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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2010
   • (edited Dec 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is a variant of Dave Carchedi's question over on MO: for $C$ an $\infty$-category and $S$ a set of morphisms in it, we have the reflective localization $C_0 \hookrightarrow C$ given by the subcategory of all objects such that homming an $s \in S$ into them gives an equivalence. But one can also look at the entire sequence $$ C_0 \hookrightarrow C_1 \hookrightarrow C_2 \hookrightarrow \cdots \hookrightarrow C $$ where $C_n$ is the full subcategory on those objects such that homming an $s \in S$ into them produces an $(n-2)$-truncated $\infty$-functor. So for instance if $C$ is $\infty$-presheaves and $S$ is suitably left-exact, then $C_0$ is $\infty$-sheaves and $C_1$ is separated $\infty$-presheaves. I know some reasons to be interested in $C_1$. This sort of implies some reasons to be interested in the other $C_n$. 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.)

   Here is a variant of Dave Carchedi’s question over on MO:

   for $C$ an $\infty$-category and $S$ a set of morphisms in it, we have the reflective localization $C_0 \hookrightarrow C$ given by the subcategory of all objects such that homming an $s \in S$ into them gives an equivalence.

   But one can also look at the entire sequence

   $$C_0 \hookrightarrow C_1 \hookrightarrow C_2 \hookrightarrow \cdots \hookrightarrow C$$

   where $C_n$ is the full subcategory on those objects such that homming an $s \in S$ into them produces an $(n-2)$-truncated $\infty$-functor.

   So for instance if $C$ is $\infty$-presheaves and $S$ is suitably left-exact, then $C_0$ is $\infty$-sheaves and $C_1$ is separated $\infty$-presheaves.

   I know some reasons to be interested in $C_1$. This sort of implies some reasons to be interested in the other $C_n$. 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.)

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexlink to what I guess is the question: [The plus construction for stacks of n-types](http://mathoverflow.net/questions/47615/the-plus-construction-for-stacks-of-n-types).

   link to what I guess is the question: The plus construction for stacks of n-types.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
   • (edited Dec 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe question I am referring to is <a href="http://mathoverflow.net/questions/48567/advantages-of-diffeological-spaces-over-general-sheaves">Advantages of Diffeological Spaces over General Sheaves</a>

   The question I am referring to is Advantages of Diffeological Spaces over General Sheaves

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAh, ok. I thought it might be that one, but couldn't see the relation.

   Ah, ok. I thought it might be that one, but couldn’t see the relation.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> but couldn't see the relation. Do you see it now, or should I say more?

   but couldn’t see the relation.

   Do you see it now, or should I say more?

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 8th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorDavidCarchedi
   • CommentTimeDec 9th 2010
   • PermaLink
   Author: DavidCarchedi
   Format: TextWell, David is right, in the sense that, as my question on MO stands, there is no clear link. But what Urs is asking, is something else I "secretly" want to know. However, I first want to get a good handle on the baby-question at the level of 1-categories. To gives some "concreteness" (and pardon the horrible pun) to this, diffeological spaces are concrete sheaves, which we can interpret this in the following way: Consider the functor from the terminal category * to Cartesian manifolds, which "picks out" the one-point manifold. This induces a geometric morphism from Set to sheaves on Cartesian manifolds. It is in fact, a geometric embedding, hence corresponds to a Lawvere-Tierney topology on the sheaf topos. Calling this topology J, diffeological spaces are nothing more than the J-separated objects. Ok, so, what makes these "nicer" or "closer to manifolds" than general sheaves? it doesn't seem to be that they lend themselves any better to applying concepts from differential geometry to them (with the possible exception of tangent spaces?). They certainly have underlying sets which is "nice", but, not essential for geometry. Nonetheless, there should be "something there", of why at looking at this quasi-topos of J-separated guys is a good idea. One of the answers to my MO question seems to go in the right direction, namely, that these guys are some how "tamer" than "general wild sheaves". But, I'd like to make this precise.

   Well, David is right, in the sense that, as my question on MO stands, there is no clear link. But what Urs is asking, is something else I "secretly" want to know. However, I first want to get a good handle on the baby-question at the level of 1-categories. To gives some "concreteness" (and pardon the horrible pun) to this, diffeological spaces are concrete sheaves, which we can interpret this in the following way:
   
   Consider the functor from the terminal category * to Cartesian manifolds, which "picks out" the one-point manifold. This induces a geometric morphism from Set to sheaves on Cartesian manifolds. It is in fact, a geometric embedding, hence corresponds to a Lawvere-Tierney topology on the sheaf topos. Calling this topology J, diffeological spaces are nothing more than the J-separated objects.
   
   Ok, so, what makes these "nicer" or "closer to manifolds" than general sheaves? it doesn't seem to be that they lend themselves any better to applying concepts from differential geometry to them (with the possible exception of tangent spaces?). They certainly have underlying sets which is "nice", but, not essential for geometry. Nonetheless, there should be "something there", of why at looking at this quasi-topos of J-separated guys is a good idea. One of the answers to my MO question seems to go in the right direction, namely, that these guys are some how "tamer" than "general wild sheaves". But, I'd like to make this precise.