In generality, homotopy theory is the study of mathematical contexts in which functions or rather homomorphisms are equipped with a concept of homotopy between them, hence with a concept of equivalent deformations of morphisms, and then iteratively with homotopies of homotopies between those, and so forth. Stable homotopy theory in problems in homotopy theory. The course offers an introduction to algebraic topology centered around the theory of higher homotopy groups of a topological space. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic k theory. The general construction of homotopy that we will use allows us to talk about homotopy theories in many other settings though. Rational homotopy theory alexander berglund november 12, 2012 abstract these are lecture notes for a course on rational homotopy theory given at. Homotopy theory is an outgrowth of algebraic topology and homological algebra, with relationships to higher category theory, while. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of.
Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by cohen, moore, and the author, on the exponents of homotopy groups. For instance, in homotopy theory, a solid ball of any dimension and a point are considered as equivalent, also a solid torus and a circle are equivalent from the point of view of homotopy theory. Homotopy theory of higher categories carlos simpson to cite this version. An elementary illustrated introduction to simplicial sets greg friedman texas christian university december 6, 2011 minor corrections august, 2015 and october 3, 2016 see errata at end of paper 2000 mathematics subject classi cation. See the history of this page for a list of all contributions to.
Final functors in unenriched category theory 101 8. Instead, a rather intricate blend of model theory and classical homotopy theory is required. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. Here global refers to simultaneous and compatible actions of compact lie groups. Simplicial homotopy theory modern birkhauser classics book title.
Vector bundles, projective modules, motivic homotopy theory, post nikov systems, algebraic. When studying the relationship to topological spaces now it depends on how much homotopy theory you already know. Workshop on the homotopy theory of homotopy theories 5 use the universe axiom, so that everything \large becomes small in the next universe. Homotopy colimits for quillen model categories are obtained as a particular case. Introduction to type theory and homotopy theory michael shulman january 24, 2012 147. Algebraic ktheory algebraic topology homological algebra homotopy ktheory algebra colimit homology homotopy theory. Of greater importance philosophically, however, is hottufs proposal that such types can be the. At an intuitive level, a homotopy class is a connected component of a function space. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. Homotopy and the fundamental groupoid you probably. Relative homotopy groups, homotopy fiber, long exact sequence in homotopy, whitehead theorem. In homotopy type theory, however, there may be multiple different paths, and transporting an object along two different paths will yield two different results.
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories e. Let be a category and let sf be the category of sets and set maps. This is an introduction to type theory, synthetic topology, and homotopy type theory from a categorytheoretic and topological point of view, written as a chapter for the book new spaces for mathematics and physics ed. A primer for unstable motivic homotopy theory arxiv. R denote the standard interval with its induced topology. As named before we will be using category theory to make this generalization. Slogan homotopy theoryis the study of 1categories whose objects are not just setlike but contain paths and higher paths. Buy introduction to homotopy theory fields institute monographs on free shipping on qualified orders. Abstract homotopy theory michael shulman march 6, 2012 152 homotopy theory switching gears today will be almost all classical mathematics, in set theory or whatever foundation you prefer. A model for the homotopy theory of homotopy theory charles rezk abstract. These are the notes from a minicourse in rational homotopy theory given at utrecht uni versity in.
Egbert rijke successfully defended his master thesis in utrecht a couple of weeks ago. Clark barwick, the future of homotopy theory, 2017 pdf, pdf. We will only describe one of these setting, namely the case of simplicial sets. The 2sphere s2 can be built from a point and a 2disc, but. The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. We have been using homotopy theory of spectra, so we need a way to add homotopy theory to these categories. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. For stable homotopy, the suspensions of all spaces are cohspaces. The thread of ideas started with the work of poincar. Homotopy theories and model categories purdue university. Introduction to homotopy theory is presented in nine chapters, taking the reader from basic homotopy to obstruction theory with a lot of marvelous material in between. Furthermore, the homomorphism induced in reduced homology by the inclusion xr.
Cambridge university press, 19, 2011, new mathematical monographs, 9780521516952. Theoretically there is a collection of atomic hspaces or atomic cohspaces. We develop the homotopy theory of cubical diagrams in a gradual way, starting with squares and working up to cubes and beyond. Therefore, in homotopy type theory, when applying the substitution property, it. That section contains the statement and proof of many of the major theorems of classical homotopy. Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a bit for the. Abstract homotopy theory 32715 in the last lecture we constructed various categories of spectra. That type theory wasalready inventedby type theorists, long before anyone realized it had to do with homotopy. Various ways to provide a home for global stable homotopy types have previously. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. A framework for doing mathematics that can be \compiled into any 1topos. Section x6 gives hoc a more conceptual signi cance by showing that it is equivalent to the \localization of c with respect to the class of weak equivalences.
Cubical homotopy theory university of rochester mathematics. In homotopy theory, spaces are glued together fromdiscs. If x and y are objects in ory, a branch of algebraic topology. We show that for such models, \functors between two homotopy theories form a homotopy theory, or more precisely that the category of such models has a wellbehaved internal hom. Change of base for homotopy limits and colimits 99 8. The homotopy theory of spaces studies spaces up to weak equivalence.
Its aim was to highlight areas in which category theoretic abstractions provide a particularly valuable insight into classical homotopy theoretic constructions. The underlying idea is that it should be possible to develop a purely algebraic approach to. Alex heller this memoir deals with much of the familiar structure of homotopy theory, including standard theorems on homotopy limits and localization, and gives a description of algebrasuptohomotopy designed. For our purposes the \ homotopy theory associated to c is the homotopy category hoc together with various related constructions x10.
Homotopy type theory permits logic of homotopy types. Introduction to higher homotopy groups and obstruction theory. This note contains comments to chapter 0 in allan hatchers book 5. However, this process causes some changes in the original model. Global equivariant homotopy theory studies such uniform phenomena, i. In algebraic topology, homotopy theory is the study of homotopy groups. Here we discuss the basic constructions and facts in abstract homotopy theory, then below we conclude this introduction to homotopy theory by showing that topological spaces equipped with the above system of classes continuous functions is indeed an example of abstract homotopy theory in this sense. In order to discuss this we have to introduce one more notion from category theory. In the 21th century, the new mathematical theories will start from a homotopy theory.
In contrast to previously encountered situations, model theoretic techniques are intrinsically insu. A dependently typed programming language that incorporates homotopical ideas. Homotopy theory is a subdomain of topology where, instead of considering the. He published it on the homotopy type theory blog here is a direct link to the pdf file revised. As cell complexes are built out of disks and spheres, these are. Pdf elements of homotopy theory download full pdf book. They have many applications in homotopy theory and are necessary for the proofs in section 6. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic ktheory. This category has many nice properties which are not present in the unstable homotopy category of spaces, following from the fact that the suspension functor becomes. A classi cation theorem theorem jordan, 1866 two orientable surfaces with boundaries are homeomorphic if and only if they have the same genus and the same number of boundary curves.
Introduction to combinatorial homotopy theory institut fourier. Higher homotopy groups, weak homotopy equivalence, cw complex. These notes were used by the second author in a course on simplicial homotopy theory given at the crm in february 2008 in preparation for the advanced courses on simplicial methods in higher categories that followed. The authors intention is to rely on the geometric approach by appealing to the readers own intuition to help understanding. An introduction to stable homotopy theory semester project by maximilien holmbergperoux responsible professor prof.
We take a moment to explain why this might be useful for understanding the above motivating questions. A classical 1groupoid is a structured collection ofmorphisms. Homotopy theory an introduction to algebraic topology. In homotopy theory, one studies topological spaces by way of their points, paths between points, homotopies paths between paths, homotopies between homotopies paths between paths between paths, and so on. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. A paracompact hausdorff space which is locally anr is an anr. Other examples can for example be found in hov99 and ds95. Notation and some standard spaces and constructions1 1. Algebraic homotopy cambridge studies in advanced mathematics. Simplicial homotopy theory modern birkhauser classics.
This book introduces a new context for global homotopy theory. In my first year at harvard, i had an opportunity to teach a graduatelevel topics course entitled categorical homotopy theory. The homotopy theory of simplicial sets in this chapter we introduce simplicial sets and study their basic homotopy theory. Arkowitz book is a valuable text and promises to figure prominently in the education of many young topologists.
A pair c,e is a relative category, and from the point of view of tpair homotopy theory is relative category theory. Unlike the usual encoding in terms of the set theory this one respects equivalences. A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. These groups offer more information than the homology or cohomology groups with which some students may be familiar, but are much harder to calculate.
They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing. Wu in homotopy theory, one can decompose hspaces as a product of homotopy indecomposable hspaces, and cohspaces as a wedge of homotopy indecomposable cohspaces. This book introduces a context for global homotopy theory. An elementary illustrated introduction to simplicial sets. Homotopy theory of topological spaces and simplicial sets. A good i my view from the modern language the best reference is simplicial homotopy theory by goerss and jardine. Emily riehl discusses two competing perspectives by which one typically first encounters homotopy colimits.
A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. An enhancement of constructive type theory with a homotopical extensionality axiom. All formulations give notions which are equivalent in a homotopy theoretic sense, see slide 110 below, although the objects involved look very different in detail. Categorical tools for homotopy colimit computations 97 8.
Introduction to higher homotopy groups and obstruction theory michael hutchings february 17, 2011 abstract these are some notes to accompany the beginning of a secondsemester algebraic topology course. For example, use oliviers thesis strasbourg 20 to understand the shimomura schools calculations at large primes. A homotopy equivalence weakens this by requiring that the compositions are only homotopic to the identities. In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. A foundation for mathematics whose basic objects behave like 1groupoids. The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of sn are trivial and that the third homotopy group of s2 is also. Warren t he institute for advanced study inprinceton is hosting a special program during the academic year 201220 on a new research theme that is based on recently discovered connections between homotopy theory, a branch of algebraic. The category of topological spaces and continuous maps3 2. These notes are from a series of lectures given at the workshop on the homotopy theory of homotopy theories which took place in caesarea. We describe a category, the objects of which may be viewed as models for homotopy theories. The idea of homotopy can be turned into a formal category of category theory. Homotopy type theory hott has often been described as the internal language of 1categories.
It has been noticed since the beginnings of equivariant homotopy theory that certain theories naturally exist not just for a particular group, but in a uniform way for all groups in a speci c class. In algebraic geometry and algebraic topology, branches of mathematics, a 1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. In mathematical logic and computer science, homotopy type theory hott h. In this lecture, we discuss two standard ways to bring homotopy theory into category theory. Homotopy theory of higher categories hal archive ouverte. Master thesis on rational homotopy theory, the study of homotopy without torsion. The book could also be used by anyone with a little background in topology who wishes to learn some homotopy theory. Following this line of thought, an entire stable homotopy category can be created. Homotop y equi valence is a weak er relation than topological equi valence, i. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Homotopy, homotopy equivalence, the categories of based and unbased space.
Equivalent spaces will often have equal invariants. We are always looking for ways to improve customer experience on. Homotopy theories johan leray and bruno vallette in the 20th century, any mathematical theory was starting from set theory. This additional structure allows one to do homotopy theory in the category in question and encodes the similarities between ordinary homotopy theory of topological spaces, simplicial homotopy. The focus of the conference, and subsequent papers, was on applications of innovative methods from homotopy theory in category theory, algebraic geometry, and related areas, emphasizing the work of younger researchers in these fields. Lectures on homotopy theory, volume 171 1st edition. The thesis mainly focuses on the sullivan equivalence, which models rational spaces by commutative differential graded algebras contrary to quillens dual approach which considers coalgebras. Simplicial homotopy theory modern birkhauser classics since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. Algebraic methods in unstable homotopy theory this is a comprehensive uptodate treatment of unstable homotopy. The thesis is well written and it contains several new results, but most importantly, it is a gentle yet nontrivial introduction to homotopy type theory. This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. A category c is a groupoid if every morphism in c is an. An introduction to simplicial homotopy theory andr ejoyal universit eduqu ebec a montr eal myles tierney rutgers university preliminary version, august 5, 1999. A dual theory is developed for homotopy limits in abc.
Well stick with the rst one, which will be our main example throughout this course. More generally a metrizable space is an anr if each point has a neighborhood homeomorphic to a convex set in a lctvs. Vietorisrips and degree rips complexes from topological data. It is based on a recently discovered connection between homotopy theory and type theory. Sometimes ores condition holds, so any diagram can be completed to a diagram. It presents elements of both homology theory and homotopy theory, and includes various applications. Voevodskys univalence axiom in homotopy type theory. Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence. Unstable homotopy theory in problems in homotopy theory. The theory is due to fabien morel and vladimir voevodsky. Keywords eilenbergmac lane and moore spaces hspaces and cohspaces fiber and cofiber spaces homotopy homotopy and homology decompositions homotopy groups loops and suspensions obstruction theory pushouts and pull backs.
1016 1354 1298 511 253 89 3 314 228 1214 7 1445 1617 134 133 1586 1080 1212 430 975 1383 1426 151 165 190 1178 379 611 1131 1554 787 647 1271 167 132 548 402 263 104 1021 34