created on May 23, 2021 at 15:51:29. See the history of this page for a list of all contributions to it.