4.1 Derivatives

We now embark on the journey to prove the theorems we stated at the beginning. We shall begin by explicitly calculating $P_k \int _{M_*} \mathbb {F}^{\mathrm{aug}}(V)$. Note that since the construction of $P_n F$ was done pointwise, we can do this calculation without knowing what the whole functor $P_k \int _{M_*} (-)$ is.

First note that both $\mathcal{V}$ and $\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})$ have a zero object, and $\mathbb {F}^{\mathrm{aug}}$ sends to zero object to the zero object by virtue of being a left adjoint. So

$$C_T(\mathbb {F}^{\mathrm{aug}}(V)) = \mathbb {F}^{\mathrm{aug}}(C_T(V)).$$

Using this and fixing a $(k + 1)$-cube $S$, we can compute $T_k \int _{M_*}(-)$:

$$\begin{aligned} T_k \int _{M_*}\mathbb {F}^{\mathrm{aug}}(V) & = \lim _{\emptyset \neq T \subseteq S}\int _{M_*} C_T(\mathbb {F}^{\mathrm{aug}}(V)) \\ & = \lim _{\emptyset \neq T \subseteq S}\int _{M_*} \mathbb {F}^{\mathrm{aug}}(C_T(V)) \\ & = \lim _{\emptyset \neq T \subseteq S}\bigoplus _{0 \leq i < \infty } \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} C_T(V)^{\otimes i}\\ & = \bigoplus _{0 \leq i < \infty } \lim _{\emptyset \neq T \subseteq S}\operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} C_T(V)^{\otimes i}. \end{aligned}$$

Fixing $M_*$, let us write

$$R_i(V) = \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i}.$$

We have then shown that

$$T_k \int _{M_*} \mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{0 \leq i < \infty } (T_k(R_i))(V).$$

Similarly, we have

$$T_k^j \int _{M_*} \mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{0 \leq i < \infty } (T_k^j(R_i))(V).$$

Therefore, we conclude that

$$P_k \int _{M_*}\mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{0 \leq i < \infty } (P_k(R_i)) (V).$$

But we have already concluded that $R_i$ is $i$-homogeneous (as a functor of $V$), so $P_k(R_i)$ is just $R_i$ if $i \leq k$, and $0$ otherwise. So

$$P_k \int _{M_*}\mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{0 \leq i < k} \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i}.$$

In particular, the $k$th derivative evaluated on $\mathbb {F}^{\mathrm{aug}}(V)$ is exactly

$$\operatorname{Conf}_k^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _k \wr \mathrm{O}(n)} V^{\otimes k}.$$

But since the $k$th derivative evaluated at an augmented $n$-disk algebra $A$ only depends on $L(A)$, and $L(A) = L(\mathbb {F}^{\mathrm{aug}}(L(A)))$, we know

In other words, we always have a cofiber sequence

$$\operatorname{Conf}_k^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _k \wr \mathrm{O}(n)} L(A)^{\otimes k} \to P_k \int _{M_*} A \to P_{k - 1} \int _{M_*} A.$$