1 Free algebras
If , then it has an underlying object in , i.e. an object in that comes with a map to and from the unit (whose composite is the identity). Since is assumed to be stable, this gives us a splitting
The module has a canonical action of , which deformation retracts to (by scaling then Gram–Schmidt). So the “underlying module” has a canonical action, and this defines a forgetful functor
The free functor is defined to be the left adjoint of . In the first talk, we observed
where is the configuration space of 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 that sends to , which we can think of as the indecomposables functor. Moreover, we can write an arbitrary augmented -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 as a left adjoint by identifying its right adjoint. The right adjoint should be a functor whose composition with is the identity. We can pick this to be the square zero extension functor , which one can check satisfies the hypothesis of the adjoint functor theorem, so it admits a left adjoint
This functor (and ) actually play a special categorical role:
exhibits as the stabilization of .