9
votes
Is the geometric realization of simplicial functors interesting?
If I understand your question correctly, you may find some interesting results of a similar nature in the paper by Vogt (Homotopy limits and colimits, Math. Z., 134, (1973), 11 – 52.) which was ...
- 9,597
5
votes
Accepted
Are simplicial abelian sheaves fibrant?
Fibrant in what model structure?
Simplicial abelian sheaves (and presheaves) are fibrant
in the projective model structure because
all simplicial abelian groups are fibrant.
Simplicial abelian ...
- 37.8k
4
votes
Accepted
Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category?
Section 2 of this paper of Rezk addresses exactly the question of when the localization by S yields a Cartesian model category. For that the relevant property is that that if you take the product of a ...
- 27.5k
3
votes
Accepted
Matching objects and hypercovers in topology
This follows from the construction of $(\operatorname{cosk}_m(X_\bullet))_n$ as a limit indexed by $(\Delta/[n])^{\operatorname{op}}_{\leq m}$ (in the non-augmented case) or $(\Delta_a/[n])^{\...
- 28.7k
3
votes
Accepted
Homotopy quotients, fixed points and stalks of simplicial (pre)sheaves
Taking stalks always commutes with taking homotopy orbits,
since filtered colimits of simplicial sets are also filtered homotopy colimits, and homotopy colimits commute with homotopy colimits.
Taking ...
- 37.8k
2
votes
Accepted
An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology
Some hints in the literature led me to an answer, which I find a bit surprising:
One can take $U'_\bullet=\mathrm{cosk}_0(U_\bullet)$ and the $1$-cocycle $\alpha\in Z^1(U_\bullet,A)\subseteq A(U_1)$ ...
- 2,928
2
votes
Accepted
Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
There are two ways to make this construction work.
The first way is to iterate the step $F↦F^†$ transfinitely many times.
The reason that a single iteration of $F↦F^†$ is not sufficient
is that while $...
- 37.8k
1
vote
Accepted
Injective model structure for simplicial presheaves
To answer the question as it is stated: $U$ is an object in a locally presentable category, therefore $U$ is a small object, hence the corepresentable functor of $U$ preserves $α$-filtered colimits ...
- 37.8k
1
vote
Can homotopy limits of simplicial sheaves be calculated (correctly) using sheaves of Kan complexes?
Let me preface by saying that I am a relative novice in these matters. In particular, if something below is confusing, do point out, since it is likely that it is a reflection of me screwing something ...
- 563
1
vote
Accepted
Whitehead Theorem in $\mathbb{A}^1$-homotopy theory
The condition you've stated implies that the homotopy sheaves are equivalent, and it is implied by the map being a weak equivalence, so they are equivalent. You're nullifying $\mathbb{A}^1$ in the $\...
- 19.6k
Only top scored, non community-wiki answers of a minimum length are eligible