James Tauber's Blog 2005/07/01
Path Homotopy
Previously we defined the notion of homotopy.
Two functions that are continuous deformations of one another are homotopic even if the two functions aren't paths.
But if the two functions are paths, then we can further define a stricter notion called path homotopy.
Two paths are path homotopic iff they are homotopic and they have the same start point and end point throughout the deformation.
In other words, if our paths are functions f and g from the interval [0, 1] to a topological space X, then path homotopy means not only the existence of a continuous map F : [0, 1] x [0, 1] -> X where
- F(x, 0) = f(x)
- F(x, 1) = g(x)
but also that:
- F(0, t) = f(0) = g(0)
- F(1, t) = f(1) = g(1)
for all t in [0, 1].
UPDATE: next post
by : Created on July 1, 2005 : Last modified July 1, 2005 : (permalink)
Summer of Code Kick-off
Congratulations to Elliot Cohen and all the other successful applicants to Google's Summer of Code.
I will be mentoring Elliot's project to create a Python library for Bayesian networks. Thank you to the Python Software Foundation for giving me the opportunity to do this.
The Summer of Code requires the project be hosted by a site like SourceForge. Much to my delight, Elliot is keen to use Subversion rather than CVS so we're likely going to give BerliOS a go. BerliOS uses the SourceForge code but already has support for Subversion.
I've also suggested Elliot start a blog and wiki.
by : Created on July 1, 2005 : Last modified July 1, 2005 : (permalink)