An \(\infty\)-operad \(\mathcal O\) is unital if for every color \(X \in \underline{\mathcal O}\), the \(0\)-ary mapping space \(\mathrm{Mul}_{\mathcal O}(\emptyset, X)\) is contractible. We let \(\mathrm{Op}^{\mathrm{un}}\) denote the full subcategory of the \(\infty\)-category of \(\infty\)-operads \(\mathrm{Op}\) on the unital \(\infty\)-operads.
7.1 Recollections on unital \(\infty\)-operads[00DL]
We recall some facts about \(\infty\)-operads from [Lur17] and [SY19] which we will use throughout.
For all \(n \geq 0\), the \(\infty\)-operads \(\mathbb E_n\) are unital.
For \(\infty\)-operads \(\mathcal O\) and \(\mathcal P\), recall that the \(\infty\)-category \(\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal P)\) of \(\mathcal O\)-algebras in \(\mathcal P\) admits a pointwise operad structure constructed in [Lur17, Ex. 3.2.4.4] and henceforth denoted \(\mathrm{Alg}_{\mathcal O}(\mathcal P)\).
Let \(\mathcal O\) be an \(\infty\)-operad and consider the operad maps \(\mathcal O\rightarrow\mathcal O\otimes \mathbb E_0\) and \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O) \rightarrow\mathcal O\) induced from the operad map \(\mathrm{Triv}\rightarrow\mathbb E_0\). Then the following hold:
The \(\infty\)-operad \(\mathcal O\otimes \mathbb E_0\) is unital. Moreover, \(\mathcal O\) is unital if and only if the operad map \(\mathcal O\rightarrow\mathcal O\otimes \mathbb E_0\) is an isomorphism.
The \(\infty\)-operad \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O)\) is unital. Moreover, \(\mathcal O\) is unital if and only if the operad map \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O) \rightarrow\mathcal O\) is an isomorphism.
Proof.
Part ([00DQ]) follows directly from [Lur17, Prop. 2.3.1.9]. For part ([00DR]), note that for an \(\infty\)-operad \(\mathcal P\), the fiber of \(\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_0, \mathcal P) \rightarrow\mathrm{Hom}_{\mathrm{Op}}(\mathrm{Triv}, \mathcal P) = \underline{ \mathcal P}^{\simeq}\) at a color \(X\in \underline{\mathcal P}\) is the \(0\)-ary mapping space \(\mathrm{Mul}_{\mathcal P}(\emptyset, X)\) and hence that \(\mathcal P\) is unital if and only if \(\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_0, \mathcal P) \rightarrow\mathrm{Hom}_{\mathrm{Op}}(\mathrm{Triv}, \mathcal P)\) is an isomorphism. In particular, evaluating at \(\mathcal P= \mathrm{Alg}_{\mathbb E_0}(\mathcal O)\) for an \(\infty\)-operad \(\mathcal O\), and using that \(\mathrm{Hom}_{\mathrm{Op}}(-, \mathrm{Alg}_{\mathbb E_0}(\mathcal O)) \simeq \mathrm{Hom}_{\mathrm{Op}}(- \otimes \mathbb E_0, \mathcal O)\) and part ([00DQ]), it follows that \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O)\) is unital. If \(\mathcal O\) is moreover unital, let \(\mathcal Q\) be a unital \(\infty\)-operad and consider the map \(\mathrm{Hom}_{\mathrm{Op}^{\mathrm{un}}}(\mathcal Q, \mathrm{Alg}_{\mathbb E_0}(\mathcal O))\rightarrow\mathrm{Hom}_{\mathrm{Op}^{\mathrm{un}}}(\mathcal Q, \mathcal O)\) which is equivalent to \(\mathrm{Hom}_{\mathrm{Op}^{\mathrm{un}}}(\mathcal Q\otimes \mathbb E_0, \mathcal O) \rightarrow\mathrm{Hom}_{\mathrm{Op}^{\mathrm{un}}}(\mathcal Q, \mathcal O)\). Since \(\mathcal Q\) is unital, this is an isomorphism by part ([00DQ]). This completes the proof of part ([00DR]). ◻
Evaluating lemma 7.1.3.([00DQ]) at \(\mathcal O= \mathbb E_0\) shows that \(\mathbb E_0\) is a (unital) idempotent in \(\mathrm{Op}\) (as also follows from Dunn additivity) with image \(\mathrm{Op}^{\mathrm{un}}\).
The following is an immediate consequence of lemma 7.1.3, also see [Lur17, Prop. 2.3.1.9].
The full inclusion \(\mathrm{Op}^{\mathrm{un}}\hookrightarrow \mathrm{Op}\) has left and right adjoints 
Most unital \(\infty\)-operads appearing in this paper arise from the following observation:
It follows from lemma 7.1.3 that for any unital \(\infty\)-operad \(\mathcal O\) and \(\infty\)-operad \(\mathcal P\) the operad \(\mathrm{Alg}_{\mathcal O}(\mathcal P) \simeq \mathrm{Alg}_{\mathcal O\otimes \mathbb E_0}(\mathcal P) \simeq \mathrm{Alg}_{\mathbb E_0}(\mathrm{Alg}_{\mathcal O}(\mathcal P))\) is unital. In particular, since \(\mathbb E_n\) is a unital \(\infty\)-operad for any \(n \geq 0\), it follows that for any \(\infty\)-operad \(\mathcal P\), the \(\infty\)-operads \(\mathrm{Alg}_{\mathbb E_n}(\mathcal P)\) are unital.
The underlying \(\infty\)-operad of a symmetric monoidal \(\infty\)-category \(\mathcal C\) is unital if and only if the tensor unit \(I\) of \(\mathcal C\) is an initial object. An example of such a symmetric monoidal \(\infty\)-category is given by the coCartesian tensor product on an \(\infty\)-category with finite coproducts.
The coCartesian monoidal structure on an \(\infty\)-category with finite coproducts can be generalized to a certain coCartesian unital operad structure on any \(\infty\)-category:
Example 7.1.7. ([Lur17, § 2.4.3]).
For any \(\infty\)-category \(\mathcal C\), there is a unital \(\infty\)-operad \(\mathcal C_{\sqcup}\) with underlying \(\infty\)-category \(\mathcal C\) and multi-ary mapping spaces \[\mathrm{Mul}_{\mathcal C_{\sqcup}}(X_1, \ldots, X_n; Y) \simeq \mathrm{Hom}_{\mathcal C}(X_1, Y) \times \cdots \mathrm{Hom}_{\mathcal C}(X_n, Y)\]
These coCartesian operads have the following universal characterization:
Lemma 7.1.8. ([Lur17, Prop. 2.4.3.9, Cor. 2.4.3.11], [SY19, Lem. 2.2.3]).
The assignment \(\mathcal C\mapsto \mathcal C_{\sqcup}\) induces a fully faithful functor \(\mathrm{Cat}_{\infty}\hookrightarrow \mathrm{Op}^{\mathrm{un}}\) which is right adjoint to the underlying-category functor \(\mathrm{Op}^{\mathrm{un}}\rightarrow\mathrm{Cat}_{\infty}\).
In particular, it follows that the unit of the adjunction is an operad map \(\mathcal O\rightarrow\underline{\mathcal O}_{\sqcup}\) which induces the identity on underlying \(\infty\)-categories \(\underline{\mathcal O} \rightarrow\underline{\underline{\mathcal O}_{\sqcup}} \simeq \underline{\mathcal O}\). Fixing colors \(X_1, \ldots, X_n, Y \in \underline{\mathcal O}\), this induces a map \[ \sigma \colon \mathrm{Mul}_{\mathcal O}(X_1, \ldots, X_n; Y) \rightarrow\mathrm{Hom}_{\underline{\mathcal O}}(X_1, Y) \times \cdots \times \mathrm{Hom}_{\underline{\mathcal O}}(X_n, Y).\] The components \(\mathrm{Mul}_{\mathcal O}(X_1, \ldots, X_n; Y) \rightarrow\mathrm{Hom}_{\underline{\mathcal O}}(X_i, Y)\) of this map may be thought of as inserting units in all but the \(i\)-th slot.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2