homotopy theory, (∞,1)-category theory, homotopy type theory
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An -connected space is a generalisation of the pattern:
For the general concept see at n-connected object of an (infinity,1)-topos.
A topological space is -connected or -simply connected if its homotopy groups are trivial up to degree .
More explicitly, is precisely -connected if every continuous map to from the -sphere extends to a continuous map to from the -disk. Then is -(simply) connected if is precisely -connected for .
Any space is -simply connected.
A space is -simply connected precisely if it has an element; that is if it is inhabited.
A space is -simply connected precisely if it is path-connected.
A space is -simply connected precisely if it is simply connected.
A space is -simply connected precisely if it is weakly contractible.
The traditional terminology is ‘-connected’, but this violates the rule that ‘-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.
An -connected topological space is precisely an n-connected object in the (∞,1)-topos ∞Gpd, presented by the model category Top of topological spaces.
Last revised on October 28, 2019 at 22:18:31. See the history of this page for a list of all contributions to it.