Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorZhen Lin
   • CommentTimeApr 26th 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexThere seem to be some misleading remarks at [[Čech model structure on simplicial presheaves]]. > Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology. [Marc Hoyois](http://mathoverflow.net/a/150431/11640) seems to says the opposite: there is no deep relation between "Čech" in "Čech cohomology" and in "Čech model structure". > [...] the corresponding Čech cover morphism . > > Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects. The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don't know how to prove it otherwise. Of course, injective-cofibrancy is trivial. > this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be Based on the discussion [here](http://nforum.mathforge.org/discussion/4457/), it seems that the Čech model structure is _not_ site-independent, even though it can be defined on the category of simplicial _sheaves_. A very strange state of affairs...

   There seem to be some misleading remarks at Čech model structure on simplicial presheaves.

   Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.

   Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.

   […] the corresponding Čech cover morphism .

   Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.

   The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.

   this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be

   Based on the discussion here, it seems that the Čech model structure is not site-independent, even though it can be defined on the category of simplicial sheaves. A very strange state of affairs…

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 26th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for catching that bit about Cech cohomology. I thought I had fixed that long ago, but apparently I didn't in this entry. I have just removed that remark now. Yes, the full story is that for Cech cohomology to compute the correct hom-spaces (in addition to the site being "small enough" that the coefficient complex of sheaves satisfies proper descent) that the Cech nerve needs to be cofibrant in the local projective Cech model structure. This is a generalized version of the familiar condition that the cover needs to be a "good cover" for Cech cohomology to compute sheaf cohomology.

   Thanks for catching that bit about Cech cohomology. I thought I had fixed that long ago, but apparently I didn’t in this entry. I have just removed that remark now.

   Yes, the full story is that for Cech cohomology to compute the correct hom-spaces (in addition to the site being “small enough” that the coefficient complex of sheaves satisfies proper descent) that the Cech nerve needs to be cofibrant in the local projective Cech model structure. This is a generalized version of the familiar condition that the cover needs to be a “good cover” for Cech cohomology to compute sheaf cohomology.