Operads and higher algebra

This part develops the machinery needed to construct and manipulate coherent algebraic structures in \(\infty \)-categories. Its central notion is that of an \(\infty \)-operad, which packages the operations and higher coherences belonging to a given type of algebraic structure. The need for such a framework already appears for symmetric monoidal \(\infty \)-categories. In Definition 8.1.1, we defined these as commutative monoids in \(\Cat _{\infty }\). This definition is conceptually clean but rather implicit: for example, even for something as natural as the cartesian monoidal structure on \(\An \), it is not a priori clear how to write down the corresponding commutative monoid in \(\Cat _{\infty }\). The operadic framework will make such structures accessible to explicit construction and manipulation.

Our approach differs from the one in Lurie’s foundational text [Lurie (2017)] in that we define \(\infty \)-operads as functors to the span category \(\Span (\Fin )\) of finite sets, rather than to the category \(\Fin _*\) of finite pointed sets. While the two approaches are equivalent (see Section 17.4), the span-based perspective has the advantage that all the required conditions can be formulated in terms of finite products, which is conceptually cleaner.

The part is organized as follows. We start in Chapter 12 with the classical theory of colored operads, gradually motivating the \(\infty \)-categorical definition; a reader already comfortable with colored operads may skim this chapter. The span category \(\Span (\Fin )\) plays a central role in our approach, and its construction as an \(\infty \)-category requires some care; this is carried out in Chapter 13. With these preparations in place, Chapter 14 introduces \(\infty \)-operads, symmetric monoidal \(\infty \)-categories, algebras, lax monoidal functors, and monoidal Bousfield localizations; it is the most important chapter in this part. We then turn to applications: Chapter 15 equips any \(\infty \)-category with finite products or coproducts with its natural symmetric monoidal structure, and Chapter 16 develops Day convolution and uses it to construct the tensor product on the \(\infty \)-category of spectra, one of the main goals of this book. In Chapter 17, we construct the envelope \(\Env (\Oo )\) of an \(\infty \)-operad, which freely promotes it to a symmetric monoidal \(\infty \)-category; this chapter also establishes the comparison with Lurie’s definition. The general operadic Day convolution and stabilization constructions are then developed in Chapter 18. Next, Chapter 19 develops the abstract theory of associative algebras and their modules, on which the theory of ring spectra in Part I (Chapter 8) is based, and concludes by constructing connective complex K-theory as a commutative ring spectrum. Finally, Chapter 20 develops monoidal Dwyer–Kan localizations and applies them to derived categories and Eilenberg–MacLane modules.

Prerequisites. This part makes heavy use of the theory of cocartesian fibrations and the straightening/unstraightening equivalence from Chapter 23. The construction of span categories in Chapter 13 also relies on the presentation of \(\infty \)-categories by complete Segal animae from Chapter 24. Readers unfamiliar with either topic are encouraged to consult the corresponding chapter in Part III as needed.