An internal category in an elementary topos is called an (intrinsic) stack if the indexed category that it represents, , is a stack for the regular topology of . 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 . However, if is a weak equivalence and is a stack, then the induced functor is an equivalence.
We say that 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).
If satisfies the axiom of choice, then every internal category is a stack, so ASC holds trivially.
If is a Grothendieck topos, then internal stack completions can be constructed using the small object argument, as fibrant replacements in the model structure on constructed by Joyal and Tierney, so it satisfies ASC.
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]
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