We will now combine the results from the previous sections to upgrade the tensor product \(X \otimes Y\) of spectra from Subsection 4.4.3 to a symmetric monoidal structure \((\Sp ,\otimes ,\S )\) on the \(\infty \)-category of spectra.
Let us start by outlining the strategy. By Corollary 16.5.11, we have an equivalence \(\Sp \simeq \Exc _*(\An ^{\fin }_*,\An )\) between the \(\infty \)-category of spectra and the \(\infty \)-category of reduced excisive functors from finite pointed animae to animae. In particular, we may regard \(\Sp \) as a full subcategory of the functor category \(\Fun (\An ^{\fin }_*,\An )\). By Proposition 16.2.7, this functor category admits the Day convolution monoidal structure. We will show that \(\Sp \) is a symmetric monoidal Bousfield localization of it.
As the first step in this plan, we need to equip \(\An ^{\fin }_*\) with a monoidal structure. The smash product on \(\An _*\) from Definition 16.3.5 restricts to \(\An _*^{\fin }\) by Lemma 14.5.2. Indeed, for fixed \(X\in \An _*^{\fin }\), the functor \(X\wedge -\) preserves finite colimits, so the fact that \(\An _*^{\fin }\) is generated under finite colimits by \(S^0\) reduces closure under smash product to \(X\wedge S^0\iso X\).
Combining everything, we can now construct the tensor product on \(\Sp \):
Theorem 16.6.1. The \(\infty \)-category \(\Sp \) admits a unique symmetric monoidal structure such that the reduced excisive approximation functor \[ P_1^{\mathrm {red}}\colon \Fun (\An _*^{\fin }, \An ) \to \Exc _*(\An _*^{\fin }, \An ) \simeq \Sp \] is symmetric monoidal with respect to Day convolution on its source. This structure has the following properties:
- (1)
-
Its monoidal unit is the sphere spectrum \(\S \), and the unreduced suspension spectrum functor \(\S [-]\colon (\An ,\times )\to (\Sp ,\otimes )\) is strongly monoidal.
- (2)
-
Its tensor product preserves small colimits separately in both variables.
- (3)
-
Its underlying bifunctor agrees with the tensor product of spectra characterized in Proposition 4.4.16.
Proof. Equip \(\An _*^{\fin }\) with the smash product monoidal structure from Definition 16.3.5, and equip \(\An \) with the cartesian monoidal structure. The category \(\An _*^{\fin }\) is small by Notation 16.5.2. Corollary 16.2.12 therefore gives the Day convolution monoidal structure on \(\Fun (\An _*^{\fin }, \An )\).
By Corollary 16.5.13, the inclusion \[ \iota \colon \Sp \simeq \Exc _*(\An ^{\fin }_*,\An ) \hookrightarrow \Fun (\An ^{\fin }_*,\An ) \] admits the left adjoint \[ P_1^{\mathrm {red}}:=P_1\circ (-)^{\mathrm {red}}. \] We show that this Bousfield localization satisfies the internal-hom criterion of Lemma 14.5.6. By Proposition 16.2.13, the Day convolution monoidal structure on \(\Fun (\An ^{\fin }_*,\An )\) admits an internal hom given by \[ \iHom _{\Fun (\An ^{\fin }_*,\An )}(F,G)(X) \simeq \int _{Y \in \An ^{\fin }_*} \iHom _{\An }(F(Y), G(X \wedge Y)). \] Here the internal hom in \(\An \) agrees with the hom anima: \[ \iHom _{\An }(A,B)\simeq \Hom _{\An }(A,B). \] By Lemma 14.5.6, it now remains to show that this internal hom is reduced and excisive whenever \(G\) is.
We first check that it is reduced. If \(X\) is the zero object of \(\An _*^{\fin }\), then so is \(X \wedge Y\), hence \(G(X \wedge Y)\) is terminal by reducedness of \(G\). Since \(\iHom _{\An }(F(Y),-)\) preserves terminal animae and an end of terminal animae is terminal, this shows that the internal hom is reduced.
We now check that it is also excisive. Note that \(- \wedge Y\) preserves pushouts, and so the functor \(G(- \wedge Y)\colon \An ^{\fin }_* \to \An \) sends pushouts to pullbacks for all \(Y\). The same then holds true for \(\iHom _{\An }(F(Y), G(- \wedge Y))\), and thus also for the end-expression since its defining limit commutes with pullbacks. We conclude that \(\iHom _{\Fun (\An ^{\fin }_*,\An )}(F,G)\) is excisive.
The internal-hom criterion, together with Proposition 14.5.3, now supplies the symmetric monoidal structure on the full subcategory of reduced excisive functors and the strong symmetric monoidal refinement of \(P_1^{\mathrm {red}}\). The ambient Day convolution structure is presentably symmetric monoidal by Proposition 22.5.5, so part (3) of Proposition 14.5.3 shows that the tensor product on \(\Sp \) preserves small colimits separately in both variables.
To see uniqueness, suppose that another symmetric monoidal structure made \(P_1^{\mathrm {red}}\) symmetric monoidal. For local objects \(X_i\) and \(Y\), adjunction and symmetric monoidality give \[ \Mm _{\Sp }(\{X_i\};Y) \simeq \oDay (\Mm _{\An _*^{\fin }},\Mm _{\An })(\{\iota (X_i)\};\iota (Y)). \] Its multimorphism operad is therefore the full suboperad of the ambient Day convolution operad on the local objects, which is determined by the localization. The remaining assertions are proved in Proposition 16.6.2, Corollary 16.6.3. □
Proposition 16.6.2. The unreduced suspension spectrum functor \(\S [-]\colon \An \to \Sp \) lifts to a strongly monoidal functor \(\S [-]\colon (\An , \times ) \to (\Sp , \otimes )\). In particular, the monoidal unit of \(\Sp \) is the sphere spectrum \(\S \).
Proof. We will show that \(\Omega ^{\infty }\colon \Sp \to \An \) is lax symmetric monoidal and that it has a strong monoidal left adjoint.
Under the equivalence \(\Sp \simeq \Exc _*(\An ^{\fin }_*,\An )\), we may identify \(\Omega ^{\infty }\) with the composite \[ \Sp \hookrightarrow \Fun (\An ^{\fin }_*,\An ) \xrightarrow {\ev _{S^0}} \An . \] The first functor is lax symmetric monoidal by construction of the tensor product of \(\Sp \). The second is lax symmetric monoidal by the precomposition statement of Proposition 16.2.10, applied to the symmetric monoidal functor \(S^0\colon *\to \An _*^{\fin }\).
The left adjoint of the first functor is symmetric monoidal by construction of the tensor product on \(\Sp \). Since \(\An _*^{\fin }\) is small and the cartesian product of animae preserves small colimits separately in both variables, Lemma 16.2.11 shows that left Kan extension along \(S^0\colon *\to \An _*^{\fin }\) is symmetric monoidal. The left adjoint \(\S [-]\) of the displayed composite is therefore symmetric monoidal. □
Corollary 16.6.3. The tensor product functor \(- \otimes -\colon \Sp \times \Sp \to \Sp \) defined by Theorem 16.6.1 agrees with the one characterized by Proposition 4.4.16.
Proof. By parts (1) and (2) of Theorem 16.6.1, the tensor product preserves colimits in both variables, and the relation \(\S \otimes \S \simeq \S \) holds since \(\S \) is the monoidal unit. These are precisely the properties that determine the tensor product uniquely in Proposition 4.4.16, so the two agree. □
16.6.1 The multiplicative infinite loop space machine
The Day convolution structures constructed above are compatible with the familiar passage from animae to spectra. We record the complete sequence, since it is the multiplicative input used for connective complex K-theory in Section 19.7.
Proposition 16.6.4. The lax symmetric monoidal functors \[ \Sp \xrightarrow {\Omega ^{\infty }} \CGrp (\An ) \hookrightarrow \CMon (\An ) \xrightarrow {\fgt } \An _* \xrightarrow {\fgt } \An \] admit strongly monoidal left adjoints: \[ (\An ,\times ) \xrightarrow {(-)_+} (\An _*,\wedge ) \xrightarrow {F^{\CMon }} (\CMon (\An ),\otimes ) \xrightarrow {(-)^{\grp }} (\CGrp (\An ),\otimes ) \xhookrightarrow {\bB ^{\infty }} (\Sp ,\otimes ). \] Here, \((-)_+\) adjoins a disjoint basepoint, \(F^{\CMon }\) is the free commutative monoid functor, \((-)^{\grp }\) is group completion, and \(\bB ^{\infty }\) is infinite delooping.
Proof. The functor \[ (\An ,\times )\longrightarrow \bigl (\Ar (\An ),\square \bigr ), \qquad X\longmapsto (\emptyset \to X), \] is symmetric monoidal: the pushout-product formula identifies its tensor comparison with \(\emptyset \to X\times Y\), and its unit is \(\emptyset \to *\). Its composite with the symmetric monoidal cofiber functor of Lemma 16.3.4 is \((-)_+\), so adjoining a basepoint is strongly monoidal.
The same conclusion for \(F^{\CMon }\) follows from the bilinear universal property of Lemma 16.4.3. Indeed, for pointed animae \(X,Y\) and a commutative monoid \(M\), there are natural equivalences \[ \begin {aligned} \Hom _{\CMon (\An )}(F^{\CMon }(X)\otimes F^{\CMon }(Y),M) &\simeq \Nat _{\Span (\Fin )^2} \bigl (F^{\CMon }(X)(-)\times F^{\CMon }(Y)(-),M(-\times -)\bigr ) \\ &\simeq \Hom _{\An _*}(X\wedge Y,\fgt (M)) \\ &\simeq \Hom _{\CMon (\An )}(F^{\CMon }(X\wedge Y),M). \end {aligned} \] The middle equivalence is obtained by applying the free commutative monoid adjunction in each variable; the functors \(S\mapsto M(S\times T)\) remain commutative monoids because \(-\times T\) preserves the finite products of \(\Span (\Fin )\). The analogous nullary argument identifies the monoidal units. The group completion functor is strongly monoidal because it is the symmetric monoidal Bousfield localization of Proposition 16.4.1.
It remains to consider infinite delooping. Write \(F^{\CGrp }:=(-)^{\grp }F^{\CMon }\). The objects \(F^{\CGrp }(X)\) generate \(\CGrp (\An )\) under colimits, and infinite delooping preserves colimits because it identifies \(\CGrp (\An )\) with the full subcategory \(\Sp _{\geq 0}\subseteq \Sp \) of connective spectra. Under this identification, \[ \bB ^\infty F^{\CGrp }(X)\simeq \Sigma ^\infty X. \] Here \(\Sigma ^\infty X\) denotes the reduced suspension spectrum of the pointed anima \(X\). Both functors \[ (G,H)\longmapsto \bB ^\infty (G\otimes H) \qquadtext {and}\qquad (G,H)\longmapsto \bB ^\infty G\otimes \bB ^\infty H \] preserve colimits separately in \(G\) and \(H\). On free generators, the canonical comparison is the strong monoidal comparison for reduced suspension spectra. This follows from Proposition 16.6.2: using the cofiber description of reduced suspension spectra from Remark 4.4.3, the pushout-product formula identifies this comparison with the isomorphism \[ \Sigma ^\infty X\otimes \Sigma ^\infty Y \simeq \Sigma ^\infty (X\wedge Y). \] The comparison is therefore an equivalence for all \(G\) and \(H\). The same argument treats the unit and the coherence maps, proving that \(\bB ^\infty \) is strongly monoidal. Finally, the right adjoints acquire the displayed lax symmetric monoidal structures by Proposition 14.3.6. □
Corollary 16.6.5. Infinite delooping induces a symmetric monoidal equivalence \[ \bB ^{\infty }\colon \bigl (\CGrp (\An ),\otimes \bigr )\xrightarrow {\ \simeq \ }\bigl (\Sp _{\geq 0},\otimes \bigr ). \]
Proof. The underlying functor is an equivalence by Theorem 5.4.6, and it is symmetric monoidal by Proposition 16.6.4. □
16.6.2 Graded homotopy groups
The ordinary Day convolution tensor product on \(\Ab ^{\Z }=\Fun (\Z ,\Ab )\) is \[ (A\otimes B)_k=\bigoplus _{n+m=k}A_n\otimes B_m. \] Its usual symmetry merely interchanges the two factors. For homotopy groups, the natural symmetry is instead the following signed variant.
Definition 16.6.6 (Koszul symmetry). Keeping the same underlying Day convolution monoidal structure on \(\Ab ^{\Z }\), we replace its ordinary symmetry by the Koszul symmetry, which on homogeneous elements is given by \[ A_n\otimes B_m\longrightarrow B_m\otimes A_n, \qquad a\otimes b\longmapsto (-1)^{nm}b\otimes a. \] We refer to the resulting symmetric monoidal category as the category of graded abelian groups with the Koszul symmetry.
Since \(\Ab ^{\Z }\) is a \(1\)-category, it is enough to check the ordinary hexagon and involutivity axioms. These follow directly from the fact that \((n,m)\mapsto (-1)^{nm}\) is a symmetric bicharacter: bilinearity gives the two hexagon identities, while \((-1)^{nm}(-1)^{mn}=1\) gives involutivity.
Observation 16.6.7 (Graded homotopy groups). The functor \[ \pi _*\colon \Sp \longrightarrow \Ab ^{\Z } \] admits a lax symmetric monoidal refinement when the target is equipped with the Koszul symmetry.
Indeed, for all \(n,m\in \Z \) the tensor product of representatives defines natural pairings \[ \pi _n(X)\otimes \pi _m(Y)\longrightarrow \pi _{n+m}(X\otimes Y), \qquad [x]\otimes [y]\longmapsto \bigl [\S ^{n+m}\simeq \S ^n\otimes \S ^m\xrightarrow {x\otimes y}X\otimes Y\bigr ]. \] Associativity and unitality are inherited from the symmetric monoidal structure on \(\Sp \). Interchanging \(x\) and \(y\) introduces the symmetry \[ \S ^n\otimes \S ^m\longrightarrow \S ^m\otimes \S ^n, \] which acts by \((-1)^{nm}\) on \(\S ^{n+m}\). For \(n,m\geq 0\), this is the sign of the block permutation interchanging \(n\) and \(m\) suspension coordinates; the general case follows by invertibility of suspension. This iterates the basic sign calculation of Warning 4.2.11. Thus the symmetry compatibility is exactly the Koszul rule. Associativity, unitality, and symmetry give all coherence conditions because the target is a \(1\)-category, so the pairings assemble into the claimed lax symmetric monoidal structure.
With the ordinary, unsigned Day symmetry, the same construction is only lax monoidal. Consequently, the homotopy groups of an associative ring spectrum form a graded ring, while those of a commutative ring spectrum form a graded-commutative ring: \[ xy=(-1)^{nm}yx \qquad (x\in \pi _nR,\ y\in \pi _mR). \]
Exercises
Exercise 16.1 (The tensor-product filtration). Let \(F,G\colon (\Z ,\leq )\to \Ab \) be filtered abelian groups, and put \(A:=\colim _iF_i\) and \(B:=\colim _jG_j\). Show that the canonical map \[ (F\otimes _{\Day }G)_n\longrightarrow A\otimes B \] has image \[ \sum _{i+j\leq n}\operatorname {im}\bigl (F_i\otimes G_j\longrightarrow A\otimes B\bigr ). \] Explain why these images define the usual tensor-product filtration on \(A\otimes B\).
Exercise 16.2 (A concrete description of the smash product). Let \(X\) and \(Y\) be pointed animae. Use Lemma 16.3.4 to construct a natural isomorphism \[ X\wedge Y\simeq (X\times Y)/(X\vee Y), \] where the right-hand side denotes the pushout which collapses \(X\vee Y\subseteq X\times Y\) to the basepoint. Deduce that \(S^m\wedge S^n\simeq S^{m+n}\) for \(m,n\geq 0\).
Exercise 16.3 (Tensor products of abelian groups). Recover the usual universal property of \(A\otimes _{\Z }B\) for abelian groups. Then compute \[ \Z /m\otimes _{\Z }\Z /n \] and identify the result as a cyclic group.
Exercise 16.4 (Excisive approximation of the identity). Let \(U\colon \An _*^{\fin }\to \An \) be the forgetful functor, which is reduced. Identify its excisive approximation at a finite pointed anima \(X\) with \[ \colim _n\Omega ^n\Sigma ^nX. \] Relate this anima to \(\Omega ^\infty \Sigma ^\infty X\).
Exercise 16.5 (Monoid animae and ring spectra). Use the symmetric monoidal functor \[ \Sigma ^\infty _+\colon (\An ,\times )\longrightarrow (\Sp ,\otimes ) \] to construct an associative ring spectrum \(\Sigma ^\infty _+M\) from a monoid anima \(M\), and a commutative ring spectrum when \(M\) is commutative. For a discrete monoid \(M\), identify the induced multiplication on \(\pi _0(\Sigma ^\infty _+M)\) with that of the monoid ring \(\Z [M]\).
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