Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits.
An $n$-cube in $\mathcal{C}$, hence an (∞,1)-functor $\Box^n \longrightarrow \mathcal{C}$, is called strongly homotopy co-cartesian or just strongly co-cartesian, if all its 2-dimensional square faces are homotopy pushout diagrams in $\mathcal{C}$.
An $n$-cube in $\mathcal{D}$, hence an (∞,1)-functor $\Box^n \longrightarrow \mathcal{D}$, 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 $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is $n$-excisive for $n \in \mathbb{N}$ (or polynomial of degree at most $n$) if whenever $X$ is a strongly cocartesian $(n + 1)$-cube in $\mathcal{C}$, def. , then $F(X)$ is a cartesian cube in $\mathcal{D}$, def. .
A 1-excisive (∞,1)-functor is often just called excisive (∞,1)-functor for short.
An $(\infty,1)$-functor which is $n$-excisive for some $n \in \mathbb{N}$ is also called a polynomial (∞,1)-functor (not to be confused with other concepts having the same name). It has degree $k$ when the smallest value of $n$ for which it is $n$-excisive is $k$.
This notion is comparable to how a polynomial of degree at most $n$ is determined by its values on $n + 1$ distinct points.
Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and a terminal object, and let $\mathcal{D}$ be a Goodwillie-differentiable (∞,1)-category.
For $n \in \mathbb{N}$
for the full sub-(∞,1)-category of the (∞,1)-functor (∞,1)-category on those (∞,1)-functors which are $n$-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 $P_n$ is in (Lurie, constrution 6.1.1.27).
For $n = 1$ this reflection is spectrification.
($n$-excisive functors form an $\infty$-topos)
For $\mathcal{D} = \mathbf{H}$ an (∞,1)-topos, then for all $n \in \mathbb{N}$ we have that
is an (∞,1)-topos. (For $n \gt 1$ this is in general not a hypercomplete (∞,1)-topos, even if $\mathbf{H}$ 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 $Exc^n(\mathcal{C}, \mathbf{H}) \hookrightarrow PSh(\mathcal{C}^{op}, \mathbf{H})$ is the Weiss topology on $\mathcal{C}^{op}$.
As $n$ ranges, the tower of $n$-excisive approximations of an $(\infty,1)$-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 $F$, then $F$ is analogous to an analytic function and is called an analytic (∞,1)-functor.
In the situation of def. , the functors $F$ for which $P_{n-1}F \simeq \ast$ (hence the $P_{n-1}$ anti-modal types) are called $n$-reduced (∞,1)-functors.
(e.g. Lurie, def. 6.1.2.1)
A polynomial $\infty$-functor of degree $k$ — that is, a $k$-excisive functor which is not $n$-excisive for any $n\lt k$ — is a homogeneous polynomial if its approximation by an $(k-1)$ degree polynomial is trivial.
(…)
Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and with terninal object. Let $\mathcal{D}$ be a pointed Goodwillie-differentiable (∞,1)-category. Write $\mathcal{C}^{\ast/}$ for the pointed objects in $\mathcal{C}$.
Then for all natural numbers $n \geq 1$ composition with the forgetful functor $\mathcal{C}^{\ast/} \to \mathcal{C}$ induces an equivalence of (∞,1)-categories
Write $\infty Grpd_{fin}$ for the (∞,1)-category of finite homotopy types, hence those freely generated by finite (∞,1)-colimits from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the pointed finite homotopy types.
For $\mathbf{H}$ an (∞,1)-topos we have that
$Exc^0(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq \mathbf{H}$ is the collection of constant functors, hence the original (∞,1)-topos itself;
$Exc^1(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq T\mathbf{H}$ is the collection of parameterized spectra in $\mathbf{H}$, hence the tangent (∞,1)-topos of $\mathbf{H}$.
Hence one might refer to the tower of toposes
with
the tower of “Goodwillie jet (∞,1)-categories” of $\mathbf{H}$.
see (Lurie, def. 1.4.2.8 and around p. 823)
The notion of $n$-excisive functors was introduced in
The Taylor tower formed by $n$-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.