Day convolution and the tensor product of spectra

Let \(C\) and \(D\) be symmetric monoidal \(\infty \)-categories. It is natural to wonder whether the \(\infty \)-category \(\Fun (C,D)\) of functors from \(C\) to \(D\) inherits a symmetric monoidal structure. The Day convolution, first introduced by Day (1971) for classical symmetric monoidal categories, provides an affirmative answer to this question under suitable assumptions on \(C\) and \(D\).

Let us first recall the classical construction. Let \((C, \otimes _C, \unit _C)\) and \((D, \otimes _D, \unit _D)\) be symmetric monoidal 1-categories, and assume that \(C\) is small and that \(D\) admits all small colimits. For any two functors \(F, G\colon C \to D\), their Day convolution \(F \otimes _{\Day } G\) is a functor \(C \to D\) defined as the left Kan extension of the composite functor \[ C \times C \xrightarrow {F \times G} D \times D \xrightarrow {\otimes _D} D \] along the tensor product functor \(\otimes _C\colon C \times C \to C\). More explicitly, for an object \(X \in C\), the value of \(F \otimes _{\Day } G\) is given by the colimit \[ (F \otimes _{\Day } G)(X) = \colim _{X_0 \otimes _C X_1 \to X} F(X_0) \otimes _D G(X_1), \] where the colimit is taken over the relative slice category \((\otimes _C)_{/X}\), whose objects are morphisms \(X_0\otimes _CX_1\to X\) and whose morphisms are compatible maps of pairs \((X_0,X_1)\). Two properties of this construction are relevant for us:

(a)

If the tensor product \(\otimes _D\colon D \times D \to D\) preserves small colimits in each variable separately, then the Day convolution equips the functor category \(\Fun (C, D)\) with a symmetric monoidal structure.

(b)

Under this monoidal structure, the category of commutative algebra objects \(\CAlg (\Fun (C, D))\) is equivalent to the category of lax symmetric monoidal functors from \(C\) to \(D\).

The goal of this chapter is to discuss an \(\infty \)-categorical analogue of Day convolution. A first \(\infty \)-categorical version was constructed by Glasman (2016) and subsequently generalized by Lurie [Lurie (2017), Section 2.2.6]. We will prove the form needed here using the Hinich–Winges model for Day convolution between symmetric monoidal \(\infty \)-categories. The description of the Hinich–Winges model in Theorem 16.2.2 is the principal input about Day convolution that we take as a black box from the literature. We use the presentability results of Chapter 22 when constructing reflective subcategories.

Our main application is the symmetric monoidal tensor product on the \(\infty \)-category \(\Sp \) of spectra, upgrading the bifunctor \(- \otimes -\colon \Sp \times \Sp \to \Sp \) from Proposition 4.4.16. To this end, we use an equivalent description of \(\Sp \) as the \(\infty \)-category of certain reduced excisive functors from \(\An _*^{\fin }\) to \(\An \). We can then exhibit \(\Sp \) as a symmetric monoidal Bousfield localization of a Day convolution category.

Section 16.1 characterizes Day convolution operads by a universal property, and Section 16.2 constructs them for symmetric monoidal \(\infty \)-categories; basic examples follow in Section 16.3. In Section 16.4 we use Day convolution to construct tensor products of commutative monoids and groups, leading to general notions of rings and semirings. We then prove the reduced excisive model of spectra in Section 16.5, and use it in Section 16.6 to equip \(\Sp \) with its symmetric monoidal structure; that section closes with the multiplicative infinite loop space machine and the lax symmetric monoidal graded homotopy functor.

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