Basics on ∞-operads

This chapter develops the basic theory of \(\infty \)-operads on which the rest of the part depends. The definition itself was already motivated in Definition 12.4.3; we now supply the vocabulary and constructions that make it usable, building on the span categories of Chapter 13.

The central construction is the multimorphism operad \(\Mm _C\) of a symmetric monoidal \(\infty \)-category \(C\), whose multimorphism animae are the \(\Hom _C(x_1 \otimes \dots \otimes x_n, y)\). It turns symmetric monoidal \(\infty \)-categories into \(\infty \)-operads, and thereby supplies both of the notions we will use constantly: an \(\Oo \)-algebra in \(C\) is an operad map \(\Oo \to \Mm _C\), and a lax symmetric monoidal functor \(C \to D\) is an operad map \(\Mm _C \to \Mm _D\). Neither has to be defined by hand as a list of coherence data; both fall out of the formalism. From this chapter onwards we make constant use of cocartesian fibrations and the straightening/unstraightening equivalence of Chapter 23.

In Section 14.1 we give the formal definition, establish a useful characterization in terms of cocartesian morphisms, and discuss the first examples. Section 14.2 constructs \(\Mm _C\) and introduces \(\Oo \)-algebras, and Section 14.3 treats lax symmetric monoidal functors, including the (lax) symmetric monoidality of the left and right adjoints of a (lax) symmetric monoidal functor. In Section 14.4 we generalize this discussion to \(\Oo \)-monoids and \(\Oo \)-monoidal \(\infty \)-categories for an arbitrary \(\infty \)-operad \(\Oo \). Finally, Section 14.5 applies the operadic description to construct symmetric monoidal structures on suitable subcategories and Bousfield localizations, as needed in Part I.

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