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Suspension objects are canonically cogroup objects up to homotopy, via their “pinch map”. In particular this is the case for positive dimensional n-spheres.
Let be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write for the reduced suspension functor.
Then the pinch map
exhibits a cogroup structure on the image of in the homotopy category .
This is equivalently the group-structure of the first (fundamental) homotopy group of the values of the functor co-represented by :
All n-spheres, regarded as their homotopy types in ∞Grpd suspensions for : , hence they carry cogroup structure in the classical homotopy category. Under the -adjunction, this cogroup structure turns into the group structure on all homotopy groups in positive degree.
Last revised on March 7, 2018 at 17:58:12. See the history of this page for a list of all contributions to it.