The large \(\infty \)-categories that occur in stable homotopy theory are usually built from small data. Presheaf categories are the basic examples: they are large enough to admit all small colimits, but every presheaf is assembled from the small collection of representable presheaves. Presentable \(\infty \)-categories are obtained from presheaf categories by imposing a small collection of relations.
The usual treatment of presentability uses regular cardinals, accessible categories, and accessible functors. We will not develop that language here. Instead, we take the presentation by generators and relations as our definition. We prove the elementary localization and compactness arguments that fit naturally with this presentation and record the remaining consequences needed elsewhere in the book as black boxes. This is equivalent to the standard definition; see [Lurie (2009), Theorem 5.5.1.1 and Proposition 5.5.4.15]. We freely use the Yoneda lemma and the elementary presheaf formalism from Section 1.8.
Sections
Presheaves and free cocompletions
Presheaves and partial free cocompletions.
Presentable categories and adjoint functors
Presentations, closure properties, and the adjoint functor theorem.
Compact objects and Ind-categories
Compact objects, compact generation, and Ind-completions.
Tensor products of presentable ∞-categories
Tensor products of categories with prescribed colimits.
Presentably symmetric monoidal categories
Presentably symmetric monoidal categories, monoidal Ind-completions, and module categories.
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