Tensor product spectra

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. □