free algebras lift to cospectrum objects

Related: Formal geometry of free algebras

Idea/content

One way to read the equivalence $Sp(Al{g}_{O,A/})\simeq Mo{d}_A^O$ is that trivial/square-zero extension algebras lift to spectrum objects. Koszul-dual to this is an analogous structure on free algebras.

Construction of the lift

Let $V^{\otimes }$ be a presentably-symmetric monoidal stable $\infty$-category. The functor

$$Fre{e}^{op}:V^{op}\to Al{g}_{\mathcal{O}}(V{)}^{op}$$

is a right adjoint out of a stable $\infty$-category, hence by universal property of stabilization it factors as

$$V^{op}\stackrel{\tilde{F}}{\to }Sp(Al{g}_{\mathcal{O}}(V{)}^{op})\stackrel{{\Omega }^{\infty }}{\to }Al{g}_{\mathcal{O}}(V{)}^{op}$$

Applications

This is used by Glasman-Lawson to study power operations.