Monoidal topology. A categorical approach to order, metric, and topology (Q2877685)

The search for a satisfactory notion of convergence has been one of the majorNEWLINEinterests in topology from scratch: \textit{M. Fréchet} [C. R. Acad. Sci., Paris 139, 848--850 (1905; JFM 35.0389.02); Rend. Circ. Mat. Palermo 22, 1--74 (1906; JFM 37.0348.02)]NEWLINEintroduced metric spaces and considered sequential convergence in anNEWLINEabstract manner. \textit{E. H. Moore} [Nat. Acad. Proc. 1, 628--632 (1915; JFM 45.0426.03)] and \textit{E. H. Moore} and \textit{H. L. Smith} [Am. J. Math. 44, 102--121 (1922; JFM 48.1254.01)] considered a moreNEWLINEgeneral type of convergence based on directed sets, which was to be calledNEWLINEnets, as coined by \textit{J. L. Kelley} in [Duke Math. J. 17, 277--283 (1950; Zbl 0038.27003)]. \textit{G. Birkhoff} [Bull. Am. Math. Soc. 41, 636 (1935; JFM 61.0641.20)] and \textit{H. Cartan} [C. R. Acad. Sci., Paris 205, 595--598 (1937; Zbl 0017.24305); ibid. 205, 777--779 (1937; Zbl 0018.00302)] introduced the notion of filterNEWLINEconvergence. As is well known, the idea of filter convergence was central inNEWLINE[Zbl 0027.14301]. The history of the axiomatization of convergence in topologyNEWLINEculminated in the Manes-Barr characterization of a topological space in termsNEWLINEof an abstract ultrafilter convergence relation in [Zbl 0186.02901; Zbl 0289.54003; Zbl 0204.33202]. This is one of the two main streams leadingNEWLINEto this book.NEWLINENEWLINENEWLINENEWLINEThe other stream leading to this book emerged out from theNEWLINEepoch-making paper [``Metric spaces, generalized logic, and closed categories'', Rend. Semin. Mat. Fis. Milano 43, 135--166 (1974; Zbl 0335.18006)], where \textit{F. W. Lawvere} described metric spaces asNEWLINEcategories enriched over the extended non-negative half-line adorned with hisNEWLINEunique characterization of Cauchy completeness. The two streams wereNEWLINEgeneralized by a monad \(\mathbb{T}\) replacing the ultrafilter monad and aNEWLINEquantale or, more generally, a monoidal closed category \(\mathcal{V}\) replacing the half-line. It was again Lawvere in 2000 who was the first toNEWLINEproposet that approach spaces in [Zbl 0891.54001] are to be described in termsNEWLINEof \(\mathcal{V}\)-multicategories in place of \(\mathcal{V}\)-categories,NEWLINEsuggesting a merger of \(\mathbb{T}\) and \(\mathcal{V}\). Later, \textit{M. M. Clementino} and \textit{D. Hofmann} [Appl. Categ. Struct. 11, No. 3, 267--286 (2003; Zbl 1024.18003)], following a suggestion by Janelidze, gave aNEWLINElax-algebraic description of approach spaces by using a numerical extension ofNEWLINEthe ultrafilter monad. Finally,\textit{M. M. Clementino} and \textit{W. Tholen} [J. Pure Appl. Algebra 179, No. 1--2, 13--47 (2003; Zbl 1015.18004)]NEWLINEsucceeded in combining the two parameters efficiently.NEWLINENEWLINENEWLINENEWLINEThe book under review consists of five chapters. The first chapter by \textit{R. Lowen} and \textit{W. Tholen} is anNEWLINEintroduction, and the second chapter, ``Monoidal structures'' by \textit{G. J. Seal} and \textit{W. Tholen}, is a succinct introduction to categoryNEWLINEtheory as well as the theory of ordered sets. The core of the book consists surely ofNEWLINEthe last three chapters.NEWLINENEWLINENEWLINENEWLINEChapter III, ``Lax algebras'' by \textit{D. Hofmann}, \textit{G. J. Seal} and \textit{W. Tholen}, deals with lax-algebraic description of topology, introducesNEWLINEthe central category \(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\)NEWLINEwhose objects are called \(\left( \mathbb{T},\mathcal{V}\right) \)-algebras orNEWLINE\(\left( \mathbb{T},\mathcal{V}\right) \)-spaces. As \(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\) fails to be Cartesian closed, the fourthNEWLINEsection introduces its quasitopos extension \(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Grh}\) whose objects are called quasitopological spaces, inNEWLINEorder to redeem exponentiability. The idea of quasitopological space is to beNEWLINEtraced back to [Zbl 0031.28101]. The final section of the chapter is inspiredNEWLINEby the equivalence between ordered compact Hausdorff spaces in [Zbl 0035.35402] and stably compact spaces in [Zbl 0452.06001] as well as theNEWLINEsimilar correspondence in the context of multicategories in [Zbl 0960.18004].NEWLINEThe exposition is largely influenced by [Zbl 1171.54025; Zbl 1173.18001].NEWLINENEWLINENEWLINENEWLINEChapter IV, ``Kleisli monoids'' by \textit{D. Hofmann}, \textit{R. Lowen}, \textit{R. Lucyshyn-Wright} and \textit{G. J. Seal}, embarks upon another description of \(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\) as the category \(\mathbb{T}\mathrm{-Mon}\) of monoids in the hom-set of a Kleisli category. After the well-knownNEWLINEisomorphismNEWLINENEWLINE\[NEWLINE\mathbb{F}\mathrm{-Mon}\cong\mathrm{Top}NEWLINE\]NEWLINENEWLINEfor the filter monad \(\mathbb{F}\), the first section gives an isomorphismNEWLINENEWLINE\[NEWLINE\mathbb{T}\mathrm{-Mon}\cong\left( \mathbb{T},\mathbf{2}\right)NEWLINE\mathrm{-Cat}NEWLINE\]NEWLINENEWLINEThe second section investigates sufficient conditions for a monad morphismNEWLINE\(\alpha:\mathbb{S}\rightarrow\mathbb{T}\) to give rise to an isomorphismNEWLINE\(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\rightarrow\left(\mathbb{S},\mathcal{V}\right) \mathrm{-Cat}\), as long as \(\mathbb{S}\) andNEWLINE\(\mathbb{T}\) are endowed with adequate lax extensions. Merging \(\mathbb{T}\)NEWLINEand \(\mathcal{V}\) into one entity, the third section introduces a new monadNEWLINE\(\mathfrak{\Pi=\Pi}\left( \mathbb{T},\mathcal{V}\right) \), for which we haveNEWLINENEWLINE\[NEWLINE\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\cong\left(NEWLINE\Pi,\mathbf{2}\right) \mathrm{-Cat}.NEWLINE\]NEWLINENEWLINEThe fourth section identifies injective \(\left( \mathbb{T},\mathbf{2}NEWLINE\right) \)-categories with \(\mathbb{T}\)-algebras by recalling the fact thatNEWLINEthe forgetful functor \(\mathrm{Set}^{\mathbb{T}}\rightarrow\left(\mathbb{T},\mathbf{2}\right) \mathrm{-Cat}\) is monadic of Kock-Zöberlein type. The fifth section is based largely upon [Zbl 1239.06003].NEWLINENEWLINENEWLINENEWLINEThe first two sections of Chapter V, ``Lax algebras as spaces'' by \textit{M. M. Clementino}, \textit{E. Colebunders} and \textit{W. Tholen}, explore such topological properties asNEWLINEseparation, regularity, normality, extremal disconnectedness and compactnessNEWLINEin the context of \(\left( \mathbb{T},\mathcal{V}\right)\)-categories asNEWLINEtopological spaces. Emphasis is put on properties arising naturally in theNEWLINE\(\left( \mathbb{T},\mathcal{V}\right) \)-setting such as the symmetricNEWLINEdescriptions of Hausdorff separation and compactness. The power of \(\left(\mathbb{T},\mathcal{V}\right) \)-spaces stems from their equationalNEWLINEdescription as Eilenberg-Moore algebras. The central role of proper and openNEWLINEmaps is highlightened in the third section, where proper and open maps inNEWLINE\(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\) are also consideredNEWLINEequationally. Closure of properties under direct products, such as theNEWLINETychonoff theorem [JFM 55.0963.01] and its generalization called theNEWLINEKuratowski-Mrówka theorem [Zbl 0003.10504; Zbl 0093.36305], is oneNEWLINEof the outstanding themes in this section. The fourth section is concernedNEWLINEwith an axiomatic approach to objects as spaces in a category provided with aNEWLINEclass of proper maps. The last section investigates the notion ofNEWLINEconnectedness in \(\left( \mathbb{T},\mathcal{V}\right) \mathrm{-Cat}\).NEWLINENEWLINENEWLINENEWLINEAll in all, the book is well written, and it will remain the standard textbookNEWLINEon the area for a long time.