nLab axiom of stack completions

Axiom of stack completion

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).

Examples

• If $\mathcal{E}$ satisfies the axiom of choice, then every internal category is a stack, so ASC holds trivially.

• If $\mathcal{E}$ is a Grothendieck topos, then internal stack completions can be constructed using the small object argument, as fibrant replacements in the model structure on $\mathrm{Cat}(\mathcal{E})$ constructed by Joyal and Tierney, so it satisfies ASC.

References

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

• André Joyal, Myles Tierney, Strong stacks and classifying spaces, Lecture Notes in Mathematics 1488, Springer (1991) [doi:10.1007/BFb0084222]