One colored infinity operads via symmetric sequences?

Question

The question

One standard approach to the theory of 1-colored (symmetric) operads in classical 1-categorical theory is via monoids in symmetric sequences with respect to the composition product. Has this been worked out for $\infty$-operads in the sense of Higher Algebra?

More details

More concretely, I am not even interested in the composition product, but in something that seems rather obvious but still requires a proof. For a 1-colored $\infty$-operad $\mathcal{P}$ (my operads are actually reduced) we can associate a symmetric sequence of spaces $\{\mathcal{P}(n)\}_{n\ge0}$ of $n$-arry operations. From this sequence, one can construct an endofunctor of spaces

$$T_{\mathcal{P}}(X) = \coprod_{n\ge0}(\mathcal{P}(n)\times X^n)//\Sigma_n$$

which is the monad obtained from the free forgetful adjunction $\mathcal{S}\leftrightarrows Alg_{\mathcal{P}}(\mathcal{S})$ (so far, everything can be extracted from HA).

Now let $\mathcal{P}\to \mathcal{Q}$ be a map of 1-colored $\infty$- operads. This induces a forgetful functor of algebras in spaces $Alg_{\mathcal{Q}}(\mathcal{S}) \to Alg_{\mathcal{P}}(\mathcal{S})$ from which we get by abstract nonsense a canonical map $T_{\mathcal{P}}(X)\to T_{\mathcal{Q}}(X)$. I need to know that this map is indeed the obvious thing one would expect. That is, that it is induced from the map of symmetric sequences $\{\mathcal{P}(n)\}_{n\ge0} \to \{\mathcal{Q}(n)\}_{n\ge0}$ which is itself induced from the map of operads.

This is definitely true, but I am looking for a rigorous (and preferably short) argument for it. My hope is that a comprehensive treatment of 1-colored $\infty$-operads in terms of symmetric sequences will include enough stuff to deduce this from.

Remarks on the literature:

In Higher Algebra section 6.3 the approach via symmetric sequences is mentioned, but not developed formally. This whole thing (and much more) follows morally from this paper which identifies 1-colored $\infty$-operads with analytic $\infty$-monads, but the identification is not so transparent (to me), especially when working with $\infty$-operads using the HA model. There are model category approaches to this, but again the comparisons are usually not so simple and as a first choice I would prefer to stay in the quasi-category model.

Answer 1

As far as I have understood your question, you ask the following question:

(When I write category in the following, I always mean $\infty$-category.)

Let $\mathcal{C}$ be a symmetric monoidal category compatible with small colimits (i.e. $\mathcal{C}$ admits small colimits that are preserved by the tensorproduct in each variable).

Denote $\mathrm{Sym}(\mathcal{C})$ the category of symmetric sequences in $\mathcal{C}$ viewed as a monoidal category with the composition product.

An associative algebra $\mathcal{O} $ in $\mathrm{Sym}(\mathcal{C})$ gives rise to a monad $\mathrm{T}_{ \mathcal{O} }, $ i.e. an associative algebra in $\mathrm{Fun}( \mathcal{C}, \mathcal{C})$ (endofunctors of $\mathcal{C}$) via a monoidal functor $\mathrm{Sym}(\mathcal{C}) \to \mathrm{Fun}( \mathcal{C}, \mathcal{C})$ that sends $ \mathcal{O} $ to $ \mathrm{T}_{\mathcal{O}},$ where $ \mathrm{T}_{\mathcal{O}}(X) := \coprod_{n\ge0}(\mathcal{O}(n)\otimes \mathrm{X}^n)//\Sigma_n.$

Given a monad $\mathrm{T} $ denote $ \mathrm{Alg}_{ \mathrm{T}} ( \mathcal{C} )$ its category of algebras.

Given a monoid $\mathcal{O} $ in $\mathrm{Sym}(\mathcal{C})$ its category $ \mathrm{Alg}_{ \mathcal{O} } ( \mathcal{C} )$ is the category of left modules over $ \mathcal{O} $ in $\mathrm{Sym}(\mathcal{C})$ on objects of $\mathrm{Sym}(\mathcal{C})$ that belong to $\mathcal{C} \subset \mathrm{Sym}(\mathcal{C}).$

You ask how one can show the existence of an equivalence $ \mathrm{Alg}_{ \mathcal{O} } ( \mathcal{C} )\simeq \mathrm{Alg}_{ \mathrm{T}_{ \mathcal{O} } } ( \mathcal{C} )$ over $ \mathcal{C} $ natural in $\mathcal{O} \in \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) . $

To answer this question, some preparations:

Denote $\Sigma \simeq \coprod_{ \mathrm{n} \geq 0 } \mathrm{B}( \Sigma_{\mathrm{n}} )$ the category of finite sets and isomorphisms. Then one can define $\mathrm{Sym}(\mathcal{C}) = \mathrm{Fun}(\Sigma, \mathcal{C}) \simeq \prod_{ \mathrm{n} \geq 0 } \mathrm{Fun}(\mathrm{B}( \Sigma_{\mathrm{n}}), \mathcal{C}). $

We have a small colimits preserving fully faithful functor $\iota: \mathcal{C} \subset \prod_{ \mathrm{n} \geq 0 } \mathrm{Fun}(\mathrm{B}( \Sigma_{\mathrm{n}}), \mathcal{C})$ that is the identity on the first factor and the constant functor with image the initial object of $ \mathcal{C} $ on every other factor.

As next I define the composition product via the Day-convolution on $\mathrm{Fun}(\Sigma, \mathcal{C}):$

We endow $\Sigma$ with the symmetric monoidal structure given by the coproduct of finite sets.

Then $\Sigma$ is the free symmetric monoidal category on the contractible category, i.e. for every symmetric monoidal category $ \mathcal{E} $ evaluation at the set with one element $1 \in \Sigma $ yields an equivalence $$ \mathrm{Fun}^\otimes (\Sigma, \mathcal{E}) \simeq \mathcal{E}, $$ where $\mathrm{Fun}^\otimes (\Sigma, \mathcal{E}) $ denotes the category of symmetric monoidal functors $\Sigma \to \mathcal{E}.$

We endow the functor-category $\mathrm{Sym}(\mathcal{C}) = \mathrm{Fun}(\Sigma, \mathcal{C}) $ with the Day-convolution symmetric monoidal structure.

The Day-convolution symmetric monoidal structure on $ \mathrm{Fun}(\Sigma, \mathcal{C}) $ restricts to the full subcategory $\iota: \mathcal{C} \subset \mathrm{Fun}(\Sigma, \mathcal{C}) $ so that $\iota$ gets a symmetric monoidal functor.

We have a unique small colimits preserving symmetric monoidal functor $\mathcal{S} \to \mathcal{C} $ starting at the category of spaces $\mathcal{S}$ (endowed with the cartesian symmetric monoidal structure) that sends the contractible space to the tensorunit $ \mathbb{1}_{ \mathcal{C}} $ of $ \mathcal{C} $ and gives rise to a symmetric monoidal functor $ \mathrm{Fun}(\Sigma, \mathcal{S}) \to \mathrm{Fun}(\Sigma, \mathcal{C})$ on Day-convolutions.

Composing with the symmetric monoidal Yoneda-embedding $ \Sigma \simeq \Sigma^\mathrm{op} \subset \mathrm{Fun}(\Sigma, \mathcal{S}) $ we get a symmetric monoidal functor $\phi: \Sigma \to \mathrm{Fun}(\Sigma, \mathcal{C}) $ that sends $\mathrm{n} \in \Sigma $ to the symmetric sequence concentrated in degree $\mathrm{n} $ with value $\Sigma_{\mathrm{n}} \times \mathbb{1}_{ \mathcal{C}}. $

$ \phi $ satisfies the following universal property:

The Day-convolution $ \mathrm{Fun}(\Sigma, \mathcal{C})$ is a symmetric monoidal category compatible with small colimits and we have a small colimits preserving symmetric monoidal functor $ \iota: \mathcal{C} \to \mathrm{Fun}(\Sigma, \mathcal{C}). $

For every symmetric monoidal category $ \mathcal{D} $ compatible with small colimits equipped with a small colimits preserving symmetric monoidal functor $ \mathcal{C} \to \mathcal{D} $ composition with $\phi: \Sigma \to \mathrm{Fun}(\Sigma, \mathcal{C}) $ yields an equivalence $$ \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathcal{D}) \to \mathrm{Fun}^\otimes (\Sigma, \mathcal{D})\simeq \mathcal{D}, $$ where $ \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathcal{D}) $ denotes the category of symmetric monoidal functors $\mathrm{Fun}(\Sigma, \mathcal{C}) \to \mathcal{D}$ compatible with the symmetric monoidal functors from $\mathcal{C}.$

So the equivalence $\mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathcal{D}) \to \mathrm{Fun}^\otimes (\Sigma, \mathcal{D})\simeq \mathcal{D}$ evaluates at $\phi(1)\in \mathrm{Fun}(\Sigma, \mathcal{C})$, i.e. at the symmetric sequence concentrated in degree $1 $ with value $ \mathbb{1}_{ \mathcal{C}}. $

Let $\mathrm{Z} \in \mathcal{D} $ corresponding to $ \bar{\mathrm{Z}} \in \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathcal{D}) .$

Then for every $\mathrm{X} \in \mathrm{Fun}(\Sigma, \mathcal{C})$ we have a canonical equivalence $$\bar{\mathrm{Z}}( \mathrm{X} ) \simeq \coprod_{ \mathrm{n} \geq 0 } \iota(\mathrm{X}_{ \mathrm{n}}) \otimes_{\Sigma_{\mathrm{n}} } \mathrm{Z}^{ \otimes \mathrm{n}}.$$

Taking $\mathcal{D}= \mathrm{Fun}(\Sigma, \mathcal{C})$ we get a canonical equivalence $$ \mathrm{Fun}(\Sigma, \mathcal{C}) \simeq \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathrm{Fun}(\Sigma, \mathcal{C})).$$

The category $\mathrm{Fun}(\Sigma, \mathcal{C}) \simeq \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathrm{Fun}(\Sigma, \mathcal{C})) $ carries a monoidal structure given by composition.

Given a monoidal category $\mathcal{B} $ denote $\mathcal{B}_\mathrm{rev }$ the reverse monoidal structure on $ \mathcal{B} $ with $\mathrm{X} \otimes_{ \mathcal{B}_\mathrm{rev } } \mathrm{Y} = \mathrm{Y} \otimes_{ \mathcal{B} } \mathrm{X}.$

One can define the composition product on $\mathrm{Fun}(\Sigma, \mathcal{C})$ to be the reverse monoidal structure of the monoidal structure on $ \mathrm{Fun}(\Sigma, \mathcal{C}) \simeq \mathrm{Fun}^\otimes_{ \mathcal{C} } (\mathrm{Fun}(\Sigma, \mathcal{C}) , \mathrm{Fun}(\Sigma, \mathcal{C})) $ given by composition.

Denote $\mathrm{Sym}(\mathcal{C})$ the category $\mathrm{Fun}(\Sigma, \mathcal{C})$ endowed with the composition product.

For $\mathrm{X}, \mathrm{Z} \in \mathrm{Sym}(\mathcal{C})$ their composition product $\mathrm{X} \circ \mathrm{Z} $ is by definition $$ (\bar{\mathrm{Z}} \circ \bar{\mathrm{X}}) (\phi(1)) \simeq \bar{\mathrm{Z}} (\mathrm{X}) \simeq \coprod_{ \mathrm{n} \geq 0 } \iota(\mathrm{X}_{ \mathrm{n}}) \otimes_{\Sigma_{\mathrm{n}} } \mathrm{Z}^{ \otimes \mathrm{n}}. $$

Let $\mathrm{Y} \in \mathcal{C} $. Then $\mathrm{X} \circ \iota(\mathrm{Y} ) \simeq \coprod_{ \mathrm{n} \geq 0 } \iota(\mathrm{X}_{ \mathrm{n}}) \otimes_{\Sigma_{\mathrm{n}} } \iota(\mathrm{Y})^{ \otimes \mathrm{n}} \simeq \iota(\coprod_{ \mathrm{n} \geq 0 } \mathrm{X}_{ \mathrm{n}} \otimes_{\Sigma_{\mathrm{n}} } \mathrm{Y}^{ \otimes \mathrm{n}}). $

As a monoidal category $\mathrm{Sym}(\mathcal{C})$ acts on itself from the left. So by the line above the left action of $\mathrm{Sym}(\mathcal{C})$ on $\mathrm{Sym}(\mathcal{C})$ restricts to a left action of $\mathrm{Sym}(\mathcal{C})$ on $ \mathcal{C} \subset \mathrm{Sym}(\mathcal{C})$.

Given a monoid $\mathcal{O}$ of $\mathrm{Sym}(\mathcal{C})$ denote $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C}):= \mathrm{LMod}_{ \mathcal{O} } ( \mathcal{C}) $ the category of left modules in $\mathcal{C}$ over $\mathcal{O}$ with respect to the left action of $\mathrm{Sym}(\mathcal{C})$ on $ \mathcal{C}. $

By the theory of endomorphism objects Higher Algebra 4.7.2 the category $\mathrm{Fun}( \mathcal{C}, \mathcal{C}) $ admits a monoidal structure given by composition that acts on $ \mathcal{C}$ in a universal way:

This implies that the left action of $\mathrm{Sym}(\mathcal{C})$ on $ \mathcal{C} $ is the pullback of the universal left action of $\mathrm{Fun}( \mathcal{C}, \mathcal{C})$ on $ \mathcal{C}$ along a unique monoidal functor $\mathrm{T}: \mathrm{Sym}(\mathcal{C}) \to \mathrm{Fun}( \mathcal{C}, \mathcal{C})$.

For every $\mathrm{Y} \in \mathcal{C} $ and $\mathcal{O} \in \mathrm{Sym}(\mathcal{C})$ one has $\mathrm{T}(\mathcal{O} )( \mathrm{Y}) \simeq \coprod_{ \mathrm{n} \geq 0 } \mathcal{O}_{ \mathrm{n}} \otimes_{\Sigma_{\mathrm{n}} } \mathrm{Y}^{ \otimes \mathrm{n}}. $

So $\mathrm{T}: \mathrm{Sym}(\mathcal{C}) \to \mathrm{Fun}( \mathcal{C}, \mathcal{C})$ induces a functor $\mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \to \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C}))$ that sends an associative algebra in $ \mathrm{Sym}(\mathcal{C})$ to its associated monad on $\mathcal{C}$.

As the left action of $\mathrm{Sym}(\mathcal{C})$ on $ \mathcal{C} $ is the pullback of the universal left action of $\mathrm{Fun}( \mathcal{C}, \mathcal{C})$ on $ \mathcal{C}$ along $\mathrm{T}: \mathrm{Sym}(\mathcal{C}) \to \mathrm{Fun}( \mathcal{C}, \mathcal{C})$, the map $$ \Phi: \mathrm{LMod}( \mathcal{C}) \to \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \times \mathcal{C} $$ of cartesian fibrations over $\mathrm{Alg}(\mathrm{Sym}(\mathcal{C}))$ is the pullback of the map $$ \Psi: \mathrm{LMod}( \mathcal{C}) \to \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C})) \times \mathcal{C} $$ of cartesian fibrations over $\mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C})) $ along the functor $ \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \to \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C}))$.

Denote ${\mathrm{Cat}_\infty}_{/ \mathcal{C} }$ the category of small $(\infty$-) categories over $\mathcal{C}. $

$\Psi$ classifies a functor $\alpha: \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C})) \to {\mathrm{Cat}_\infty}_{/ \mathcal{C} }$ that sends a monad to its category of algebras.

$\Phi $ classifies a functor $\beta: \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \to {\mathrm{Cat}_\infty}_{/ \mathcal{C} }$ that sends an associative algebra in $ \mathrm{Sym}(\mathcal{C})$ to its category of algebras.

As $ \Phi $ is the pullback of $ \Psi$ along the functor $ \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \to \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C}))$, the functor $\beta$ factors as $$\mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) \to \mathrm{Alg}(\mathrm{Fun}( \mathcal{C}, \mathcal{C})) \xrightarrow{\alpha} {\mathrm{Cat}_\infty}_{/ \mathcal{C} },$$ which provides a canonical equivalence $$ \mathrm{Alg}_{ \mathcal{O} } ( \mathcal{C} )\simeq \mathrm{Alg}_{ \mathrm{T}( \mathcal{O} ) } ( \mathcal{C} ) $$ over $ \mathcal{C} $ natural in $\mathcal{O} \in \mathrm{Alg}(\mathrm{Sym}(\mathcal{C})) . $

Answer 2

A quick and simple answer to your question is to consider the Cisinski-Heuts-Moerdijk-Weiss model of $\infty$-operads via dendroidal sets. This is known to be Quillen equivalent to the Lurie model.

In the dendroidal model, an $\infty$-operad is a presheaf $XX$ on $\Omega$, the category of rooted trees and blowup maps, and evaluation on corollas $C_n$ make the symmetric sequence structure evident.

For another direction, we can consider the lax operads of Brinkmeier, which are algebras over the Boardman-Vogt W-construction of the operad of $\mathfrak{C}$-coloured operads. In this case, the symmetric sequence is actually relaxed to hold only up to coherent homotopy.