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Here is a variant of Dave Carchedi’s question over on MO:
for C an ∞-category and S a set of morphisms in it, we have the reflective localization C0↪C given by the subcategory of all objects such that homming an s∈S into them gives an equivalence.
But one can also look at the entire sequence
C0↪C1↪C2↪⋯↪Cwhere Cn is the full subcategory on those objects such that homming an s∈S into them produces an (n−2)-truncated ∞-functor.
So for instance if C is ∞-presheaves and S is suitably left-exact, then C0 is ∞-sheaves and C1 is separated ∞-presheaves.
I know some reasons to be interested in C1. This sort of implies some reasons to be interested in the other Cn. But the question is:
what (if any) is a good general abstract characterization of interesting properties that such filtrations have? (Vague question, I know, but still.)
link to what I guess is the question: The plus construction for stacks of n-types.
The question I am referring to is Advantages of Diffeological Spaces over General Sheaves
Ah, ok. I thought it might be that one, but couldn’t see the relation.
but couldn’t see the relation.
Do you see it now, or should I say more?
I know that David’s question is about the category of sheaves being filtered by ’niceness’ of its objects, and I can imagine a higher version of this, but I can’t see how it relates to your question. There’s no pressure to clarify, unless you think it will lead to better insight.
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