Goodwillie filtration and factorization homology — Free algebras

1 Free algebras

If AAlgnaug(V)A \in \mathrm{Alg}_ n^{\mathrm{aug}}(\mathcal{V}), then it has an underlying object in 1/V/1{}^{1/}\mathcal{V}_{/1}, i.e. an object in VV that comes with a map to and from the unit 11 (whose composite is the identity). Since V\mathcal{V} is assumed to be stable, this gives us a splitting

A=1V. A = 1\oplus V.

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

U ⁣:Algnaug(V)ModO(n)(V)IVV \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 Faug\mathbb {F}^{\mathrm{aug}} is defined to be the left adjoint of UU. In the first talk, we observed

Theorem 1
MFaug(V)=i0Confifr(M)ΣiO(n)Vi, \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 Confifr(M)\operatorname{Conf}_ i^{{\mathrm{fr}}}(M_*) is the configuration space of ii 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:Algnaug(V)ModO(n)(V)L: \mathrm{Alg}_ n^{\mathrm{aug}}(\mathcal{V})\to \operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V}) that sends Faug(V)\mathbb {F}^{\mathrm{aug}}(V) to VV, which we can think of as the indecomposables functor. Moreover, we can write an arbitrary augmented nn-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 LL as a left adjoint by identifying its right adjoint. The right adjoint should be a functor ModO(n)(V)Algnaug(V)\operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V}) \to \mathrm{Alg}_ n^{\mathrm{aug}}(\mathcal{V}) whose composition with UU is the identity. We can pick this to be the square zero extension functor tt, which one can check satisfies the hypothesis of the adjoint functor theorem, so it admits a left adjoint

L ⁣:Algnaug(V)ModO(n)(V). L\colon \mathrm{Alg}_ n^{\mathrm{aug}}(\mathcal{V})\to \operatorname{Mod}_{\mathrm{O}(n)}(\mathcal{V}).

This functor LL (and tt) actually play a special categorical role:

Theorem 2

The adjunction

\begin{useimager} 
    \[
      \begin{tikzcd}
        \Algn \ar[r, yshift=2, "L"] & \Mod_{\O(n)}(\mathcal{V}) \ar[l, yshift=-2, "t"]
      \end{tikzcd}
    \]
  \end{useimager}

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

This will be very important when we get to the Goodwillie calculus part of the story.