用户: Scavenger/Infinity Category and Higher Algebra/Tensor Product of Infinity Operads

Let ${\mathcal{O}}^{\otimes }$ and ${\mathcal{O}}^{\prime \otimes }$ be $\infty$-operads. We will introduce a new $\infty$-operad ${\mathcal{O}}^{\prime \prime \otimes }$, called the tensor product of ${\mathcal{O}}^{\otimes }$ and ${\mathcal{O}}^{\prime \otimes }$. The major advantage of ${\mathcal{O}}^{\prime \prime \otimes }$ we want is the existence of an equivalence ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime \prime }}(\mathcal{C})\simeq {\mathrm{Alg}}_{\mathcal{O}}({\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}))$ for every symmetric monoidal infinity category ${\mathcal{C}}^{\otimes }$, which we will explain.

Definition 0.1 ([1] 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 0.2. 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 ${\mathcal{O}}^{\prime \prime \otimes }$.

We let $\mathrm{BiFunc}({\mathcal{O}}^{\otimes },{\mathcal{O}}^{\prime \otimes };{\mathcal{O}}^{\prime \prime \otimes })$ denote the full subcategory of ${\mathrm{Fun}}_{N({\mathrm{Fin}}_{\ast })}({\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes },{\mathcal{O}}^{\prime \prime \otimes })$ spanned by the bifunctors.

It is easy to see that given a bifunctor of $\infty$-operads $f:{\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\prime \prime t}$ and another $\infty$-operad ${\mathcal{C}}^{\otimes }$, there is a natural map ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime \prime }}(\mathcal{C})\to \mathrm{BiFunc}({\mathcal{O}}^{\otimes },{\mathcal{O}}^{\prime \otimes };{\mathcal{O}}^{\prime \prime \otimes })$. We say that $f$ exhibits ${\mathcal{O}}^{\prime \prime \otimes }$ as a tensor product of ${\mathcal{O}}^{\otimes }$ and ${\mathcal{O}}^{\prime \otimes }$ if the above map is an equivalence for any $\infty$-operad ${\mathcal{C}}^{\otimes }$.

Example 0.3. Let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad. Then there is a unique bifunctor of $\infty$-operads ${\mathrm{Comm}}^{\otimes }\times {\mathcal{O}}^{\otimes }\to {\mathrm{Comm}}^{\otimes }$.

It is clear that the tensor product of two $\infty$-operads, if exists, is uniquely determined up to equivalence of $\infty$-operads (that is, equivalence of $\infty$-categories over $N({\mathrm{Fin}}_{\ast })$ which and whose inverse are maps of $\infty$-operads).

Proposition 0.4 ([1] Proposition 2.2.5.6). Tensor product of two $\infty$-operads exists.

Example 0.5 ([1] Theorem 5.1.2.2). Let $k,k^{\prime }$ be nonnegative integers. [1] Construction 5.1.2.1 gives a map ${\mathbb{E}}_k^{\otimes }\times {\mathbb{E}}_{k^{\prime }}^{\otimes }\to {\mathbb{E}}_{k+k^{\prime }}^{\otimes }$ which exhibits ${\mathbb{E}}_{k+k^{\prime }}^{\otimes }$ as a tensor product of ${\mathbb{E}}_k^{\otimes }$ and ${\mathbb{E}}_{k^{\prime }}^{\otimes }$.

We now proceed to present the desired equivalence of algebra objects. We first explain the tensor product structure on algebra objects.

Construction 0.6 ([1] 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 ${\mathcal{O}}^{\otimes }$.

0.7 ([1] Remark 3.2.4.2). In the context of Construction 0.6, 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 ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }/{\mathcal{O}}^{\prime \prime }}(\mathcal{C})$.

We collect the crucial properties of Construction 0.6.

Proposition 0.8 ([1] Proposition 3.2.4.3). In the contex of Consturction 0.6, 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 0.9 ([1] 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 0.3. 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 0.8, 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 0.8.

0.10. Let ${\mathcal{C}}^{\otimes }$ be a symmetric monoidal $\infty$-category. Let $f:{\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes }\to {\mathcal{O}}^{\prime \prime \otimes }$ be a bifunctor of $\infty$-operads which exhibits ${\mathcal{O}}^{\prime \prime \otimes }$ as a tensor product of ${\mathcal{O}}^{\otimes }$ and ${\mathcal{O}}^{\prime \otimes }$. Let ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}{)}^{\otimes }$ be the symmetric monoidal $\infty$-category as in Example 0.9. By the definition of ${\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}{)}^{\otimes }$, we have a map ${\mathrm{Alg}}_{\mathcal{O}}({\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}{)}^{\otimes })\to {\mathrm{Fun}}_{/\mathrm{Com}{\mathrm{m}}^{\otimes }}({\mathcal{O}}^{\otimes },{\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}{)}^{\otimes })\cong {\mathrm{Fun}}_{/{\mathrm{Comm}}^{\otimes }}^{\prime }({\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes },{\mathcal{C}}^{\otimes })$, where ${\mathrm{Fun}}_{/{\mathrm{Comm}}^{\otimes }}^{\prime }({\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes },{\mathcal{C}}^{\otimes })$ means the full subcategory of ${\mathrm{Fun}}_{/{\mathrm{Comm}}^{\otimes }}({\mathcal{O}}^{\otimes }\times {\mathcal{O}}^{\prime \otimes },{\mathcal{C}}^{\otimes })$ spanned by those functors satisfying the second condition of Construction 0.6, and this is an inclusion of a full subcategory. It is not hard to show that the image of this map consists of those functors that are bifunctors of $\infty$-operads. It follows that we have an equivalence of $\infty$-categories ${\mathrm{Alg}}_{\mathcal{O}}({\mathrm{Alg}}_{{\mathcal{O}}^{\prime }}(\mathcal{C}{)}^{\otimes })\cong \mathrm{BiFunc}({\mathcal{O}}^{\otimes },{\mathcal{O}}^{\prime \otimes };{\mathcal{C}}^{\otimes })\stackrel{\sim }{\leftarrow }{\mathrm{Alg}}_{{\mathcal{O}}^{\prime \prime }}(\mathcal{C})$.

References

[1]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/