Adjunctions

Nearly every construction in this book is specified by a universal property, and in practice this means that it is produced by an adjoint functor: free algebras, localizations, Kan extensions and (co)limits are all of this form. Adjunctions are therefore the tool we reach for most often, and the goal of this chapter is to give a detailed treatment of them, together with the various related notions that the main text calls upon.

We start in Section 21.1 by recalling the definition and providing various equivalent characterizations. In Section 21.2 we discuss the relation between (co)limits and adjunctions, and deduce various useful consequences about limits and colimits. After introducing slice categories in Section 21.3, we discuss Kan extensions and their pointwise formulas in Section 21.4. In Section 21.5 we briefly introduce the notion of cofinality, and in Section 21.6 the filtered and sifted \(\infty \)-categories, whose colimits commute with finite limits and finite products respectively. Section 21.7 proves a comparison theorem for categories of structured objects, the monad-free mechanism underlying the Barr–Beck–Lurie monadicity theorem. Finally, in Section 21.8 we introduce Bousfield localizations, give various examples, and provide a criterion for recognizing them.

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