4.2 Convergence

We next want to study the problem of convergence. For $A = \mathbb {F}^{\mathrm{aug}}(V)$ 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.

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

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

$$\operatorname*{colim}_{j \in \mathcal{J}} \int _{M_*}A_j = \operatorname*{colim}_{j \in \mathcal{J}} \operatorname*{colim}_{U_+ \hookrightarrow M_*} A_j(U_+) = \operatorname*{colim}_{U_+ \hookrightarrow M_*} \operatorname*{colim}_{j \in \mathcal{J}} A_j(U_+).$$

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

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

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

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

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

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

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

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

We can then state the following theorem:

Under these assumptions, we will show that

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

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

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

Since the $t$-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 $A$ satisfies the conditions of the theorem, so does $\mathbb {F}^{\mathrm{aug}}(A)$. So it suffices to prove it for the free case.

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

We then use the fact that if $V \geq \ell$, then $V^{\otimes i} \geq i\ell$; and if $M_*$ is compact and connected, then $\operatorname{Conf}_i(M_*)$ is $i$-connected.