Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Nerve complex
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==The Čech nerve== Given an [[open cover]] <math>C=\{U_i: i\in I\}</math> of a topological space <math>X</math>, or more generally a cover in a [[Grothendieck topology|site]], we can consider the pairwise [[Pullback_(category_theory)|fibre products]] <math>U_{ij}=U_i\times_XU_j</math>, which in the case of a topological space are precisely the intersections <math>U_i\cap U_j</math>. The collection of all such intersections can be referred to as <math>C\times_X C</math> and the triple intersections as <math>C\times_X C\times_X C</math>. By considering the natural maps <math>U_{ij}\to U_i</math> and <math>U_i\to U_{ii}</math>, we can construct a [[simplicial object]] <math>S(C)_\bullet</math> defined by <math>S(C)_n=C\times_X\cdots\times_XC</math>, n-fold fibre product. This is the '''Čech nerve.'''<ref>{{Cite web|title=Čech nerve in nLab|url=https://ncatlab.org/nlab/show/%C4%8Cech+nerve|access-date=2020-08-07|website=ncatlab.org}}</ref> By taking connected components we get a [[simplicial set]], which we can realise topologically: <math>|S(\pi_0(C))|</math>.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)