1.1. From the Publisher.

This volume presents a self-contained theory of certain singular coverings of toposes, including branched coverings. This book is distinguished from classical treatments of the subject by its unexpected connection with a topic from functional analysis, namely, distributions. Although primarily aimed at topos theorists, this book may also be used as a textbook for advanced graduate courses introducing topos theory with an emphasis on geometric applications.

1.2. From the Publisher.

The self-contained theory of certain singular coverings of toposes called complete spreads, that is presented in this volume, is a field of interest to topologists working in knot theory, as well as to various categorists. It extends the complete spreads in topology due to R H Fox (1957) but, unlike the classical theory, it emphasises an unexpected connection with topos distributions in the sense of F W Lawvere (1983). The constructions, though often motivated by classical theories, are sometimes quite different from them. Special classes of distributions and of complete spreads, inspired respectively by functional analysis and topology, are studied. Among the former are the probability distributions; the branched coverings are singled out amongst the latter.

This volume may also be used as a textbook for an advanced one-year graduate course introducing topos theory with an emphasis on geometric applications. Throughout the authors emphasise open problems. Several routine proofs are left as exercises, but also as 'exercises' the reader will find open questions for possible future work in a variety of topics in mathematics that can profit from a categorical approach.

1.3. From the Preface.

This book gives an introduction to a theory of complete spreads, which basically uses the same strategy employed by R H Fox for dealing with branched coverings, and which we carry out in close connection with (and parallel to) a theory of distributions in the sense of F W Lawvere.

Rather than elucidating the concepts of toposes, distributions, and complete spreads in this preface, we give a preliminary taste of these concepts by including certain quotations which authoritatively describe the original settings and motivations behind them, and let the reader explore these concepts and their interplay in the often new guises in which we present them in this book, itself based on our own work on these topics during the past ten years.

Toposes. Quote from: P T Johnstone, Sketches of an Elephant. A Topos Theory Compendium (2002).

The original notion of a topos, as a 'generalized space' suitable for supporting the exotic cohomology theories required in algebraic geometry, sprang from the fertile brain of Alexandre Grothendieck in the early 1960's, and was developed in his Séminaire de Géométrie Algébrique du Bois-Marie particularly during the academic year 1963-64. The duplicated notes of that seminar circulated widely among algebraic geometers and category-theorists over the next decade, until Springer-Verlag did the world a service by publishing a revised and expanded edition in three volumes of Lecture Notes in Mathematics in 1972. But by then the subject had already been 'reborn' in its second incarnation, as an elementary theory having links with higher-order intuitionistic logic, through the collaboration of Bill Lawvere and Myles Tierney during 1969-70 (which, in turn, built upon Lawvere's work on providing a categorical foundation for mathematics, which had been developing since the early 1960's)

Distributions. Quote from: F W Lawvere, Comments on the Development of Topos Theory (2000).

Following up on a 1966 Oberwolfach talk where I had proposed a theory of distributions (not only in but) on presheaf toposes, in 1983 at Aarhus I posed several questions concerning distributions on $S$-toposes [where $S$ is an elementary topos, thought of to be $Set$, the category of sets and functions]. The base for the definition and questions is a pair of analogies with known theories (commutative algebra and measure theory) for variable quantities, coupled with the fact that there are many important examples of variable $S$-'quantities' where the domains of variation are $S$-toposes. The intensively variable quantities are taken to be the sheaves on the topos, i.e., simply the objects in the category. Of course, the term 'topos' means 'place' or 'situation', but Grothendieck treats the general situation by dealing instead with the category of Set-valued quantities which vary continuously over it, as an affine $k$-scheme is described by dealing with the $k$-algebra of functions on it. (. . . ) Then we follow the lead of analysis and define a distribution or extensively variable quantity on an $S$-topos to be a continuous linear functional, or generalised point, i.e., a functor to $S$ which preserves $S$-colimits, but not necessarily the finite limits.

Complete Spreads. Quote from: R H Fox, Covering Spaces with Singularities (1957).

The principal object of this note is to formulate as a topological concept the idea of a 'branched covering space'. This topological concept encompasses the combinatorial concept used by Heegard, Tietze, Alexander, Reidemeister and Seifert. This has as a consequence that the knot-invariants defined by Seifert (the linking invariants of the cyclic coverings) are invariants of the topological type of the knot (i.e., unaltered by an orientation-preserving auto-homeomorphism of 3-space). Without the developments of this note I am unable to see any simple proof that these invariants are invariants of anything more than the combinatorial type of the knot. It appears that the best way to look at a branched covering is as a completion of an unbranched covering. This completion process appears in its simplest form if it is applied to a somewhat wider class of objects. It is for this reason that I introduce the concept of a spread (a concept that encompasses, in particular, the 'branched and folded coverings' of Tucker.)

1.4. Acknowledgments.

We are grateful to Bill Lawvere for suggesting that we write a book on distributions on toposes and complete spread geometric morphisms based on our work, and for his continued support.

1.5. Review by: Carsten Butz.
Mathematical Reviews MR2258907 (2007m:18001).

The book develops the theory of topos-theoretic analogues of distributions and of spreads, i.e., clopen generated maps. Throughout the text, many illustrating examples are given, and open research problems are emphasised.

The first and most elementary part studies (Lawvere) distributions and complete spreads. Distributions and complete spreads with locally connected domain form equivalent categories, which, among other things, establishes that every geometric morphism with locally connected domain can be factored as a pure geometric morphism followed by a complete spread. Moreover, this factorisation is stable under pullbacks. In the second part, KZ-monads on 2-categories are studied, which provide the abstract framework for the symmetric monad, a locally full and faithful complete KZ-monad. The symmetric topos classifies Lawvere distributions, and complete spreads are recovered as fibrations for this monad, local homeomorphisms as opfibrations. The third part of the book deals with further aspects of the theory of distributions and complete spreads, resulting in more examples and further relations to topology, algebraic geometry and even computer science. Complete spreads are localic maps of toposes, and the (internal) locale over the co-domain (called the display locale) is investigated. Next the symmetric monad is shown to be the composite of the lower bag domain classifier and the probability distribution classifier. Finally, evidence is given regarding why complete spreads indeed generalise the notion of singular coverings.

2.1. From the Publisher.

This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.

2.2. From the Preface.

The subject of synthetic differential geometry has its origins in lectures and papers by F William Lawvere. It extends the pioneering work of Charles Ehresmann and André Weil to the setting of a topos. It is synthetic (as opposed to analytic) in that the basic concepts of the differential calculus are introduced by axioms rather than by definition using limits or other quantitative data. It attempts to capture the classical concepts of differential geometry in an intuitive fashion using the rich structure of a topos (finite limits, exponentiation, subobject classifier) in order to conceptually simplify both the statements and their proofs. The fact that the intrinsic logic of any topos model of the theory is necessarily Heyting (or intuitionistic) rather than Boolean (or classical) plays a crucial role in its development. It is well adapted to the study of classical differential geometry by virtue of some of its models.

This book is intended as a natural extension of synthetic differential geometry (SDG), in particular of the book by Anders Kock to (a subject that we here call) synthetic differential topology (SDT). Whereas the basic axioms of SDG are the representability of jets (of smooth mappings) by tiny objects of an algebraic nature, those of SDT are the representability of germs (of smooth mappings) by tiny objects of a logical sort introduced by Jacques Penon. In both cases, additional axioms and postulates are added to the basic ones in order to develop special portions of the theory.

In a first part we include those portions of topos theory and of synthetic differential geometry that should minimally suffice for a reading of the book. As an illustration of the benefits of working synthetically within topos theory we include in a second part a version of the theory of connections and sprays as well as one of the calculus of variations. The basic axioms for SDT were introduced by the authors of this book and are the contents of the third part of this book. The full force of SDT is employed in the fourth part of the book and consists of an application to the theory of stable germs of smooth mappings including Mather's theorem and Morse theory on the classification of singularities. The fifth part of the book recalls the notion of a well adapted model of SDG in the sense of E J Dubuc and H Bergeron, and extends it to one of SDT. In this same part, and under the assumption of the existence of a well adapted model of SDT, a theory of unfoldings is given as a particular case of the general theory, unlike what is done in the classical case. The sixth part of the book is devoted to exhibiting one such well adapted model of SDT, namely a Grothendieck topos G constructed by Eduardo Dubuc using the algebraic theory of $C^{∞}$-rings and germ determined (or local) ideals. On account of the existence of a well adapted model of SDT, several classical results can be recovered. In these applications of SDG and SDT to classical mathematics, it should be noted that not only do they profit from the rich structure of a topos, not available when working in the category of smooth manifolds, but also that the results so obtained are often of a greater generality and conceptual simplicity than their classical counterparts.

2.3. Review by: David Michael Roberts.
Mathematical Reviews MR3753634.

Synthetic techniques in geometry are venerable, dating as far back as Euclid. The main idea is that one axiomatises properties of the geometry involved, without specifying the construction of a model of that geometry in an a priori existing theory. This latter idea is more recent, a typical example being Descartes' coordinate geometry. Synthetic differential geometry (SDG), the precursor to the main topic of the present book, is the idea of specifying abstract properties of a category of manifold-like objects so that one can do differential geometry, or more prosaically, multivariable calculus, without first constructing the real numbers, Euclidean spaces or second countable Hausdorff locally Euclidean topological spaces. More precisely, SDG sees the world of geometry through the lens of synthetic infinitesimals, where there is a subobject $D$ of the 'real line' object $\mathbb{R}$ such that maps $D \rightarrow M$ correspond to tangent vectors on an object $M$, and similarly with higher and infinite jets. Synthetic Differential Topology (SDT), on the other hand, can access more information, namely germ-level information, and not just jet-level information: it can deal with local neighbourhoods, not just infinitesimal neighbourhoods of points.

The book under review is the first textbook account of SDT, collecting several decades of results into a unified account. While the book consists of six parts, I-VI, one could view the book as consisting of three main sections: Synthetic Differential Geometry, Synthetic Differential Topology, and models, each of which is split into two parts.

Part I is essentially a recap of the required topos theory, and the basics of SDG. It is here that the book is weakest, as it has to summarise a significant amount of background information; not so much the quantity, but the conceptual density of the topos-theoretic approach to differential geometry. As usual with SDG, the approach is to work in the internal logic of a topos satisfying synthetic axioms and postulates, which means that the reasoning is constructive. This is not as onerous as it sounds, because it amounts to working with statements that are geometrically meaningful, rather than just set-theoretically meaningful. ...
...
Once the rough start of Part I is past, however, the material generally picks up. Part II deals with two topics in classical geometry, namely the Ambrose-Singer theorem and the calculus of variations, more specifically a treatment of geodesics and the Euler-Lagrange equations. In proving the Ambrose-Singer theorem in SDG, the text deals with synthetic notions of connections on vector bundles, sprays, flows, torsion and so forth. Since toposes are cartesian closed, they contain all spaces of smooth functions, and so one can work with calculus on infinite-dimensional space on an equal footing with the finite-dimensional objects. ...
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The last third or so of the book deals with showing that analytic models of the axioms hold. Part V deals with a general theory of well-adapted models of SDT together with a short chapter on unfoldings of germs of functions.