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David Carchedi

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Name David Carchedi
Member for 3 years
Seen 16 hours ago
Website
Location Bonn
Age 31
I am a research postdoc at the Max Planck Institute for Mathematics. I have recently finished my PhD at Utrecht University under the advisement of Ieke Moerdijk. My thesis was entitled "Categorical Properties of Topological and Differentiable Stacks". Some of my current research is in this or similar areas, but I am also branching out into derived geometry, and higher stacks. Furthermore, I have an interest in (higher) operads / Dendroidal sets.
Jun
4
 awarded ● Nice Question
Jun
3
 revised Are reflective subcategories of complete infinity categories complete?
added 51 characters in body
Jun
3
 comment Are reflective subcategories of complete infinity categories complete?
@Dylan: The proposition says something much simpler: left adjoints preserves colimits and right adjoints preserve limits. I need to knwo that limits exist before I can show they are preserved.
Jun
1
 accepted Co-completeness of differential stacks?
May
30
 comment Co-completeness of differential stacks?
P.S. Exactly what types of colimits are you wanting to exist/compute?
May
30
 comment Co-completeness of differential stacks?
Just because there are 2-colimits of topological groupoids does not mean that there are 2-colimits of topological stacks. The Yoneda embedding generally fails to preserve colimits!
May
30
 asked Is there a name for (pre)sheaves satisfying this transitivity condition?
May
30
 answered Co-completeness of differential stacks?
May
20
 comment Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.
(ofc, the notion of homotopy colimit I mean, is only well defined up to weak equivalence).
May
20
 comment Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.
I didn't realize the phrase "homotopy colimit" had any ambiguity. I am thinking about the model category structure on $Top$ (i.e. the colimit in the associated infinity category). At any rate, is $M \wr M$ notation for the "action category of $M$ on itself", i.e. the Grothendieck construction of the functor $M \to Set$ sending $*$ to $M$ and each $m \in M$ to the morphism $m:M \to M$ induced by composition? If so, this is what I hoped would work! Anyhow, thanks for your references, and for the spectral sequence! (Btw, where can I find a reference for this SS?)
May
20
 asked Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.
May
20
 comment `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$.
@Lennart: What is a reference for this neat fact? Thanks!
May
18
 comment Is the site of (smooth) manifolds hypercomplete?
Very nice Marc! I'm glad this works for the continuous case too, where one does not have good covers.
May
18
 comment Reference request: sheaves on closed sets
P.S. "I don't need anything about injective resolutions or making the category of sheaves into an abelian category."- Well, you have it even if you don't need it :)
May
18
 comment What is the theory of local rings and local ring homomorphisms?
You may want to read about the concept of a "geometry" introduced in Lurie's DAG V. It was invented precisely to get around this problem.
May
17
 comment What are the smooth manifolds in the topos of sheaves on a smooth manifold?
@Dmitry: I'm curious why you are thinking about this. What is your motivation?
May
17
 comment Vector fields on a simplicial manifold.
P.S. depending on your motivation, you may want to take a section over a hypercover of $M_\bullet,$ e.g. if you are trying to model a vector field on the associated higher differentiable stack.
May
17
 accepted Vector fields on a simplicial manifold.
May
17
 comment Vector fields on a simplicial manifold.
You may also be interested in this: arxiv.org/abs/0810.0979
May
17
 comment Is the site of (smooth) manifolds hypercomplete?
@Andre: But we are saved by the existence of good covers :)
May
17
 answered Is the site of (smooth) manifolds hypercomplete?
May
17
 comment Reference request: sheaves on closed sets
The upshot is, your definition is equivalent to being a sheaf on $K(X)$ for a certain Grothendieck topology (which I only described implicitly), so it lies in the same world of ordinary sheaves- in topos theory. For many things, you can probably dance around without talkig about this topology, and just use the definition you gave, except, if you want to define Cech cohomology, you will have to bite the bullet and define the covers explicitly. However, you should probably use sheaf cohomology anyway, instead.
May
17
 answered Vector fields on a simplicial manifold.
May
17
 revised Is the site of (smooth) manifolds hypercomplete?
edited tags; edited tags
May
17
 comment Is the site of (smooth) manifolds hypercomplete?
@Andre: You get the same theory of sheaves, but maybe not of infinity-sheaves. You can have a subsite whose topos of sheaves is equivalent to the whole topos of sheaves, but whose topos of infinity-sheaves is not equivalent to the whole infinity topos of infinity sheaves. This can't happen if everything is hypercomplete, however, this is what he is trying to prove, so we go in a circle.
May
17
 comment Is the site of (smooth) manifolds hypercomplete?
It suffices to show that the contractible opens generate the infinity topos under colimits in a canonical way, i.e. that it is a dense sub -infinity category.
May
17
 revised Reference request: sheaves on closed sets
added category theory tag
May
17
 revised Reference request: sheaves on closed sets
fixed a big mistake
May
17
 answered Reference request: sheaves on closed sets
May
17
 comment Reference request: sheaves on closed sets
Could you spell out what you mean by (satisfying the sheaf property for finite unions of compact sets, plus an extra "continuity" condition)?
May
16
 comment Grothendieck fibrations and classifying spaces
@Ronnie: You can still tell me the reference though, for "general knowledge". Thanks!
May
16
 comment Grothendieck fibrations and classifying spaces
@Ronnie: Thanks. Unfortunately, I have a specific goal in mind, and in the case I care about, $\mathcal{C}$ is not a groupoid and has a very interesting homotopy type.
May
16
 revised Grothendieck fibrations and classifying spaces
added 212 characters in body
May
16
 answered Grothendieck fibrations and classifying spaces
May
15
 comment Grothendieck fibrations and classifying spaces
(However, I'm not sure how computationally tractable this is, e.g. for computing homotopy groups, cohomology groups, etc.)
May
15
 comment Grothendieck fibrations and classifying spaces
Yes, Benjamin, thanks. That is right. That is certainly one result in this direction. This also implies that $\BC$ is the homotopy colimit of constant functor from $\mathcal{C}^{op}$ to spaces, with value the terminal object.
May
15
 asked Grothendieck fibrations and classifying spaces
May
13
 awarded ● Popular Question
May
10
 awarded ● Nice Question
Apr
23
 comment How does the machinery of left-exact comonads generalize from sheaves to stacks?
I'll try to come back and answer this later when I have more time (it's late here), but for now: since you get a morphism between the toposes of sheaves on the two sites, you get a morphism between their 2-toposes of stacks, since there is a full and faithful embedding of the bicategory of toposes into the the tricategory of 2-toposes.
Apr
16
 revised Constructing a stack (gerbe) from a connected groupoid
added 58 characters in body
Apr
15
 answered Constructing a stack (gerbe) from a connected groupoid
Apr
12
 revised universal families and maps to quotient stacks
added 16 characters in body
Apr
11
 comment universal families and maps to quotient stacks
By the way, in some more detail, there is a canonical map $$\left[X//G\right] \to \left[pt//G\right]=BG$$, which is faithful. This is how you get a $G$-torsor out of a map $$S \to \left[X//G\right]$$ (it is classified by the composition into $\left[pt//G\right].$) I should be a bit careful, since I am used to working with topological stacks, but I think everything should work.
Apr
11
 comment universal families and maps to quotient stacks
What do you mean by "universal"? To me, universal means that every family is a pullback of it, which is not true here, but is locally- that's why I said locally universal. If instead, you mean canonical, then sure.
Apr
11
 accepted universal families and maps to quotient stacks
Apr
11
 answered universal families and maps to quotient stacks
Apr
4
 comment Are there non-categorical notions in topos theory?
(sorry, $\mathcal{T}\left(E,\mathcal{O}\right)$ is equivalent to $\phi\left(E\right)$, of course).
Apr
4
 comment Are there non-categorical notions in topos theory?
I.e. the functor $$\mathcal{T} \to \mathbf{Cat}$$ sending $E$ to $$Nat\left(\mathbf{FinSet}^{op},\mathcal{T}\left(1,E\right)\right)$$ is corepresentable by an object $\mathcal{O}$ such that for each $E$ in $\mathcal{T}$, $\mathcal{T}\left(\mathcal{O},E\right)$ is equivalent to $\phi\left(E\right)$. Nice!
Apr
4
 comment Are there non-categorical notions in topos theory?
I edited the title. I was merely following the terminology to which I was exposed (yes through the internet). Anyway, from my understanding, the terminology "evil" is only applied to notions which are from category theory, so I think there is little notion of offending non-category theorists.