1 Free algebras
If A∈Algnaug(V), then it has an underlying object in 1/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 V is assumed to be stable, this gives us a splitting
A=1⊕V.
The module V has a canonical action of Emb(Rn,Rn), which deformation retracts to O(n) (by scaling then Gram–Schmidt). So the “underlying module” V has a canonical O(n) action, and this defines a forgetful functor
U:Algnaug(V)I⊕V→ModO(n)(V)↦V
The free functor Faug is defined to be the left adjoint of U. In the first talk, we observed
Theorem
1
∫M∗Faug(V)=i≥0⨁Confifr(M∗)Σi≀O(n)⨂V⊗i,
where Confifr(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:Algnaug(V)→ModO(n)(V) that sends Faug(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 ModO(n)(V)→Algnaug(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:Algnaug(V)→ModO(n)(V).
This functor L (and t) actually play a special categorical role:
Theorem
2
The adjunction
exhibits ModO(n)(V) as the stabilization of Algnaug(V).
This will be very important when we get to the Goodwillie calculus part of the story.