Category theory in context (Q2829627)

This is a carefully and thoroughly prepared textbook on category theory, leading the reader to its core. Each section is usually concluded with relevant exercises. A synopsis of the book goes as follows. NEWLINENEWLINEChapter 1 introduces the basic language of category theory, defining \textit{categories}, \textit{functors} and \textit{natural transformations} while introducing the \textit{principle of duality}, \textit{equivalence} of categories and the method of proof by diagram chasing. NEWLINEChapter 2 studies the notions of universal property, generalized element and representation by the \textit{Yoneda lemma}.NEWLINEChapter 3 deals with \textit{limits} and \textit{colimits} with the underlying philosophy that, thanks to the Yoneda lemma, the set-theoretical constructions of limits and colimits suffice to prove general formulae for limits and colimits in any category.NEWLINENEWLINEChapter 4, consisting of six sections, addresses \textit{adjunctions}. \S 4.1 and \S 4.2 give two equivalent definitions of an adjunction. \S 4.3 explores the forms of adjunction arising between pairs of contravariant functors or between \(\left( n+1\right) \)-tuples of functors with \(n\)-variables. \S 4.4 develops the basic calculus of adjunction. \S 4.5 establishes that right adjoint functors preserve limits while left adoint functors preserve colimits.NEWLINE\S 4.6 gives sufficient conditions for a functor to have an adjoint.NEWLINENEWLINEChapter 5, consisting of six sections, introduces the categorical approach to universal algebra. the notion of \textit{algebra} being given a precise meaning in relation to a \textit{monad}. \S 5.1 gives the general definition of a monad together with a range of examples, where any adjunction is shown to give rise to a monad. \S 5.2 investigates the converse question of constructing an adjunction presenting a given monad, with two universal solutions, namely, the \textit{Kleisli} and \textit{Eilenberg-Moore} categories, the latter of which is also called the category of algebras.NEWLINE\S 5.3 is concerned with \textit{monadicity}, while \S 5.5 establishes the \textit{monadicity theorem} on the lines of [\textit{J. M. Beck}, Repr. Theory Appl. Categ. 2003, No. 2, 1--59 (2003; Zbl 1022.18004)]. \S 5.4 establishes that every algebra admits a canonical presentation as a quotient of free algebras. \S 5.6 explains the author's interest in recognizing categories of algebras.NEWLINENEWLINEChapter 6, consisting of five sections, is concerned with \textit{Kan extensions}. \S 6.1 gives the general definition and some basic examples. \S 6.2 presents a formula defining left Kan extensions as certain colimits and right Kan extensions as certain limits. It is shown in \S 6.3 that Kan extensions defined by (co)limit formulae are \textit{pointwise} Kan extensions. \S 6.4 investigates \textit{total derived functors} as certain Kan extensions and introduces an abstract general framework constructing \textit{point-set-level} lifts of these derived functors. It is shown in \S 6.5 how simple special cases of Kan extensions can be used to reduce adjunctions, limits, colimits and monads with a generalization of the Yoneda lemma.NEWLINENEWLINEEpilogue, consisting of five sections, is concerned wtih theorems in category theory. \S E.1 extracts some of the main theorems from the text. \S E.2 addresses coherence for symmetric monoidal categories [\textit{S. MacLane}, Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)]. \S E.3 introduces a new universal property of the unit interval by \textit{T. Leinster} [Adv. Math. 226, No. 4, 2935--3017 (2011; Zbl 1214.03049); \url{https://www.mta.ca/~cat-dist/catlist/1999/realcoalg}]. \S E.4 is concerned with Giraud's theorem [\textit{M. Artin} et al., Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas. (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne, B. Saint-Donat. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0234.00007)]. \S E.5 explores the embedding theorem for abelian categories by \textit{P. Freyd} [Repr. Theory Appl. Categ. 2003, No. 3, xxiii, 1--164 (2003; Zbl 1041.18001)].