# Augmented algebras over $\infty$-operads via the envelope

Let $$\mathcal{O}^\otimes$$ be an $$\infty$$-operad and $$\mathcal{C}^\otimes$$ a symmetric monoidal $$\infty$$-category, both in the sense of Lurie's Higher Algebra. By augmented $$\mathcal{O}^\otimes$$-algebras in $$\mathcal{C}^\otimes$$ I mean either of the following: $$\mathrm{Alg}^{\rm aug}_{\mathcal{O}}(\mathcal{C}^\otimes) := \mathrm{Alg}_{\mathcal{O}}(\mathcal{C}^\otimes)_{/1} \simeq \mathrm{Alg}_\mathcal{O}(\mathcal{C}^\otimes_{/1})$$ In the latter case $$\mathcal{C}^\otimes_{/1}$$ has the symmetric monoidal structure coming from the commtative algebra structure of $$1$$. I would like to find an "augmented envelope": a symmetric monoidal $$\infty$$-category $$\mathrm{Env}_{\rm aug}^\otimes(\mathcal{O}^\otimes)$$ such that: $$\mathrm{Fun}^{\rm strong -\otimes}(\mathrm{Env}_{\rm aug}^\otimes(\mathcal{O}^\otimes), \mathcal{C}^\otimes) \simeq \mathrm{Alg}^{\rm aug}_{\mathcal{O}}(\mathcal{C}^\otimes) .$$

The usual envelope $$\mathrm{Env}^\otimes(\mathcal{O}^\otimes)$$ has a similar property for non-augmented algebras. It is a symmetric monoidal $$\infty$$-category whose underlying $$\infty$$-category is the subcategory $$\mathcal{O}_{\rm act}^\otimes \subset \mathcal{O}^\otimes$$ and the symmetric monoidal structure can be thought of as "concatenating sequences of colors". In HA.2.2.4.7 Lurie remarks that this symmetric monoidal structure can be extended to all of $$\mathcal{O}^\otimes$$, but that this won't be needed. I'm fairly convinced that this symmetric monoidal structure should be the augmented envelope I'm looking for: the inert maps should exactly give all the augmentations.

There is also a more category theoretic way of saying what I'm looking for. The envelope $$\mathrm{Env}^\otimes$$ is the left-adjoint to the forgetful functor $$\mathrm{Op}_\infty \leftarrow \mathrm{Cat}_\infty^\otimes$$ from symmetric monoidal $$\infty$$-categories to $$\infty$$-operads. Similarly, the augmented envelope should be the left-adjoint to the forgetful functor $$\mathrm{Op}_\infty \longleftarrow \mathrm{Cat}_\infty^{\otimes, 1\rm -term}$$ where the $$\mathrm{Cat}_\infty^{\otimes,1\rm -term} \subset \mathrm{Cat}_\infty^\otimes$$ is the full subcategory on those $$\mathcal{C}^\otimes$$ such that $$1 \in \mathcal{C}$$ is terminal. I think this amounts to the same as above because sending $$\mathcal{C}^\otimes$$ to $$\mathcal{C}^\otimes_{/1}$$ should be a colocalisation onto the full subcategory $$\mathrm{Cat}_\infty^{\otimes,1\rm -term} \subset \mathrm{Cat}_\infty^\otimes$$.

Question: Has such an augmented envelope been constructed or has someone written something about this extended symmetric monoidal structure on $$\mathcal{O}^\otimes$$ that Lurie mentions?

• I should say that I do have a guess for how to construct the augmented envelope. Let $\mathrm{Fin}_*^\vee$ be the symmetric monoidal category of finite pointed sets equipped with the wedge product. Then I think that $\mathrm{Env}_{\rm aug}^\otimes(\mathcal{O}^\otimes) := \mathcal{O}^\otimes \times_{\mathrm{Fin}_*} \mathrm{Fin}_*^\vee$ is the correct thing. Here the pullback is taken along the functor $\bigvee: \mathrm{Fin}_*^\vee \to \mathrm{Fin}_*$ and resulting category is equipped with a functor to $\mathrm{Fin}_*$ coming from the structure functor $\mathrm{Fin}_*^\vee \to \mathrm{Fin}_*$. Jan 21, 2022 at 11:08