用户: Scavenger/Infinity Category and Higher Algebra/Smash Product of Spectra

Our goal in this notes is to construct a symmetric monoidal structure on the $\infty$-category of spectra, namely the smash product.

1The Main Construction

Construction 1.1 ([3] Notation 4.8.1.2). Let $\varkappa$ be an inaccessible cardinal. We construct an $\infty$-category ${\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }$ as the nerve of the following locally Kan simplicial category:

(i) The objects are finite sequences $[X_1,\cdots ,X_n]$, where each $X_i$ is a $\varkappa$-$\infty$-category.

(ii) Given a pair of objects $[X_1,\cdots ,X_n]$ and $[Y_1,\cdots ,Y_m]$, the morphism space from the former object to the latter is ${\coprod }_{\alpha :⟨n⟩\to ⟨m⟩}({\prod }_{j\in ⟨m{⟩}^{\circ }}\mathrm{Fun}({\prod }_{i\in {\alpha }^{-1}(j)}{X}_i,Y_j{)}^{\simeq })$.

Now we assume $\varkappa$ is $1$-inaccessible. Let $P$ be the set of $\varkappa$-sets whose elements are $\varkappa$-simplicial sets, partially ordered by inclusion. We define a subcategory $\mathcal{M}$ of ${\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }\times N(P)$ as follows:

(i) An object $([X_1,\cdots ,X_n],\mathcal{K})$ belongs to $\mathcal{M}$ if and only if each of the $\infty$-category $X_i$ admits $K$-indexed colimits for every $K\in \mathcal{K}$.

(ii) Let $f:([X_1,\cdots ,X_n],{\mathcal{K}}_1)\to ([Y_1,\cdots ,Y_m],{\mathcal{K}}_2)$ be a morphism, covering a map $\alpha :⟨n⟩\to ⟨m⟩$. It belongs to $\mathcal{M}$ if and only if each of the associated functors ${\prod }_{i\in {\alpha }^{-1}(j)}{X}_i\to Y_j$ preserves $K$-indexed colimits separately in each variable for every $k\in {\mathcal{K}}_1$.

Notice that this subcategory can also be realized as the nerve of a locally Kan simplicial category.

If $\mathcal{K}\in P$, Let ${\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa }$ denote the fiber product $\mathcal{M}{\times }_{N(P)}\{\mathcal{K}\}$.

Proposition 1.2 ([3] Proposition 4.8.1.3, Corollary 4.8.1.4). Let $\varkappa$ be an $1$-inaccessible cardinal. Then the functor $\mathcal{M}\to N({\mathrm{Fin}}_{\ast })\times N(P)$ is a coCartesian fibration.

Let $\mathcal{K}$ be a $\varkappa$-set whose elements are $\varkappa$-simplicial sets. Then the category ${\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa }$ is a symmetric monoidal $\infty$-category. Let ${\mathcal{K}}_1\subseteq {\mathcal{K}}_2$ be two $\varkappa$-sets of $\varkappa$-simplicial sets, then the inclusion ${\mathrm{Cat}}_{\infty }({\mathcal{K}}_2{)}^{\otimes ,\varkappa }\to {\mathrm{Cat}}_{\infty }({\mathcal{K}}_1{)}^{\otimes ,\varkappa }$ is a map of $\infty$-operads.

Remark 1.3. Let $\varkappa$ be an inaccessible cardinal. Then ${\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }$ is a symmetric monoidal $\infty$-category, and there is an evident functor ${\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }\to {\mathrm{Cat}}_{\infty }^{\varkappa }$, bringing $[X_1,\cdots ,X_n]$ to ${\prod }_{i=1}^n{X}_i$, which is a Cartesian structure (in the sense of [3] Definition 2.4.1.1).

Symmetric Monoidal $\infty$-Categories as Commutative Algebra Objects

We briefly review how a symmetric monoidal $\infty$-category can be identified with a commutative algebra object of the symmetric monoidal $\infty$-category ${\mathrm{Cat}}_{\infty }^{\otimes }$.

Definition 1.4 ([3] Definition 2.4.2.1). Let $\mathcal{C}$ be an $\infty$-category and let ${\mathcal{O}}^{\otimes }$ be an $\infty$-operad. An ${\mathcal{O}}^{\otimes }$-monoid in $\mathcal{C}$ is a functor of $\infty$-categories $M:{\mathcal{O}}^{\otimes }\to \mathcal{C}$ with the following property: for every object $X\in {\mathcal{O}}_{⟨n⟩}^{\otimes }$ corresponding to objects $X_1,\cdots ,X_n\in \mathcal{O}={\mathcal{O}}_{⟨1⟩}^{\otimes }$, the natural maps $M(X)\to M(X_i)$ exhibit $M(X)$ as a product of $M(X_i)$ in the infinity category $C$. We let ${\mathrm{Mon}}_{\mathcal{O}}(\mathcal{C})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{O}}^{\otimes },\mathcal{C})$ spanned by the ${\mathcal{O}}^{\otimes }$-monoids in $\mathcal{C}$.

In the special case where ${\mathcal{O}}^{\otimes }$ is the commutative $\infty$-operad, we will refer to ${\mathcal{O}}^{\otimes }$-monoids as commutative monoids.

Proposition 1.5 ([3] Example 2.4.2.4). Let $\varkappa$ be an inaccessible cardinal. Let ${\mathcal{O}}^{\otimes }$ be a $\varkappa$-$\infty$-operad. A functor $M:{\mathcal{O}}^{\otimes }\to {\mathrm{Cat}}_{\infty }^{\varkappa }$ is an ${\mathcal{O}}^{\otimes }$ monoid if and only if the corresponding coCartesian fibration ${\mathcal{C}}^{\otimes }\to {\mathcal{O}}^{\otimes }$ is an ${\mathcal{O}}^{\otimes }$-monoidal $\infty$-category.

In the case where $\mathcal{C}$ has finite products, the notion of ${\mathcal{O}}^{\otimes }$-monoids is equivalent to that of ${\mathcal{O}}^{\otimes }$-algebra:

Proposition 1.6 ([3] Proposition 2.4.2.5). Let ${\mathcal{C}}^{\otimes }$ ba a symmetric monoidal $\infty$-category, $\pi :{\mathcal{C}}^{\otimes }\to \mathcal{D}$ a Cartesian structure (in the sense of [3] Definition 2.4.1.1), and ${\mathcal{O}}^{\otimes }$ an $\infty$-operad. Then composition with $\pi$ induces an equivalence of $\infty$-categories ${\mathrm{Alg}}_{\mathcal{O}}(\mathcal{C})\to {\mathrm{Mon}}_{\mathcal{O}}(\mathcal{D})$.

Combining Proposition 1.5, Proposition 1.6 and Remark 1.3, we obtain the following:

Proposition 1.7. Let $\varkappa$ be an inaccessible cardinal. Then there are equivalences of $\infty$-categories:

${\mathrm{Cat}}_{\infty }^{\mathrm{Comm},\varkappa }\simeq {\mathrm{Mon}}_{\mathrm{Comm}}({\mathrm{Cat}}_{\infty }^{\varkappa })\stackrel{\sim }{\leftarrow }\mathrm{CAlg}({\mathrm{Cat}}_{\infty }^{\varkappa })$

where ${\mathrm{Cat}}_{\infty }^{\mathrm{Comm},\varkappa }$ is the $\infty$-category of $\varkappa$-symmetric monoidal $\infty$-categories.

1.8. Let $\varkappa$ be an $1$-inaccessible cardinal, let $\mathcal{K}$ be a $\varkappa$-set of $\varkappa$-simplicial sets. Then the inclusion ${\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa }\to {\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }$ gives rise to an inclusion $\mathrm{CAlg}({\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa })\to \mathrm{CAlg}({\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa })$, identifying $\mathrm{CAlg}({\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa })$ with the subcategory of ${\mathrm{Cat}}_{\infty }^{\mathrm{Comm},\varkappa }$ spanned by the symmetric monoidal $\infty$-categories whose symmetric monoidal structure is compatible with $\mathcal{K}$-colimits (that is, the underlying $\infty$-category admits $\mathcal{K}$-colimits, and the tensor product functor $\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ preserves $\mathcal{K}$-colimits separately in each variable), and the symmetric monoidal functors between these symmetric monoidal $\infty$-categories whose restriction to the underlying $\infty$-categories preserves $\mathcal{K}$-colimits.

The coCartesian fibration $\mathcal{M}{\times }_{N(P)}\{\varnothing \to \mathcal{K}\}\to N({\mathrm{Fin}}_{\ast })\times {\Delta }^1$ provides a left adjoint to the inclusion ${\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa }\to {\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa }$ (this adjunction is actually an adjunction of inner fibrations over $N({\mathrm{Fin}}_{\ast })$), and this left adjoint is a symmetric monoidal functor. This symmetric monoidal functor is given on objects by $\mathcal{C}\to {\mathcal{P}}_{\varnothing }^{\mathcal{K}}(\mathcal{C})$ (see [4] Proposition 5.3.6.2 for the definition of ${\mathcal{P}}_{\varnothing }^{\mathcal{K}}(\mathcal{C})$).

Consequently, the inclusion $\mathrm{CAlg}({\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\otimes ,\varkappa })\to \mathrm{CAlg}({\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa })$ admits a left adjoint, bringing a symmetric monoidal $\infty$-category $\mathcal{C}$ to ${\mathcal{P}}_{\varnothing }^{\mathcal{K}}(\mathcal{C})$. This left adjoint is symmetric monoidal (with respect to the natural symmetric monoidal structure on the $\infty$-category of commutative algebra objects in a symmetric monoidal $\infty$-category, which is coCartesian), and consequently preserves initial objects.

Let ${\varkappa }_0<\varkappa$ be an inaccessible cardinal. We specialize to the case where $\mathcal{K}$ is the set of ${\varkappa }_0$-simplicial sets. The initial object of $\mathrm{CAlg}({\mathrm{Cat}}_{\infty }^{\otimes ,\varkappa })$ is given by the symmetric monoidal $\infty$-category ${\Delta }^0$, and since ${\mathcal{P}}_{\varnothing }^{\mathcal{K}}({\Delta }^0)\simeq {\mathcal{S}}^{{\varkappa }_0}$, we obtain a symmetric monoidal $\infty$-category ${\mathcal{C}}^{\otimes }$ and an equivalence ${\mathcal{S}}^{{\varkappa }_0}\simeq \mathcal{C}$, bringing ${\Delta }^0\in {\mathcal{S}}^{{\varkappa }_0}$ to the unit object of $\mathcal{C}$ (the last assertion can be seen by contemplating at the unit of the adjunction constructed above).

We now fix an $1$-inaccessible cardinal $\varkappa$ and an inaccessible cardinal ${\varkappa }_0<\varkappa$. From now on when we talk about presentable $\infty$-categories we always means the notion with size parameter ${\varkappa }_0$. Let $\mathcal{K}$ be the set of ${\varkappa }_0$ simplicial sets.

Proposition 1.9 ([3] Proposition 4.8.1.15). The full subcategory ${\mathrm{Pr}}^{L,\varkappa }\subseteq {\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\varkappa }$ of presentable $\varkappa$-$\infty$-categories (and functors preserving ${\varkappa }_0$-colimits between them) is closed under tensor products in ${\mathrm{Cat}}_{\infty }(\mathcal{K}{)}^{\varkappa }$, and therefore inherits a symmetric monoidal structure by virtue of [3] Proposition 2.2.1.1.

Proposition 1.10 ([3] Proposition 4.8.2.18 or [2] Remark 4.1.13). Let ${\mathrm{Cat}}_{\infty }^{LPr,\sigma ,\varkappa }$ denote the full subcategory of ${\mathrm{Pr}}^{L,\varkappa }$ spanned by the stable $\infty$-categories. By virtue of [3] Corollary 1.4.4.5 the inclusion ${\mathrm{Cat}}_{\infty }^{LPr,\sigma ,\varkappa }\subseteq {\mathrm{Pr}}^{L,\varkappa }$ admits a left adjoint given by $\mathcal{C}\to \mathrm{Sp}(\mathcal{C})$, and the unit of this adjunction is given by ${\Sigma }^{\infty }$. The reflective subcategory ${\mathrm{Cat}}_{\infty }^{LPr,\sigma ,\varkappa }\subseteq {\mathrm{Pr}}^{L,\varkappa }$ satisfies the condition of [3] Definition 2.2.1.6, and therefore ${\mathrm{Cat}}_{\infty }^{LPr,\sigma ,\varkappa }$ inherits a symmetric monoidal structure and the left adjoint is a symmetric monoidal functor, by virture of [3] Proposition 2.2.1.9.

The various adjunctions constructed above can all be viewed as adjunctions of inner fibrations over $N({\mathrm{Fin}}_{\ast })$, and therefore give adjunctions between $\infty$-categories of commutative algebra objects. Now we can now prove our main theorem:

Proposition 1.11. Let $\mathrm{Sp}=\mathrm{Sp}({\mathcal{S}}^{{\varkappa }_0})$ be the $\infty$-category of spectra. Then there exists a symmetric monoidal $\infty$-category ${\mathcal{C}}^{\otimes }$ and an equivalence $\mathrm{Sp}\simeq \mathcal{C}$, bringing the sphere spectrum to the unit object of $\mathcal{C}$, such that, for any symmetric monoidal $\infty$-category ${\mathcal{D}}^{\otimes }$ satisfying the following conditions:

(i) $\mathcal{D}$ is stable and presentable.

(ii) The tensor product functor $\mathcal{D}\times \mathcal{D}\to \mathcal{D}$ preserves ${\varkappa }_0$ colimits separatly in each variable.

the $\infty$-category ${\mathrm{Fun}}^{\otimes ,\mathrm{colim}}(\mathcal{C},\mathcal{D})$ of symmetric monoidal functors from $\mathcal{C}$ to $\mathcal{D}$ whose restriction to the underlying $\infty$-categories preserves ${\varkappa }_0$-colimits is a contractible Kan complex. If moreover the essentially unique functor $\mathrm{Sp}\to \mathcal{D}$ preserving ${\varkappa }_0$-colimits and bringing the sphere specturm to the unit object of $\mathcal{D}$ is an equivalence, then the essentially unique symmetric monoidal functor ${\mathcal{C}}^{\otimes }\to {\mathcal{D}}^{\otimes }$ is an equivalence.

Proof. Simply choose $\mathcal{C}$ to be the initial object of $\mathrm{CAlg}({\mathrm{Cat}}_{\infty }^{LPr,\sigma ,\varkappa })$.

$\square$

2Properties and Miscellany

Proposition 2.1. Consider the category $\mathrm{GAb}:=\mathrm{Fun}(\mathbb{Z},\mathrm{Ab})$ of graded abelian groups, whose elements we denote as $(A_i{)}_{i\in \mathbb{Z}}$. This category admits a symmetric monoidal structure, given by $((A_i{)}_{i\in \mathbb{Z}}\otimes (B_j{)}_{j\in \mathbb{Z}}{)}_k={⨁}_{i+j=k}(A_i\otimes B_j)$ (and the isomorphism $(A_i{)}_{i\in \mathbb{Z}}\otimes (B_j{)}_{j\in \mathbb{Z}}\simeq (B_j{)}_{j\in \mathbb{Z}}\otimes (A_i{)}_{i\in \mathbb{Z}}$ brings $A_i\otimes B_j$ to $(-1{)}^{ij}{B}_j\otimes A_i$), so the commutative algebra objects of this symmetric monoidal category are the graded commutative rings. Then the homotopy group functors ${\pi }_i$ induces a functor $\mathrm{Sp}\to \mathrm{GAb}$, given by $X\to ({\pi }_i(X){)}_{i\in \mathbb{Z}}$. This functor can be naturally promoted to a lax symmetric monoidal functor.

Proof. I don’t know how to prove this rigorously, so I’ll only give a sketch of a seemingly possible method.

Since the tensor product on $\mathrm{Sp}$ is exact in each variable and the sphere spectrum $S$ is the unit object, there are natural isomorphisms $S[i]\otimes S[j]\simeq S[i+j]$ for $i,j\in \mathbb{Z}$. Therefore for $X,Y\in \mathrm{Sp}$ we have a natural map ${\pi }_i(X)\otimes {\pi }_j(Y)\simeq {\mathrm{Mor}}_{h\mathrm{Sp}}(S[i],X)\otimes {\mathrm{Mor}}_{h\mathrm{Sp}}(S[j],Y)\to {\mathrm{Mor}}_{h\mathrm{Sp}}(S[i]\otimes S[j],X\otimes Y)\simeq {\mathrm{Mor}}_{h\mathrm{Sp}}(S[i+j],X\otimes Y)\simeq {\pi }_{i+j}(X\otimes Y)$. It then left to prove that these maps together with the natural isomorphism $\mathbb{Z}\simeq {\pi }_0(S)$ constitute a lax symmetric monoidal functor.

The only subtlety is to prove that the flip isomorphism $S[1]\otimes S[1]\simeq S[1]\otimes S[1]$ is multiplication by $-1$ instead of identity. To see this, we observe, using the constructions developed in section 1, that there are symmetric monoidal functors $\mathcal{S}\to {\mathcal{S}}_{\ast }\stackrel{{\Sigma }^{\infty }}{\to }\mathrm{Sp}$, where the first map is the left adjoint to the projection functor, $\mathcal{S}$ is equipped with the product symmetric monoidal structure, and ${\mathcal{S}}_{\ast }$ is equipped with the smash product symmetric monoidal structure. The flip isomorphism of $S[1]$ then results from the filp isomorphism of $S^1$, and then one can reduce to considering homotopy categories $h{\mathcal{S}}_{\ast }$ and $h\mathrm{Sp}$ and use some classical techniques...

$\square$

Proposition 2.2 ([3] Lemma 7.1.1.7). The canonical $t$-structure on $\mathrm{Sp}$ is compatible with the symmetric monoidal sturcture, in the sense of [3] Example 2.2.1.3. Consequently, ${\mathrm{Sp}}^{♡}$ inherits a symmetric monoidal structure by virture of [3] Example 2.2.1.10. Proposition 2.1 shows that this symmetric monoidal structure can be identified with the tensor product of abelian groups, through the homotopy group functor ${\pi }_0$.

References

[1]	Jacob Lurie, “Kerodon”, https://kerodon.net/
[2]	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
[3]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HA.pdf
[4]	Jacob Lurie, “Higher Topos Theory”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HTT.pdf