Internal Homs of (Higher) Operads and $(\infty, 2)$-Categories

Question

While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-categories. Here is one such situation where they seem to be tacitly appearing in Haugseng's An allegedly somewhat friendly introduction to $\infty$-operads, p. 33.

There is a theorem by Lurie (Theorem 3.4.4) stating the following:

Theorem. The $\infty$-category of $\infty$-operads $\mathsf{Opd}_{\infty}$ admits a symmetric monoidal structure with the Boardman-Vogt tensor product which preserves colimits in each variable.

The immediate corollary (Corollary 3.4.5) is:

Corollary. The $\infty$-category $\mathsf{Opd}_{\infty}$ admits internal Homs $\mathsf{ALG}_{\mathscr{O}}(\mathscr{P})$ with natural equivalences $$\mathsf{Alg}_{\mathscr{O} \otimes \mathscr{P}}(\mathscr{Q}) \simeq \mathsf{Alg}_{\mathscr{O}}(\mathsf{ALG}_{\mathscr{P}}(\mathscr{Q})).$$

From this Haugseng then deduces that the underlying $\infty$-category of $\mathsf{ALG}_{\mathscr{O}}(\mathscr{P})$ is $\mathsf{Alg}_{\mathscr{O}}(\mathscr{P})$.

I'm a little confused about how the corollary arises. First, I don't know if $\mathsf{Opd}_{\infty}$ is presentable (is it?) - but if it is, then the theorem is essentially saying that $\mathsf{Opd}_{\infty}$ is presentably symmetric monoidal and we can take $\mathsf{ALG}_{\mathscr{O}}(-)$ as a right adjoint of $- \otimes \mathscr{O}$ by the Adjoint Functor Theorem. But from this I don't know how to deduce the equivalence. We work inside $\mathsf{Fun}^{\mathsf{Opd}_{\infty}}(-,-)$ if we work with $\mathsf{Alg}$ but an $(\infty, 1)$-adjunction only makes statements about the $\mathsf{Hom}$ anima. So it feels like one needs a $(\infty, 2)$-adjunction to make statements about $\infty$-functor category functors.

Are there some references or explanations to clear up these confusions of mine? While this fact is mentioned in the introduction of Lurie's Higher Algebra 2.2.5, I can't find any proofs of it there.

Answer 1

I can sketch a proof using only $(\infty,1)$-categorical machinery. There are details missing, but the key idea is there. As I said half a year ago in a comment, it follows from general Yoneda nonsense that $$ \mathsf{ALG}_{\mathscr{O}\otimes\mathscr{P}}(\mathscr{Q})\simeq\mathsf{ALG}_\mathscr{O}(\mathsf{ALG}_\mathscr{P}(\mathscr{Q})), $$ and hence it suffices to show that $\mathsf{ALG}_\mathscr{O}(\mathscr{P})_{\langle 1\rangle}\simeq\mathsf{Alg}_\mathscr{O}(\mathscr{P})$ as $\infty$-categories. It suffices to produce for every $n\geq 0$ an equivalence $$ \mathsf{Cat}_\infty([n],\mathsf{ALG}_\mathscr{O}(\mathscr{P})_{\langle 1\rangle})\simeq\mathsf{Cat}_\infty([n],\mathsf{Alg}_\mathscr{O}(\mathscr{P})) $$ that collectively are natural in $[n]$. Write $\widehat{(-)}\colon\mathsf{Cat}_\infty\to\mathsf{Op}_\infty$ for the usual inclusion functor, which is left adjoint to $\mathscr{O}\mapsto \mathscr{O}_{\langle 1\rangle}$. The $\infty$-operad $\widehat{[n]}\to\mathbb{F}_*$ satisfies the following: if a morphism in it does not live over the identity map on $\langle 1\rangle$, then it is inert.

We find a natural equivalence $$ \mathsf{Cat}_\infty([n],\mathsf{ALG}_\mathscr{O}(\mathscr{P})_{\langle 1\rangle})\simeq\mathsf{Op}_\infty(\widehat{[n]}\otimes\mathscr{O},\mathscr{P})\simeq\mathsf{Op}_\infty^\mathrm{bf}(\widehat{[n]},\mathscr{O};\mathscr{P}), $$ where the latter is the space of $\infty$-operad bifunctors $\widehat{[n]}\times\mathscr{O}\to\mathscr{P}$ as defined in Definition 3.4.1 of the document by Haugseng that you mentioned above. This is the space of commutative squares $$ \require{AMScd} \begin{CD} \widehat{[n]}\times\mathscr{O} @>{\varphi}>>\mathscr{P}\\ @VVV @VVV\\ \mathbb{F}_*\times \mathbb{F}_* @>>{\mu}> \mathbb{F}_* \end{CD} $$ where $\mu$ is the smash product of finite pointed sets, and we require $\varphi$ to send pairs of inert morphisms to inert morphisms in $\mathscr{P}$. It is not hard to see that $\mathsf{Cat}_\infty([n],\mathsf{Alg}_\mathscr{O}(\mathscr{P}))$ is equivalent to the space of commutative diagrams $$ \require{AMScd} \begin{CD} {[n]}\times\mathscr{O} @>{\psi}>>\mathscr{P}\\ @VVV @VVV\\ \{\langle 1\rangle\}\times \mathbb{F}_* @>>{\mu}> \mathbb{F}_* \end{CD} $$ (where the bottom horizontal arrow is an identity) where we require $\psi$ to send morphisms of the form $(\mathrm{id},g)$ to an inert in $\mathscr{P}$ whenever $g$ is inert in $\mathscr{O}$. We have thus reduced the task to showing that these two spaces of commutative squares are equivalent, via restriction along the inclusion $\{\langle 1\rangle\}\times \mathbb{F}_*\hookrightarrow \mathbb{F}_*\times\mathbb{F}_*$. (This restriction map is natural in $[n]$, so we don't have to worry about naturality anymore.)

We know that $\varphi$ needs to send pairs of inerts to inerts, and that any morphism $f$ in $\widehat{[n]}$ not living over $\mathrm{id}_{\langle 1\rangle}$ is inert. Given a pair $(f,g)$ of morphisms in $\widehat{[n]}\times\mathscr{O}$, where $f$ is as in the previous sentence and $g$ is arbitrary, we can decompose $(f,g)\simeq(\mathrm{id},g)\circ(f,\mathrm{id})$. It is already determined what $\varphi$ does to $(f,\mathrm{id})$ as this is a pair of inert morphisms, and what $\varphi$ does to $(\mathrm{id},g)$ is determined by what $\varphi$ does to morphisms of the form $(\mathrm{id}_j,g)$, where $j\in[n]\simeq\widehat{[n]}_{\langle 1\rangle}$. This will make the data of $\varphi$ be determined by its restriction along $\{\langle 1\rangle\}\times \mathbb{F}_*\hookrightarrow \mathbb{F}_*\times\mathbb{F}_*$ (i.e. its restriction to a functor $[n]\times\mathcal{O}\to\mathscr{P}$ over $\mathbb{F}_*$). You need to work a bit harder to make this argument precise, which requires you to talk in an $\infty$-categorical manner about the factorization system on a product of $\infty$-categories I used here, and the defining property of inert morphisms in $\widehat{[n]}$.

This establishes that the restriction operation is an isomorphism on the $\pi_0$ of the spaces of commutative squares we are looking at. By running the same argument on diagrams of the form $$ \require{AMScd} \begin{CD} [k]\times\widehat{[n]}\times\mathscr{O} @>{H}>>\mathscr{P}\\ @VVV @VVV\\ \{\langle 1\rangle\}\times\mathbb{F}_*\times \mathbb{F}_* @>>{\mu}> \mathbb{F}_* \end{CD} $$ (satisfying certain conditions on $H$) we find that our restriction map between spaces of commutative squares induces isomorphisms on all homotopy groups, and hence is an equivalence of spaces.