nLab
split hypercover
Contents
Context
Model category theory
model category , model $\infty$ -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$ -categories

Model structures

for $\infty$ -groupoids

for ∞-groupoids

for equivariant $\infty$ -groupoids

for rational $\infty$ -groupoids

for rational equivariant $\infty$ -groupoids

for $n$ -groupoids

for $\infty$ -groups

for $\infty$ -algebras

general $\infty$ -algebras

specific $\infty$ -algebras

for stable/spectrum objects

for $(\infty,1)$ -categories

for stable $(\infty,1)$ -categories

for $(\infty,1)$ -operads

for $(n,r)$ -categories

for $(\infty,1)$ -sheaves / $\infty$ -stacks

$(\infty,1)$ -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 $[C^{op}, sSet]_{proj,loc}$ over a site $C$ .

It is a hypercover satisfying an extra condition that roughly says that it is degreewise freely given by representables.

Definition
Regard $X \in C$ under the Yoneda embedding as an object $X \in [C^{op}, sSet]_{proj,loc}$ . Then a morphism $(Y \to X) \in [C^{op}, sSet]$ is a split hypercover of $X$ if

$Y$ is a hypercover in that

$Y$ is degreewise a coproduct of representables ,

$Y = \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_n} U_{i_n} \;\,,
\;\;\; with \{U_{i_n} \in C\}$ ;

with $Y \to X$ regarded as a presheaf of augmented simplicial set s, for all $n \in \mathbb{N}$ the morphism $Y_{n+1} \to (\mathbf{cosk}_n Y)_{n+1}$ into the $n+1$ -cells of the $n$ -coskeleton is a local epimorphism with respect to the given Grothendieck topology on $C$

$Y$ 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 $Y$ becomes a cofibrant object in $[C^{op}, sSet]_{proj,loc}$ , 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 May 1, 2021 at 05:00:14.
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