Algebras and modules

The theory of algebras and modules is not specific to spectra, but makes sense in any suitable monoidal \(\infty \)-category. Defining the objects requires no new work: an associative algebra is an \(\Assoc \)-algebra and a left module is an \(\oLMod \)-algebra, so both are instances of notions we already have. The work lies in showing that they behave like their classical counterparts, and this is what we carry out here, in the generality required for the study of ring spectra in Part I (Chapter 8).

In Section 19.1, we introduce algebras and modules, construct limits in module categories, and establish higher-categorical analogues of basic constructions from classical algebra, such as free modules. In Section 19.2, we study the relative tensor product \(M\otimes _A N\) as the composition of bimodules and deduce the resulting restriction and extension of scalars functors. We then briefly record the alternative simplicial models for algebras and modules in Section 19.3, both as a conceptual explanation of colimits of modules and as preparation for the cut diagrams used in Section 19.4. The latter section constructs endomorphism algebras and explains their universal relation to algebra actions on a fixed object.

The remaining sections specialize the theory and apply it. In Section 19.5 we pass to stable \(\infty \)-categories and spectra, culminating in the monogenic Morita theorem, and Section 19.6 packages categories of algebraic objects by monads. Finally, Section 19.7 applies the coherent algebra developed in this part to construct connective complex K-theory as a commutative ring spectrum.

The stable and spectral consequences use the mode-theoretic universal properties of Section 18.5. They and the final application to connective complex K-theory also use results established earlier in Part I.

The theory developed here is deep and often technically demanding. We will therefore frequently refer to Lurie’s foundational text [Lurie (2017)], where these topics are treated in exhaustive detail. Whenever we invoke a result formulated for Lurie’s model of \(\infty \)-operads, we transport it through the equivalence of Proposition 17.4.8.

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