We read on p.12 that a semigroupoid is a nonempty set equipped with a partially defined binary operation, associative on so-called meaningfully defined products. So what’s a meaningfully defined product? Well, the idea is that any two elements of the set can be acted on by the semigroupoid’s defining binary operation, but only if the result is again in the semigroupoid do we say that this result, i.e. the product, is meaningfully defined. So the requirement of closure is relaxed, but the notion of meaningful definition is introduced so as to tag when it does hold. Tellingly, nine pages down the line we encounter a theorem on completing semigroupoids to semigroups, and three pages beyond that we get to groupoids: just add a notion of inverse elements (in the context of meaningfully defined products; see p. 24 for details). Thus we have before us a construct with clear ties to semigroups as well as groups, but its definitions are considerably more relaxed vis à vis the ultimately rather stringent requirement of closure.
It is interesting to contrast this approach with the definition given by Ronnie Brown on p. 201 of his book Topology and Groupoids, reviewed here in 2009: “A category whose objects form a set and in which every morphism is an isomorphism is called a groupoid.” Of course Brown’s goal in his book is expressly and emphatically topology, so his categorical approach is natural (at least nowadays, gratia Henri Cartan, Eilenberg, Mac Lane, Grothendieck, and so on), but it is striking that this characterization of a groupoid is at first glance so different from what is presented in the book currently under review.
However, a second glance takes away much of the mystery: take the collection of morphisms as your underlying set (assuming it is a set!), with composition of morphisms as the operation. Since we can only compose morphisms when the codomain of the first is the domain of the second, this gives a partially-defined operation. If all morphisms are invertible, then inverses exist when they make sense. This does not quite work for semigroupoids, however, since categories always have identity morphisms: a category is already a “monoidoid.”
Topological connections being of paramount importance for everyone concerned, we find on p. 45 of Groupoid Metrization Theory that to get a topological groupoid, one merely endows a groupoid with a topology and requires both groupoid operations to be continuous relative to that topology — all very classical, of course — and this entirely echoes what Brown gives us on p. 387 of Topology and Grouoids, i.e. that “the structure maps … are continuous for the given topology.”
All right, then, so groupoids are consonant with various topological considerations (for one thing, much of Brown’s fine book is concerned with the fundamental groupoid); what else is there? Indeed, topology being situated at the crossroads of so many mathematical journeys, what is the thrust of the book under review, particularly given its subtitle concerning “applications to analysis on quasi-metric spaces and functional analysis”?
Well, first of all, the authors (three Mitreas and a Monniaux, so to speak) emphasize that metrization theory is a at the core of what they write about, and this is reflected in their observation, in the book’s Introduction, that its first part (Ch. 1–3) is concerned with developing “a metrization theory in the abstract setting of groupoids that, among other things, contains as particular cases the Aoki-Rolewicz theorem for locally bounded topological vector spaces and a sharpened version of the Macías-Segovia metrization theorem for quasimetric spaces.” They also note that their theory “can be used to present a conceptually natural proof for the Alexandroff-Urysohn metrization theorem for uniform topological spaces.” Finally, the first part of the book is characterized by “predominantly functional analytic/algebraic” methods, which certainly fits with the book’s aforementioned subtitle.
More to the point, however, the book’s second part (Ch’s. 4–6) concerns “a multitude of applications of our metrization theory to … analysis on quasimetric spaces …, function space theory …, as well as classical functional analysis … in settings where the notions of vector space and norm are significantly weakened.” So it is that § 4.9 is concerned with “Hardy spaces on Ahlfors-regular quasimetric spaces,” § 6.2 concerns an extension of the Brikhoff-Kakutani theorem, and §§ 6.5 and 6.6 deal with the closed graph theorem and the uniform boundedness principle. Chapter 6 in fact generally deals with “functional analysis on quasi-pseudonormed groups,” to give an indication of where we have travelled.
Groupoid Metrization Theory is a work devoted to introducing and proselytizing themes in (very) modern mathematics that manifestly possess connections to any number of themes in analysis, in a pretty broad sense, and should score well across the board. It is very interesting mathematics, and deserves a lot of air-play.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.