nLab
split hypercover
Contents
Context
Model category theory
model category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of ( ∞ , 1 ) (\infty,1) -categories
Model structures
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for ( ∞ , 1 ) (\infty,1) -sheaves / ∞ \infty -stacks
( ∞ , 1 ) (\infty,1) -Topos Theory
(∞,1)-topos theory
Background
Definitions
elementary (∞,1)-topos
(∞,1)-site
reflective sub-(∞,1)-category
(∞,1)-category of (∞,1)-sheaves
(∞,1)-topos
(n,1)-topos , n-topos
(∞,1)-quasitopos
(∞,2)-topos
(∞,n)-topos
Characterization
Morphisms
Extra stuff, structure and property
hypercomplete (∞,1)-topos
over-(∞,1)-topos
n-localic (∞,1)-topos
locally n-connected (n,1)-topos
structured (∞,1)-topos
locally ∞-connected (∞,1)-topos , ∞-connected (∞,1)-topos
local (∞,1)-topos
cohesive (∞,1)-topos
Models
Constructions
structures in a cohesive (∞,1)-topos
Contents
Idea
A split hypercover is a cofibrant resolution of a representable in the projective local model structure on simplicial presheaves [ C op , sSet ] proj , loc [C^{op}, sSet]_{proj,loc} over a site C C .
It is a hypercover satisfying an extra condition that roughly says that it is degreewise freely given by representables.
Definition
Regard X ∈ C X \in C under the Yoneda embedding as an object X ∈ [ C op , sSet ] proj , loc X \in [C^{op}, sSet]_{proj,loc} . Then a morphism ( Y → X ) ∈ [ C op , sSet ] (Y \to X) \in [C^{op}, sSet] is a split hypercover of X X if
Y Y is a hypercover in that
Y Y is degreewise a coproduct of representables ,
Y = ∫ [ n ] ∈ Δ Δ [ n ] ⋅ ∐ i n U i n , with { U i n ∈ C } Y = \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_n} U_{i_n} \;\,,
\;\;\; with \{U_{i_n} \in C\} ;
with Y → X Y \to X regarded as a presheaf of augmented simplicial set s, for all n ∈ ℕ n \in \mathbb{N} the morphism Y n + 1 → ( cosk n Y ) n + 1 Y_{n+1} \to (\mathbf{cosk}_n Y)_{n+1} into the n + 1 n+1 -cells of the n n -coskeleton is a local epimorphism with respect to the given Grothendieck topology on C C
Y Y is split in that the image of the degeneracy maps identifies with a direct summand in each degree.
Properties
The splitness condition on the hypercover is precisely such that Y Y becomes a cofibrant object in [ C op , sSet ] proj , loc [C^{op}, sSet]_{proj,loc} , according to the characterization of such cofibrant objects described here .
Examples
Over the site CartSp , the Cech nerve of an open cover becomes split as a height-0 hypercover precisely if the cover is a good open cover .
References
Last revised on August 31, 2010 at 03:56:17.
See the history of this page for a list of all contributions to it.