Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
In the context of Goodwillie calculus an (∞,1)-functor is called analytic if it behaves in analogy with an analytic function in that it its Goodwillie-Taylor tower converges to it.
Let be an (∞,1)-category with finite (∞,1)-colimits.
An -cube in , hence an (∞,1)-functor , is called strongly homotopy co-cartesian or just strongly co-cartesian, if all its 2-dimensional square faces are homotopy pushout diagrams in .
An -cube in , hence an (∞,1)-functor , is called homotopy cartesian or just cartesian, if its “first” object exhibits a homotopy limit-cone over the remaining objects.
An (∞,1)-functor is stably -excisive with constants and _ – or satisfies “condition ” – if for every strongly co-Cartesian -cube in , def. , such that is -connective for for all , then is an -cube in such that the comparison map
(to the indicated homotopy limit) is -connective.
The functor is called -analytic if there is such that it satisfies the condition for all .
(e.g. Johnson 95, def. 1.1, def. 1.3)
For -analytic functors their Goodwillie-Taylor tower converges to them on -connective objects. See there.
The identity -functor on ∞Grpd is 1-analytic, def. . For this is the statement of the Blakers-Massey theorem, for this is the statment of the higher cubical BM-theorems.
(see e.g. Munson-Volic 15, example 10.1.18)
The concept is due to
Review includes
Tom Goodwillie, section 3 of The differential calculus of homotopy functors (pdf)
Brenda Johnson, The derivatives of homotopy theory, Transactions of the AMS, Volume 347, 1995 (pdf)
A textbook account is in
Last revised on January 5, 2016 at 04:37:35. See the history of this page for a list of all contributions to it.