Goodwillie filtration and factorization homologyConvergence

4.2 Convergence

We next want to study the problem of convergence. For A=Faug(V)A = \mathbb {F}^{\mathrm{aug}}(V) we can do this pretty explicitly using the description above, and for non-free algebras, our strategy is to write them as sifted colimits of free algebras.

Lemma 21

Every element of Algnaug(V)\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V}) is a sifted colimit of free augmented nn-disk algebras.

Proof
The forgetful functor Algnaug(V)1/V/1\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to {}^{1/}V_{/1} is monadic, and so every augmented nn-disk algebra AA is the geometric realization of the bar construction (Faug)+1A(\mathbb {F}^{\mathrm{aug}})^{\bullet + 1}A.
Proof

Lemma 22

M()\int _{M_*}(-) and PkM()P_k \int _{M_*}(-) preserves sifted colimits.

Proof
Recall that an augmented nn-disk algebra is a symmetric monoidal functor Diskn,+V\mathrm{Disk}_{n, +} \to \mathcal{V}. Suppose A ⁣:JAlgnaug(V)A\colon \mathcal{J} \to \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V}) is a sifted diagram of augmented nn-disk algebras. Then

colimjJMAj=colimjJcolimU+MAj(U+)=colimU+McolimjJAj(U+). \operatorname*{colim}_{j \in \mathcal{J}} \int _{M_*}A_j = \operatorname*{colim}_{j \in \mathcal{J}} \operatorname*{colim}_{U_+ \hookrightarrow M_*} A_j(U_+) = \operatorname*{colim}_{U_+ \hookrightarrow M_*} \operatorname*{colim}_{j \in \mathcal{J}} A_j(U_+).

So to show that M\int _{M_*} commutes with sifted colimits, we have to show that

colimjJAj(U+)(colimjJAj)(U+). \operatorname*{colim}_{j \in \mathcal{J}} A_j(U_+) \cong \left(\operatorname*{colim}_{j \in \mathcal{J}} A_j\right)(U_+).

Since AjA_j is symmetric monoidal, it suffices to prove this for the case where U+=R+nU_+ = \mathbb {R}^n_+. That is, we have to show that the forgetful functor Algnaug(V)V\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to \mathcal{V} preserves sifted colimits, which is a general fact about algebras over monads.

For the case of PkM()P_k \int _{M_*}(-), we observe that this is true for P0M()P_0 \int _{M_*}(-), which is constant, and then proceed by induction using the explicit formula for the kkth derivative.

Proof

Finally, we consider the problem of convergence. In the case of a free algebra, we are asking when the map

0i<Confifr(M)ΣiO(n)Vi0i<kConfifr(M)ΣiO(n)Vi \bigoplus _{0 \leq i < \infty } \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i} \to \prod _{0 \leq i < k} \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i}

is an equivalence. This is not always true. In the case where V\mathcal{V} is the category of spectra, say, a sufficient condition for this to be true is that the connectivity of Confifr(M)ΣiO(n)Vi\operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i} tends to infinity as ii \to \infty . There are (at least) two ways this can happen — either the connectivity of Confifr(M)\operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) tends to infinity, or the connectivity of ViV^{\otimes i} tends to infinity.

In general, to make sense of this argument in an arbitrary stable \infty -category, we need a tt-structure, which is a collection of reflexive subcategories Vt\mathcal{V}_{\geq t} satisfying certain properties, which we think of as the subcategory of tt-connective objects. We require this to be compatible with the tensor product, i.e. \otimes maps Vt×Vs\mathcal{V}_{\geq t} \times \mathcal{V}_{\geq s} to Vt+s\mathcal{V}_{\geq t + s}, and to be cocomplete, i.e. tVt={0}\bigcap _t \mathcal{V}_{\geq t} = \{ 0\} .

We can then state the following theorem:

Theorem 23

The map

MAlimkPkMA \int _{M_*} A \to \lim _{k \to \infty } P_k \int _{M_*} A

is an equivalence if at least one of the following two hold:

  1. The augmentation ideal ker(A1)>0\ker (A \to 1) > 0.

  2. ker(A1)0\ker (A \to 1) \geq 0 and MM_* is connected and compact.

Proof
Under these assumptions, we will show that

ker(MAPkMA)k. \ker \left(\int _{M_*} A \to P_k \int _{M_*} A\right) \geq k.

Then since taking kernels commutes with sequential limits, we know that

ker(MAlimPkMA)m for all m. \ker \left(\int _{M_*} A \to \lim P_k \int _{M_*} A\right) \geq m \text{ for all }m.

Since the tt-structure is cocomplete, the kernel is trivial. So this map is an isomorphism.

Since kernels are the same as cokernels (up to a shift), taking kernels commutes with taking shifted colimits. Moreover, if AA satisfies the conditions of the theorem, so does Faug(A)\mathbb {F}^{\mathrm{aug}}(A). So it suffices to prove it for the free case.

In the case where A=Faug(V)A = \mathbb {F}^{\mathrm{aug}}(V), the conditions ker(A1)>0\ker (A \to 1) > 0 and 0\geq 0 correspond to V>0V > 0 and V0V \geq 0.

We then use the fact that if VV \geq \ell , then ViiV^{\otimes i} \geq i\ell ; and if MM_* is compact and connected, then Confi(M)\operatorname{Conf}_i(M_*) is ii-connected.

Proof