nLab module over an algebra over an (∞,1)-operad

(under construction)

Idea

(…)

As stabilization

Let $\mathcal{O}^{\otimes}$ be a reduced coherent (∞,1)-operad and $\mathcal{C}^{\otimes}$ be a stable $\mathcal{O}$-monoidal $\infty$-category (that is, an $\mathcal{O}$-algebra in the $\infty$-category of stable $\infty$-categories and exact functors). Then, Theorem 7.3.4.13 of (Lurie) presents $\mathcal{O}$-$A$-modules as stabilization of $\mathcal{O}$-algebras over $A$.

Theorem

The stabilization $\mathrm{Sp}(\Alg_{\mathcal{O}}(\mathcal{C})_{/A})$ is canonically equivalent to $\mathrm{Mod}_A^{\mathcal{O}}$.

The associated suspension-loops adjunction is an $\mathcal{O}$-algebra version of the adjunction between cotangent complexes and split square-zero extensions. To see the theorem, begin by noting that the we may reduce to the case of the $\mathcal{O}$-monoidal unit $\mathrm{A} = 1$ by expressing $\mathcal{O}$-$A$-algebras as $\mathcal{O}$-algebras under $A$:

$$\begin{split} \mathrm{Sp}(\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})_{/A}) &\simeq \mathrm{Sp}((\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})_{/A})_*) \\ &\simeq \mathrm{Sp}((\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})_{A//A})) \\ &\simeq \mathrm{Sp}((\mathrm{Alg}^{\mathrm{Aug}}_{\mathcal{O}}(\mathrm{Mod}_A^{\mathcal{O}}\mathcal{C}))). \end{split}$$

To see this in the unital case, taking Kernels of the augmentation yields a monadic “augmentation ideal” functor

$$I\colon \mathrm{Alg}_{\mathcal{O}}^{\mathrm{aug}}(\mathcal{C}) \rightarrow \mathcal{C}$$

whose associated monad corresponds with the positive-arity $\mathcal{O}$-symmetric powers:

$$T_I(X) \simeq \bigoplus_{n > 0} \left(\mathcal{O}(n) \times X^{\otimes n} \right)_{h\Sigma_n}.$$

By stability of $\mathcal{C}$, the augmentation ideal factors through a monadic functor $\partial I\colon \mathrm{Sp}(\mathrm{Alg}^{\mathrm{aug}}(\mathcal{C}) \rightarrow \mathcal{C}$ which turns out to be the first Goodwillie derivative

$$\partial I(X) \simeq \colim_{n} \Omega^n I(\Sigma^n X).$$

Intuitively, when $n \geq 2$, the $X^{\otimes n}$ term in the monad $T_{\partial I}(X)$ is a colimit of spaces becoming arbitrarily highly connected, so it is contractible, demonstrating that $T_{\partial I}(X)$ is the identity monad.

Related concepts

• ∞-module

• (∞,1)-operad, model structure on operads

  • algebra over an (∞,1)-operad, model structure on algebras over an operad

    • module over an algebra over an (∞,1)-operad, model structure on modules over an algebra over an operad

References

Section 3.3.3 of

• Jacob Lurie, Higher Algebra