Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
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.
(e.g. Lurie, def. 6.1.1.2 with prop. 6.1.1.15)
An (∞,1)-functor is -excisive for (or polynomial of degree at most ) if whenever is a strongly cocartesian -cube in , def. , then is a cartesian cube in , def. .
A 1-excisive (∞,1)-functor is often just called excisive (∞,1)-functor for short.
An -functor which is -excisive for some is also called a polynomial (∞,1)-functor (not to be confused with other concepts having the same name). It has degree when the smallest value of for which it is -excisive is .
This notion is comparable to how a polynomial of degree at most is determined by its values on distinct points.
Let be an (∞,1)-category with finite (∞,1)-colimits and a terminal object, and let be a Goodwillie-differentiable (∞,1)-category.
For
for the full sub-(∞,1)-category of the (∞,1)-functor (∞,1)-category on those (∞,1)-functors which are -excisive.
The inclusion of def. is lex reflective, hence the inclusion functor has a left adjoint (∞,1)-functor
which moreover is left exact (preserves finite (∞,1)-limits).
This is essentially the statement of (Goodwillie 03, theorem 1.8). In the above form it appears explicitly as (Lurie, theorem 6.1.1.10). The construction of the reflector is in (Lurie, constrution 6.1.1.27).
For this reflection is spectrification.
(-excisive functors form an -topos)
For an (∞,1)-topos, then for all we have that
is an (∞,1)-topos. (For this is in general not a hypercomplete (∞,1)-topos, even if is.)
This observation is due to Charles Rezk. It appears as (Lurie, remark 6.1.1.11). See also at Joyal locus.
A site of definition of is the Weiss topology on .
As ranges, the tower of -excisive approximations of an -functor, accordding to prop. , forms a tower analogous to the the Taylor series of a smooth function. This is called the Goodwillie-Taylor tower
If this converges to , then is analogous to an analytic function and is called an analytic (∞,1)-functor.
In the situation of def. , the functors for which (hence the anti-modal types) are called -reduced (∞,1)-functors.
(e.g. Lurie, def. 6.1.2.1)
A polynomial -functor of degree — that is, a -excisive functor which is not -excisive for any — is a homogeneous polynomial if its approximation by an degree polynomial is trivial.
(…)
Let be an (∞,1)-category with finite (∞,1)-colimits and with terninal object. Let be a pointed Goodwillie-differentiable (∞,1)-category. Write for the pointed objects in .
Then for all natural numbers composition with the forgetful functor induces an equivalence of (∞,1)-categories
Write for the (∞,1)-category of finite homotopy types, hence those freely generated by finite (∞,1)-colimits from the point. Write for the pointed finite homotopy types.
For an (∞,1)-topos we have that
is the collection of constant functors, hence the original (∞,1)-topos itself;
is the collection of parameterized spectra in , hence the tangent (∞,1)-topos of .
Hence one might refer to the tower of toposes
with
the tower of “Goodwillie jet (∞,1)-categories” of .
see (Lurie, def. 1.4.2.8 and around p. 823)
The notion of -excisive functors was introduced in
The Taylor tower formed by -excisive functors was then studied in
See also
A discussion in the general abstract context of (∞,1)-category theory is in
A model structure for n-excisive functors is given in
Georg Biedermann, Boris Chorny, Oliver Röndigs, Calculus of functors and model categories, Advances in Mathematics 214 (2007) 92-115 (arXiv:math/0601221)
Georg Biedermann, Oliver Röndigs, Calculus of functors and model categories II (arXiv:1305.2834v2)
Relation to Mackey functors is discussed in
Last revised on July 11, 2023 at 10:31:29. See the history of this page for a list of all contributions to it.