Goodwillie filtration and factorization homologyFree algebras

# 1 Free algebras

If $A \in \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})$, then it has an underlying object in ${}^{1/}\mathcal{V}_{/1}$, i.e. an object in $V$ that comes with a map to and from the unit $1$ (whose composite is the identity). Since $\mathcal{V}$ is assumed to be stable, this gives us a splitting

$A = 1\oplus V.$

The module $V$ has a canonical action of $\mathrm{Emb}(\mathbb {R}^n, \mathbb {R}^n)$, which deformation retracts to $\mathrm{O}(n)$ (by scaling then Gram–Schmidt). So the “underlying module” $V$ has a canonical $\mathrm{O}(n)$ action, and this defines a forgetful functor

\begin{aligned} U\colon \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})& \to \operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V})\\ I \oplus V & \mapsto V \end{aligned}

The free functor $\mathbb {F}^{\mathrm{aug}}$ is defined to be the left adjoint of $U$. In the first talk, we observed

Theorem 1
$\int _{M_*} \mathbb {F}^{\mathrm{aug}}(V) = \bigoplus _{i \geq 0} \operatorname{Conf}_i^{{\mathrm{fr}}}(M_*) \bigotimes _{\Sigma _i \wr \mathrm{O}(n)} V^{\otimes i},$

where $\operatorname{Conf}_i^{{\mathrm{fr}}}(M_*)$ is the configuration space of $i$ points together with a framing of the tangent bundle at each point.

All the proofs in the talk will be based on this computation, and so it would be nice to have a functor $L: \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to \operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V})$ that sends $\mathbb {F}^{\mathrm{aug}}(V)$ to $V$, which we can think of as the indecomposables functor. Moreover, we can write an arbitrary augmented $n$-disk algebra as a sifted colimit of free algebras, and so it would be nice if this functor preserves colimits as well.

We will construct $L$ as a left adjoint by identifying its right adjoint. The right adjoint should be a functor $\operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V}) \to \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})$ whose composition with $U$ is the identity. We can pick this to be the square zero extension functor $t$, which one can check satisfies the hypothesis of the adjoint functor theorem, so it admits a left adjoint

$L\colon \mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})\to \operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V}).$

This functor $L$ (and $t$) actually play a special categorical role:

Theorem 2

exhibits $\operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V})$ as the stabilization of $\mathrm{Alg}_n^{\mathrm{aug}}(\mathcal{V})$.