Diagrams indexed by Grothendieck constructions

Gluing derived equivalences together

Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to other subjects such as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather, I shall emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960s, but that have been satisfactorily resolved only in the last few years. 1991 Mathematics Subject Classification 55P42, 55N20.

Category theory has been used by mathematicians for more than sixty years now. Its importance in algebraic geometry, algebraic topology, and homological algebra is indisputable. Its overall significance in mathematics and the foundations of mathematics is still a matter of debate. An historical analysis of the origins and development of category theory (CT) might shed an interesting light on various issues related to this significance. Ralf Krömer’s book constitutes the first global attempt at such an analysis. It is based on extensive original sources, and it offers an original and challenging perspective. It definitely should be read by historians and philosophers of contemporary mathematics. The book’s main thesis is that a form of pragmatism best describes the implicit philosophy of mathematicians using and developing CT from 1945 to approximately 1970. The thesis is primarily defended by looking at the history of CT, and the latter is organized by following the interactions of CT with three fields: algebraic topology, homological algebra, and algebraic geometry. This does indeed roughly follow the actual development of the field and is thus entirely justified.

Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaf-theoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective ``Gel'fand transform algebra'' over the spectrum of the initial algebra. As a result, one further considers the so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric applications; yet, the same class of algebras is still ``closed'', with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their potential applications as well.

Arithmetic combinatorics is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in nature. In recent years, however, many outstanding problems in these directions have been solved by algebraic means (and more specifically, using tools from algebraic geometry and/or algebraic topology), giving rise to an emerging set of techniques which is now known as the polynomial method . Broadly speaking, the strategy is to capture (or at least partition) the arbitrary sets of objects (viewed as points in some configuration space) in the zero set of a polynomial whose degree (or other measure of complexity) is under control; for instance, the degree may be bounded by some function of the number of objects. One then uses tools from algebraic geometry to understand the structure of this zero set, and thence to control the original sets of objects. While various instances of the polynomial method have been known for decades (e.g. Stepanov’s method, the combinatorial Nullstellensatz, or Baker’s theorem), the general theory of this method is still in the process of maturing; in particular, the limitations of the polynomial method are not well understood, and there is still considerable scope to apply deeper results from algebraic geometry or algebraic topology to strengthen the method further. In this survey we present several of the known applications of these methods, focusing on the simplest cases to illustrate the techniques. We will assume as little prior knowledge of algebraic geometry as possible.

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

Stacks were introduced in order to parametrize problems in algebraic geometry where the presence of automorphisms prevented representability by a scheme or even a sheaf (see Artin [1], Deligne–Mumford [3] and Giraud [5]). One early application was Deligne and Mumford’s use of stacks to prove the irreducibility of the space of curves of a given genus [3]. More recently stacks have also played an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and, in certain situations, stacks over the moduli stack of formal groups give rise to spectra (see Goerss [6], Goerss–Hopkins [8] and Rezk [21]) which play an important role in understanding the homotopy groups of spheres (see Goerss–Henn–Mahowald–Rezk [7] and Behrens [2]). One fundamental example is the spectrum of topological modular forms (see Hopkins [12]) which is associated to the moduli stack of elliptic curves.

We study the category of Reedy diagrams in a $\mm$-model category. Explicitly, we show that if K is a small category, V is a closed symmetric monoidal category and C is a closed V-module, then the diagram category V^K is a closed symmetric monoidal category and the diagram category C^K is a closed V^K-module. We then prove that if further K is a Reedy category, V is a monoidal model category and C is a V-model category, then with the Reedy model category structures, V^K is a monoidal model category and C^K$ is a $\mm^K-model category provided that either the unit 1 of V is cofibrant or V is cofibrantly generated.

This text discusses two subjects of quite different natures: construction methods for quotients of quasi-projective schemes either by group actions or by equivalence relations; and properties of direct images of certain sheaves under smooth morphisms. Both methods together allow to prove the central result of the text, the existence of quasi-projective moduli schemes, whose points parametrize the set of manifolds with ample canonical divisors or the set of polarized manifolds with a semi-ample canonical divisor. Starting with A. Grothendieck's construction of Hibert schemes, including the basics of D. Mumford's geometric invariant theory and an introduction to M. Artin's theory of algebraic spaces, the reader finds the tools for the construction of moduli, usually not contained in textbooks on algebraic geometry.

In the forty years of existence of derived category, it was first thought as a tool in algebraic geometry, especially in the development of duality theories that were done by Hartshorne and others. After these first moments, the theory provided a powerful homological tool for the study of linear differential equations. The basic example in the literature that can be found about this is the Riemann-Hilbert problem of associating suitable regular systems of differential equations to constructible sheaves. This studies can be found in the work of Kashiwara and Schapira. See M. Kashiwara, P. Schapira “Sheaves on manifolds" ([15]). To understand the structure of the derived category is necessary to study the axioms of triangulated categories that were introduced in the mid 1960’s by J.L.Verdier in his thesis “Des catégories dérivées des catégories abéliennes" ([21]). The role of the triangles in the derived category is a similar role of the exact sequence in the abelian category. But it is important to remember that these axioms had their origins in algebraic geometry and algebraic topology. Nowadays there are important applications of triangulated categories in areas like algebraic geometry, algebraic topology (stable homotopy theory), commutative algebra, differential geometry and representation theory of artin algebras. See, for instance, the book of D. Happel- “Triangulated categories and the representation theory of finite dimensional algebras" ([11]). The objective of this notes is to present an introdutory material to the undergraduate and graduate students that would like to know some ideas about the derived category. These are the notes a one week series of introductory lectures which I gave in the XXIII-Escola de Algebra, in Maringá, Paraná, Brazil. Firstly we introduced the concepts of additive and abelian category to show the axioms of triangulated category that are our main objective. The triangulated category obey four axioms. We first introduced the first three axioms and their consequences on chapter one and then the octahedral axioms in various equivalent forms in a separate section of the first chapter. The objective of this section is to give a model capable of making this axiom more palatable since, in general, the form that it is presented in the literature does not remind the reader of any similar structure in other fields of mathematics. So, we make the necessary efforts here to present another form of this axiom that is similar to other tools that could be seen in the abelian categories. We present in chapter one the main example of triangulated category, the homotopy category of complexes. Secondly, to understand the morphisms in the derived category I introduced the concept of localization in chapter two. To those that are starting to study localization, we present the necessary background to understand the localization of non commutative ring. We believe that with this model in mind the student will profit more from the study of localization of categories. On chapter two, the student will find the necessary information and exercises to begin to manipulate morphisms in the derived category. So, on chapter three we introduce the definition of derived category of an abelian category and we explain how one sees the original abelian category as a subcategory of its derived category. After having done all this work, it is natural to have many questions about the behavior of the derived category or its applications. Therefore, we present here a bibliography in portuguese and in english that will help the students to make further investigations. The reader that whishes to know the history and the motivation of the begining of the derived category with many details, should read the introduction of the book "Sheaves on Manifolds - M. Kashiwara and M. Schapira ([15]). Acknowledgements: I am particularly grateful to Sônia Maria Fernandes-DMAUFV, Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IMEUSP, who carefully worked through the text and sent me detailed lists of corrections, questions and remarks. These notes were writen for the first time in 2014 and were used in a minicourse which I tough in the XXIII-Escola de Algebra in Maringá, Paraná, Brazil. The last version was written during my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104 - 3.

One goal of algebraic topology is to find algebraic invariants that classify topological spaces up to homotopy equivalence. The notion of homotopy is not only restricted to topology. It also appears in algebra, for example as a chain homotopy between two maps of chain complexes. The theory of model categories,introduced by D. Quillen [Qui06], provided us with a powerful common language to represent different notions of homotopy. Quillen’s work transformed algebraic topology from the study of topological spaces into a wider setting useful in many areas of mathematics, such as homological algebra and algebraic geometry, where homotopy theoretic approaches led to interesting results. In brief, a model structure on a category C is a choice of three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying certain axioms. We can pass to the homotopy category Ho(C) associated to a model category C by inverting the weak equivalences, i.e. by making them into isomorphisms. While the axioms allow us to define the homotopy relations between classes of morphisms in C, the classes of fibrations and cofibrations provide us with a solution to the set-theoretic issues arising in general localisations of categories. Even though it is sometimes sufficient to work in the homotopy category, looking at the homotopy level alone does not provide us with enough higher order structure information. For example, homotopy (co)limits are not usually a homotopy invariant, and in order to define them we need the tools provided by the model category. This is where the question of rigidity may be asked: if we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question of rigidity has been investigated during the last decade, and an extremely small list of examples have been studied, which leaves us with a lot of open questions regarding this fascinating subject. Ou goal is to investigate one of the open questions which have not been answered before. In this thesis, we prove rigidity of the K(1)-local stable homotopy category Ho(LK(1)Sp) at p = 2. In other words, we show that recovering higher order structure information, which is meant to be lost on the homotopy level,is possible by just looking at the triangulated structure of Ho(LK(1)Sp). This new result does not only add one more example to the list of known examples of rigidity in stable homotopy theory, it is also the first studied case of rigidity in the world of Morava K-theory.

Over the last few decades triangulated categories have become increasingly important, to the extent that they can now be viewed as a unifying theory underlying major parts of modern mathematics. This 2010 collection of survey articles, written by leading experts, covers fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. These self-contained articles are a useful introduction for graduate students entering the field and a valuable reference for experts.

Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.

Seven Papers on Algebra, Algebraic Geometry and Algebraic Topology

Bounding the Independence Number of a Graph