用户: Scavenger/Infinity Category and Higher Algebra/Infinity Operads

1Basic Definitions

Definition 1.1 ([2] Notation 2.0.0.2). We define a category ${\mathrm{Fin}}_{\ast }$ as follows:

$\bullet$ Objects of ${\mathrm{Fin}}_{\ast }$ are the pointed finite sets $⟨n⟩=\{\ast ,1,2,\cdots ,n\}$, for every $n\in {\mathbb{Z}}_{\ge 0}$. We will denote the subset $\{1,2,\cdots ,n\}$ by $⟨n{⟩}^{\circ }$.

$\bullet$ Morphisms from $⟨m⟩$ to $⟨n⟩$ are the morphisms of pointed finite sets, i.e. those maps of sets which bring $\ast$ to $\ast$.

The composition law is defined in the obvious way.

Notation 1.2. A morphism $f:⟨m⟩\to ⟨n⟩$ is called inert if $\#f^{-1}(i)=1$ (here $\#$ denotes cardinality) for all $i\in ⟨n⟩$. It is called active if $f^{-1}(\ast )=\{\ast \}$. Let $n$ be a natural number and let $i\in ⟨n{⟩}^{\circ }$, we use ${\rho }^i$ to denote the unique inert morphism $⟨n⟩\to ⟨1⟩$ such that ${\rho }^i(i)=1$.

Definition 1.3 ([2] Definition 2.1.1.10, Definition 2.1.2.3). An infinity operad is a morphism of infinity categories $p:{\mathcal{O}}^{\otimes }\to N(\mathrm{Fi}{\mathrm{n}}_{\ast })$ satisfying the following conditions:

(i) For every inert morphism $f:⟨m⟩\to ⟨n⟩$ of $\mathrm{Fi}{\mathrm{n}}_{\ast }$, there exists a $p$-coCartesian morphism lifting $f$.

(ii) Let $C\in {\mathcal{O}}_{⟨m⟩}^{\otimes }$ ( $C\in {\mathcal{O}}_{⟨m⟩}^{\otimes }$ means the fiber of $p$ over the vertex $⟨m⟩$) and $C^{\prime }\in {\mathcal{O}}_{⟨n⟩}^{\otimes }$ be objects. Let $f:⟨m⟩\to ⟨n⟩$ be a morphism in $\mathrm{Fi}{\mathrm{n}}_{\ast }$. Let ${\mathrm{Hom}}_{{\mathcal{O}}^{\otimes }}^f(C,C^{\prime })$ denote the full subcategory of ${\mathrm{Hom}}_{{\mathcal{O}}^{\otimes }}(C,C^{\prime })$ spanned by those morphisms lying over $f$. Choose $p$-cocartesian morphisms $C^{\prime }\to C_i^{\prime }$ lying over the inert morphisms ${\rho }^i=⟨n⟩\to ⟨1⟩$ for $i\in ⟨n{⟩}^{\circ }$. Then the induced map ${\mathrm{Hom}}_{{\mathcal{O}}^{\otimes }}^f(C,C^{\prime })\to {\prod }_{i\in ⟨n{⟩}^{\circ }}{\mathrm{Hom}}_{{\mathcal{O}}^{\otimes }}^{{\rho }^i\circ f}(C,C_i^{\prime })$ is a homotopy equivalence.

(iii) For every finite collection of objects $C_1,\cdots ,C_n\in {\mathcal{O}}_{⟨1⟩}^{\otimes }$, there exists an object $C\in {\mathcal{O}}_{⟨n⟩}^{\otimes }$ and $p$-cocartesian morphisms $C\to C_i$ covering ${\rho }^i:⟨n⟩\to ⟨1⟩$ for $i\in ⟨n{⟩}^{\circ }$.

In this situation we denote ${\mathcal{O}}_{⟨1⟩}^{\otimes }$ by $\mathcal{O}$ and call it the underlying $\infty$-category of the $\infty$-operad $\mathcal{O}$. For $n\in {\mathbb{Z}}_{\ge 0}$, pushforwards along ${\rho }^i,1\le i\le n$ (notice that $p$ is a coCartesion fibration over ${\rho }^i$) give an equivalence of $\infty$-categories ${\mathcal{O}}_{⟨n⟩}^{\otimes }\stackrel{\sim }{\to }{\mathcal{O}}^n$.

We say that a morphism $f$ in ${\mathcal{O}}^{\otimes }$ is inert if $p(f)$ is inert and $f$ is $p$-coCartesian. It is active if $p(f)$ is active.

Definition 1.4 ([2] Definition 2.1.1.1). A colored operad $\mathcal{O}$ is the collection of the following data:

(i) A set $\{X,Y,\cdots \}$ whose elements we will refer to as objects or colors of $\mathcal{O}$.

(ii) For every finite sequence $X_1,\cdots X_n$ of objects of $\mathcal{O}$, and every object $Y$ of $\mathcal{O}$, a set ${\mathrm{Mul}}_{\mathcal{O}}(X_1,\cdots ,X_n;Y)$, which we call the set of morphisms from $X_1,\cdots ,X_n$ to $Y$.

(iii) A composition law that is required to satisfy certain conditions. (For details, see [2] Definition 2.1.1.1.)

Construction 1.5. Let $\mathcal{O}$ be a colored operad, we can construct a category ${\mathcal{O}}^{\otimes }$ equipped with a functor ${\mathcal{O}}^{\otimes }\to {\mathrm{Fin}}_{\ast }$ (see [2] Construction 2.1.1.7). This functor makes $N({\mathcal{O}}^{\otimes })$ into an infinity operad ([2] Proposition 2.1.1.27).

This construction has a generalization to simplicial colored operads.

Example 1.6. The identity map $N({\mathrm{Fin}}_{\ast })\to N({\mathrm{Fin}}_{\ast })$ exhibits $N({\mathrm{Fin}}_{\ast })$ as an $\infty$-operad, which will be called the commutative $\infty$-operad. We will also denote it by ${\mathrm{Comm}}^{\otimes }$

Definition 1.7. Let ${\mathcal{O}}^{\otimes }$ and ${\mathcal{O}}^{\prime \otimes }$ be $\infty$-operads. An $\infty$-operad map from ${\mathcal{O}}^{\otimes }$ to ${\mathcal{O}}^{\prime \otimes }$ is a map of simplicial sets over $N({\mathrm{Fin}}_{\ast })$ that brings inert morphisms to inert morphisms.

We use ${\mathrm{Alg}}_{{\mathcal{O}}^{\otimes }}({\mathcal{O}}^{\prime \otimes })$ (or simply ${\mathrm{Alg}}_{\mathcal{O}}({\mathcal{O}}^{\prime })$) to denote the full subcategory of ${\mathrm{Fun}}_{/N({\mathrm{Fin}}_{\ast })}({\mathcal{O}}^{\otimes },{\mathcal{O}}^{\prime \otimes })$ spanned by the $\infty$-operad maps.

A map of $\infty$-operads is a fibration of $\infty$-operads if its underlying map of simplicial sets is an isofibration. It a coCartesian fibration of $\infty$-operads if its underlying map of simplicial sets is a coCartesian fibration.

Notation 1.8. Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad. An ${\mathcal{O}}^{\otimes }$-monoidal $\infty$-category (or simply $\mathcal{O}$-monoidal $\infty$-category) is a coCartesian fibration of $\infty$-operads ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$. A symmetric monoidal $\infty$-category is a ${\mathrm{Comm}}^{\otimes }$-monoidal $\infty$-category.

2Model Structure for $\infty$-Operads

Definition 2.1. Recall that a marked simplicial set is a pair $(X,M)$ where $X$ is a simplicial set and $M\subseteq X_2$ is a set of edges of $X$, which contains all degenerate edges. An edge of $X$ is said to be marked if it belongs to $M$. A morphism between two marked simplicial sets $(X,M)$ and $(Y,N)$ is a map of simplicial sets $X\to Y$ that brings marked edges of $X$ to marked edges of $Y$.

If $X$ is a simplicial set, we use $X^{♯}$ to denote the marked simplicial set $(X,M)$ where $M$ consists of all edges of $X$, and $X^{♭}$ to denote the marked simplicial set $(X,M^{\prime })$ where $M^{\prime }$ consists of all degenerate edges of $X$.

Definition 2.2 ([2] Definition 2.1.4.2, Notation 2.1.4.5). An $\infty$-preoperad is a marked simplicial set over the marked simplicial set $(N({\mathrm{Fin}}_{\ast }),M)$, where $M$ consists of all inert morphisms in $N({\mathrm{Fin}}_{\ast })$. The $\infty$-preoperads form a category $\mathcal{P}{\mathrm{Op}}_{\infty }$, which is precisely the overcategory of the category of marked simplicial sets over $(N({\mathrm{Fin}}_{\ast }),M)$. Thus it inherits a simplicial enrichment as the overcategory of a simplicial category (see [2] Remark 2.1.4.4 for details).

Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad, we use ${\mathcal{O}}^{\otimes ,♮}$ to denote the $\infty$-preoperad $({\mathcal{O}}^{\otimes },M)$, where $M$ consists of all inert morphisms in ${\mathcal{O}}^{\otimes }$.

Proposition 2.3 ([2] Proposition 2.1.4.6). There exists a simplicial model structure on $\mathcal{P}{\mathrm{Op}}_{\infty }$ such that a morphism $X\to Y$ is a cofibration if and only if it induces an injection on the underlying simplicial set, a weak equivalence if and only if the induced map ${\mathrm{Mor}}_{\mathcal{P}{\mathrm{Op}}_{\infty }}(Y,{\mathcal{O}}^{\otimes ,♮})\to {\mathrm{Mor}}_{\mathcal{P}{\mathrm{Op}}_{\infty }}(X,{\mathcal{O}}^{\otimes ,♮})$ is a weak homotopy equivalence of simplicial sets for every $\infty$-operad ${\mathcal{O}}^{\otimes }$.

Moreover, if ${\mathcal{O}}^{\otimes }$ is an $\infty$-operad, then a map $X\to {\mathcal{O}}^{\otimes ,♮}$ in $\mathcal{P}{\mathrm{Op}}_{\infty }$ is a fibration if and only if it is isomorphic to ${\mathcal{C}}^{\otimes ,♮}\to {\mathcal{O}}^{\otimes ,♮}$ (in the slice category $\mathcal{P}{\mathrm{Op}}_{\infty /{\mathcal{O}}^{\otimes ,♮}}$) for some fibration of $\infty$-operads ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$.

Corollary 2.4. Let ${\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\otimes }\stackrel{f}{\leftarrow }{\mathcal{O}}^{\prime \prime \otimes }$ be maps of $\infty$-operads, let ${\mathcal{C}}^{\otimes }:={\mathcal{O}}^{\prime \otimes }{\times }_{{\mathcal{O}}^{\otimes }}{\mathcal{O}}^{\prime \prime \otimes }$ (fiber product as simplicial sets), then if $f$ is a fibration of $\infty$-operads, then ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\prime \otimes }$ is also a fibration of $\infty$-operads. If moreover $f$ is a coCartesian fibration of $\infty$-operads, then so is ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\prime \otimes }$.

3Generalized $\infty$-Operads

Definition 3.1 ([2] Definition 2.3.2.1). A generalized $\infty$-operad is an $\infty$-category ${\mathcal{O}}^{\otimes }$ equipped with a map $q:{\mathcal{O}}^{\otimes }\to N({\mathrm{Fin}}_{\ast })$ satisfying the following conditions:

(i) For every object $X\in {\mathcal{O}}^{\otimes }$ and every inert morphism $\alpha :q(X)\to ⟨n⟩$, there exists a $q$-coCartesian morphism $\overline{\alpha }:X\to Y$ such that $q(\overline{\alpha })=\alpha$.

(ii) Suppose we are given a commutative diagram $\sigma$:

in $N({\mathrm{Fin}}_{\ast })$ which consists of inert morphisms and induces a bijection of finite sets $⟨m^{\prime }{⟩}^{\circ }{\coprod }_{⟨n^{\prime }{⟩}^{\circ }}⟨n{⟩}^{\circ }\to ⟨m{⟩}^{\circ }$. Then the induced diagram

is a pullback diagram of $\infty$-categories. (This condition depends only on the coCartesian fibration ${\mathcal{O}}^{\otimes }{\times }_{N({\mathrm{Fin}}_{\ast })}({\Delta }^1\times {\Delta }^1)\to ({\Delta }^1\times {\Delta }^1)$, by virtue of, for example, [3] tag 02T8, so don’t worry.)

(iii) Let $\sigma$ be as in (ii), and suppose that $\sigma$ can be lifted to a diagram $\overline{\sigma }$

consisting of $q$-coCartesian morphisms in ${\mathcal{O}}^{\otimes }$. Then $\overline{\sigma }$ is a $q$-limit diagram.

An $\infty$-operad is a generalized $\infty$-operad by virtue of [2] Proposition 2.3.2.5.

Notation 3.2 ([2] Definition 2.3.2.2). Let $q:{\mathcal{O}}^{\otimes }\to N({\mathrm{Fin}}_{\ast })$ be a generalized $\infty$-operad. We will say that a morphism $\alpha$ in ${\mathcal{O}}^{\otimes }$ is inert if $q(\alpha )$ is inert in $N({\mathrm{Fin}}_{\ast })$ and $\alpha$ is $q$-coCartesian. Let $q^{\prime }:{\mathcal{O}}^{\prime \otimes }\to N({\mathrm{Fin}}_{\ast })$ be another generalized $\infty$-operad. A morphism of simplicial sets $f:{\mathcal{O}}^{\otimes }\to {\mathcal{O}}^{\prime \otimes }$ is a map of generalized $\infty$-operads if $q=q^{\prime }\circ f$ and $f$ carries inert morphisms in ${\mathcal{O}}^{\otimes }$ to inert morphisms in ${\mathcal{O}}^{\prime \otimes }$. It is a fibration of generalized $\infty$-operads if in addition it is an isofibration of $\infty$-categories.

It is easy to see that if $q:{\mathcal{O}}^{\otimes }\to N({\mathrm{Fin}}_{\ast })$ is a generalized $\infty$-operad, then $q$ is a fibration of generalized $\infty$-operad.

Example 3.3 ([2] Proposition 2.3.2.9 (1)). Let $\mathcal{C}$ be an $\infty$-category, then the product $\mathcal{C}\times N({\mathrm{Fin}}_{\ast })$ is a generalized $\infty$-operad.

Remark 3.4. In [1] a (seemingly) different notion, called an $\mathcal{O}$-operad family, is defined for every $\infty$-operad ${\mathcal{O}}^{\otimes }$. These two notions coincides in the sense below:

Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad and let ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ be a map of simplicial sets. Then ${\mathcal{C}}^{\otimes }$ is an $\mathcal{O}$-operad family if and only if ${\mathcal{C}}^{\otimes }$ is a generalized $\infty$-operad and ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ is a fibration of generalized $\infty$-operads, by virtue of [2] Proposition B.2.7.

Model Structure for Generalized $\infty$-Operads

Construction 3.5 ([2] Remark 2.3.2.4). Let $\mathcal{P}{\mathrm{Op}}_{\infty }$ be the category of $\infty$-preoperads, then there exists a model structure on $\mathcal{P}{\mathrm{Op}}_{\infty }$ such that:

(i) A morphism is a cofibration if and only if the underlying map of simplicial sets is a monomorphism.

(ii) An object $\overline{X}$ is fibrant if and only if it is isomorphic to ${\mathcal{O}}^{\otimes ,♮}$ for some generalized $\infty$-operad ${\mathcal{O}}^{\otimes }$, where ${\mathcal{O}}^{\otimes ,♮}$ denotes the marked simplicial set whose underlying simplicial set is ${\mathcal{O}}^{\otimes }$, and whose marked edges are the inert edges.

(iii) Let ${\mathcal{O}}^{\otimes }$ be a generalized $\infty$-operad, then a map $\overline{X}\to {\mathcal{O}}^{\otimes }$ is a fibration if and only if it is isomorphic to ${\mathcal{C}}^{\otimes ,♮}\to {\mathcal{O}}^{\otimes ,♮}$ (in the slice category $\mathcal{P}{\mathrm{Op}}_{\infty /{\mathcal{O}}^{\otimes ,♮}}$) for some fibration of generalized $\infty$-operads ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$.

Corollary 3.6. Let ${\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\otimes }\stackrel{f}{\leftarrow }{\mathcal{O}}^{\prime \prime \otimes }$ be maps of generalized $\infty$-operads, let ${\mathcal{C}}^{\otimes }:={\mathcal{O}}^{\prime \otimes }{\times }_{{\mathcal{O}}^{\otimes }}{\mathcal{O}}^{\prime \prime \otimes }$ (fiber product as simplicial sets), then if $f$ is a fibration of generalized $\infty$-operads, then ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\prime \otimes }$ is also a fibration of generalized $\infty$-operads.

4Bifunctor of $\infty$-Operads

Our goal in this section is to construct the monoidal structure on the $\infty$-categories of algebra objects.

Definition 4.1 ([2] Notation 2.2.5.1). Define a functor $\wedge :{\mathrm{Fin}}_{\ast }\times {\mathrm{Fin}}_{\ast }\to {\mathrm{Fin}}_{\ast }$ as follows:

(i) On objects, $\wedge$ is given by the formula $\wedge (⟨m⟩,⟨n⟩)=⟨mn⟩$.

(ii) Let $f:⟨m⟩\to ⟨m^{\prime }⟩$ and $g:⟨n⟩\to ⟨n^{\prime }⟩$ be morphisms in ${\mathrm{Fin}}_{\ast }$. We define $\wedge (f,g)$ to be given by the formula: for $a\in ⟨m{⟩}^{\circ }$ and $b\in ⟨n{⟩}^{\circ }$,

$$\wedge (f,g)(an+b-n)=\{\begin{array}{rlrl}& \ast & & \text{if }f(a)=\ast \text{ or }g(b)=\ast \\ & f(a)n^{\prime }+g(b)-n^{\prime }& & \text{otherwise}\end{array}$$(1)

Definition 4.2. Let ${\mathcal{O}}^{\otimes },{\mathcal{O}}^{\prime \otimes }$ and ${\mathcal{O}}^{\prime \prime \otimes }$ be $\infty$-operads. A bifunctor of $\infty$-operads is a map $f:{\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\prime \prime \otimes }$ of $\infty$-categories satisfying the following conditions.

(i) The diagramcommutes

(ii) For every inert morphisms $\alpha$ of ${\mathcal{O}}^{\otimes }$ and $\beta$ of ${\mathcal{O}}^{\prime \otimes }$, $f(a,b)$ is an inert morphism in ${\mathcal{O}}^{\prime \prime \otimes }$.

Example 4.3. Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad. Then there is a unique bifunctor of $\infty$-operads ${\mathrm{Comm}}^{\otimes }\times {\mathcal{O}}^{\otimes }\to {\mathrm{Comm}}^{\otimes }$.

Construction 4.4 ([2] Construction 3.2.4.1). Let $f:{\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\prime \prime \otimes }$ be a bifunctor of $\infty$-operads. We construct a simplicial set ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }$ over ${\mathcal{O}}^{\otimes }$ by the following universal property: Let $K\to {\mathcal{O}}^{\otimes }$ be a map of simplicial sets, then there is a bijection between ${\mathrm{Hom}}_{({\mathrm{Set}}_{\Delta }{)}_{/{\mathcal{O}}^{\otimes }}}(K,{\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes })$ and the set of commutative diagrams

such that $F$ has the following property: for every vertex $v\in K$ and every inert morphism $\alpha$ in ${\mathcal{O}}^{\prime \otimes }$, the morphism $F({\mathrm{id}}_v,\alpha )$ is inert.

Or in concrete terms, taking $K={\Delta }^n$ in the above construction, we get the $n$-simplices of ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }$ over some specific $n$-simplex of ${\mathcal{O}}^{\otimes }$, and this construction is functorial in a sense that it indeed gives a simplicial set over ${\mathcal{O}}^{\otimes }$.

4.5 ([2] Remark 3.2.4.2). In the context of Construction 4.4, let $x\in \mathcal{O}={\mathcal{O}}_{⟨1⟩}^{\otimes }$ be an object. The restriction of $f$ to $\{x\}\times {\mathcal{O}}^{\prime \otimes }$ determines a map of $\infty$-operads ${\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\prime \prime \otimes }$, and we have a canonical isomorphism (of simplicial sets) of the fiber ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}_x^{\otimes }={\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }{\times }_{{\mathcal{O}}^{\otimes }}\{x\}$ and the $\infty$-category ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C})$.

Proposition 4.6 ([2] Proposition 3.2.4.3). In the contex of Consturction 4.4, we have:

(i) The map $p:{\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }\to {\mathcal{O}}^{\otimes }$ is a fibration of $\infty$-operads.

(ii) A morphism $\alpha$ in ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }$ is inert if and only if $p(\alpha )$ is inert and for every object $x\in {\mathcal{O}}^{\prime }$, the composition ${\Delta }^1\times \{x\}\to {\Delta }^1\times {\mathcal{O}}^{\prime \otimes }\stackrel{\alpha }{\to }{\mathcal{C}}^{\otimes }$ is an inert morphism in ${\mathcal{C}}^{\otimes }$. Here ${\Delta }^1\times {\mathcal{O}}^{\prime \otimes }\stackrel{\alpha }{\to }{\mathcal{C}}^{\otimes }$ is the map given by the universal property of ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }$.

(iii) Suppose that $q$ is a coCartesian fibration of $\infty$-operads. Then $p$ is a coCartesian fibration of $\infty$-operads.

(iv) Suppose that $q$ is a coCartesian fibration of $\infty$-operads. Then a morphism $\alpha$ in ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C}{)}^{\otimes }$ is $p$-coCartesian if and only if, for every object $x\in {\mathcal{O}}^{\prime }$, the composition ${\Delta }^1\times \{x\}\to {\Delta }^1\times {\mathcal{O}}^{\prime \otimes }\stackrel{\alpha }{\to }{\mathcal{C}}^{\otimes }$ is a $q$-coCartesian morphism in ${\mathcal{C}}^{\otimes }$.

Example 4.7 ([2] Example 3.2.4.4). Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad and let ${\mathrm{Comm}}^{\otimes }\times {\mathcal{O}}^{\otimes }\to {\mathrm{Comm}}^{\otimes }$ be the bifunctor of $\infty$-operads of Example 4.3. Let ${\mathcal{C}}^{\otimes }$ be a symmetric monoidal $\infty$-category, then ${\mathrm{Alg}}_{\mathcal{O}/\mathrm{Comm}}(\mathcal{C}{)}^{\otimes }$ is a symmetric monoidal $\infty$-category by Proposition 4.6, which we will denote by ${\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C}{)}^{\otimes }$. The underlying $\infty$-category of this symmetric monoidal $\infty$-category is the $\infty$-category ${\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C})$ of algebra objects.

For every object $x\in \mathcal{O}$ we have an evaluation functor ${\mathrm{ev}}_x:{\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\to {\mathcal{C}}^{\otimes }$ defined as the composition ${\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\times \{x\}\to {\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C}{)}^{\otimes }\times {\mathcal{O}}^{\otimes }\to {\mathcal{C}}^{\otimes }$. This is a symmetric monoidal functor by virtue of Proposition 4.6.

References

[1]	Jacob Lurie, “Derived Algebraic Geometry III: Commutative Algebra”, October 21, 2009 version, Avalible for download at https://www.math.ias.edu/ lurie/papers/DAG-III.pdf
[2]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/
[3]	Jacob Lurie, “Kerodon”, https://kerodon.net