Operadic Day convolution and stabilization

In Chapter 16, we realized \(\Sp \) as the \(\infty \)-category of reduced excisive functors from finite pointed animae to animae and used Day convolution to construct its tensor product. This construction suggests a more general question. Recall from Section 4.3 that \(\Sp =\Sp (\An )\) is an instance of the stabilization \(\Sp (C)\) of an \(\infty \)-category \(C\) with finite limits, characterized as the cofree stable \(\infty \)-category generated by \(C\). If \(C\) carries a symmetric monoidal structure, when does \(\Sp (C)\) inherit one as well?

Stabilization is the last of four parallel constructions: pointed objects \(C_*\), commutative monoids \(\CMon (C)\), commutative groups \(\CGrp (C)\), and spectrum objects \(\Sp (C)\). Following Nikolaus [Nikolaus (2016)], we treat all four uniformly. Their lax symmetric monoidal universal properties are most naturally formulated in the operadic setting. Thus, for an \(\infty \)-operad \(\Oo \) with sufficient (co)limits, we will construct:

• The pointed \(\infty \)-operad \(\Oo _*\) of pointed objects in \(\Oo \);
• The semiadditive \(\infty \)-operad \(\oCMon (\Oo )\) of commutative monoid objects in \(\Oo \);
• The additive \(\infty \)-operad \(\oCGrp (\Oo )\) of commutative group objects in \(\Oo \);
• The stable \(\infty \)-operad \(\oSp (\Oo )\) of spectrum objects in \(\Oo \).

Each of these constructions comes with a universal property, exhibiting it as the cofree operad with the corresponding algebraic structure built from \(\Oo \). These constructions always produce \(\infty \)-operads, but even when \(\Oo = \Mm _C\) they need not be represented by symmetric monoidal \(\infty \)-categories. In the presentable setting they are represented by the corresponding presentably symmetric monoidal \(\infty \)-categories; for stabilization, see Corollary 18.5.5.

Applying these four constructions to the cartesian symmetric monoidal \(\infty \)-category \((\An , \times )\) recovers the symmetric monoidal structures on \(\An _*\), \(\CMon (\An )\), \(\CGrp (\An )\) and \(\Sp = \Sp (\An )\) constructed in Chapter 16.

The operadic results are genuinely more general than the presentable theory: they apply to non-presentable \(\infty \)-operads, and the stabilization result characterizes lax symmetric monoidal functors whose underlying functors preserve finite limits. Readers interested only in presentably symmetric monoidal \(\infty \)-categories may instead pass directly to Section 18.5, where the four constructions arise from smashing localizations of \(\PrL \). In that setting the operadic stabilization is represented by a symmetric monoidal structure, as we will explain in Corollary 18.5.5.

The chapter is organized as follows. We begin in Section 18.1 with the notions of operadic limit and colimit, which let us formulate the universal properties that the four constructions above should satisfy. We then construct the multiplicative Yoneda embedding in Section 18.2 and extend Day convolution to arbitrary target operads in Section 18.3. Combining these ingredients, Section 18.4 constructs the four cofree operads. The final Section 18.5 explains the common universal property of the resulting presentably symmetric monoidal \(\infty \)-categories.

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