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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.

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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})) . $

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  • $\begingroup$ This answer fails to distinguish between what is expected to be true and what has actually been proved infinity-categorically. In particular, although it is possible to define a composition product on symmetric sequences using the universal property of free cocomplete symmetric monoidal infinity-categories, as proposed here, it is not known that associative algebras in this monoidal infinity-category are equivalent to (enriched) infinity-operads in any other sense, such as Lurie's. $\endgroup$ – Rune Haugseng Apr 5 '18 at 8:03
  • $\begingroup$ (My preprint 1708.09632 gives an alternative construction of the composition product where this comparison holds, essentially by construction.) Given the composition product one can define algebras to be certain left modules in symmetric sequences, as proposed here, and it should be very straightforward to show that the resulting infinity-category is algebras for a monad with the expected formula. However, even in the unenriched case it is not known that this notion of algebras is equivalent to the usual one, as maps of infinity-operads into the symmetric monoidal infinity-category of spaces. $\endgroup$ – Rune Haugseng Apr 5 '18 at 8:09
  • $\begingroup$ I agree with Rune. This is definitely what should be true (and is known to be so 1-categorically), but I am unaware of suitable references for complete proofs in the $\infty$-categorical case. I ended up proving myself the bits I need (which are far less than the full comparison). $\endgroup$ – KotelKanim Apr 9 '18 at 7:24
  • $\begingroup$ I never did mention the word "operad" and only talked about associative algebras with respect to the composition product. What parts of my answer haven't been proved formally? $\endgroup$ – Hadrian Heine Apr 9 '18 at 11:23
  • $\begingroup$ You are right. The thing is that what I needed is algebras over an operad in the sense of Lurie. If you define an operad as an associative monoid in symmetric sequences, then your answer gives (an outline of) a proof, but it is the comparison of the two models which I hoped to avoid as it was not worked out yet by anyone as far as I know. $\endgroup$ – KotelKanim May 16 '18 at 5:59

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