Homotopy theories and model categories purdue university. Slogan homotopy theoryis the study of 1categories whose objects are not. 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. Modern classical homotopy theory, jeffrey strom graduate studies. Homotop y equi valence is a weak er relation than topological equi valence, i. Global equivariant homotopy theory studies such uniform phenomena, i. For our purposes the \homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. Modern foundations for stable homotopy theory 3 homotopy theory, and one wants a category in which all of the usual constructions on spaces are present and, up to homotopy, the suspension functor is an equivalence. Does anyone know a good historical account of the transitional works between classical homotopy theory and the modern approach. Assign to a space x an algebraic object a group or ring and to any map of spaces x y an appropriate homomorphism. American mathematical society, providence, ri, 2011.
A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important. Editorial committee davidcoxchair rafemazzeo martinscharlemann gigliolasta. Get a printable copy pdf file of the complete article 296k, or click on a page image below to browse page by page. Sh shk be the functor induced by sending a space to the constant presheaf of spaces on smk.
Three models for the homotopy theory of homotopy theories. One of the fundamental constructions of classical homotopy theory is the postnikov tower of a space x. The pointset topological background required is fairly minimal, and is used only to set up the machinery of homotopy theory that is the main focus of the book. Modern foundations for stable homotopy theory university of chicago.
They form the rst four chapters of a book on simplicial homotopy theory. Modern physics formalized in modal homotopy type theory. Modern classical homotopy theory graduate studies in mathematics 9780821852866. This is meant to expand on the notion of fundamental group and is a powerful tool that can be. In homotopy theory, spaces are glued together fromdiscs. This book consists of notes for a second year graduate course in advanced topology given by professor whitehead at m. 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. We can mimic classical commutative algebra in this category. Modern classical homotopy theory mathematical association of. In topology, two continuous functions from one topological space to another are called homotopic from greek. The 2sphere s2 can be built from a point and a 2disc, but. Using terminology of dugger, this model category c s s is a presentation for the homotopy theory of homotopy theories, since it is a localization of a category of diagrams of spaces. Of course, theres a classical obstruction theory to the existence of a section s.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. A pair c,e is a relative category, and from the point of view of tpair homotopy theory is relative category theory. I was hoping whiteheads own 50 years of homotopy theory would do the job and it would be perfect to bookend with the treatise,but its not really about that. The first 20 or so chapters of the book are largely accessible to someone who understands what groups and rings are and has a solid understanding of the. This book introduces a new context for global homotopy theory. 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. Propositional equality is interpreted as homotopy and type isomorphism as homotopy equivalence.
The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. Full text is available as a scanned copy of the original print version. That section contains the statement and proof of many of the major theorems of classical homotopy. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. One thing thats special about this tower is that each map has continue reading. Homotopy theory is an outgrowth of algebraic topology and homological. They were of the view that efficiency of the organisation can be increased by making human beings efficient. Cambridge university press, 19, 2011, new mathematical monographs, 9780521516952. In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
Higher homotopy groups, weak homotopy equivalence, cw complex. To understand homotopy theories, and then the homotopy theory of them, we. They have many applications in homotopy theory and are necessary for the proofs in section 6. Homotopy, homotopy equivalence, the categories of based and unbased space. It is based on a recently discovered connection between homotopy the ory and type theory. These groups offer more information than the homology or cohomology groups with which some students may be familiar, but are much harder to calculate. In homotopy geometric region is called a homotopy class. The starting point is the classical homotopy theory of topological spaces. Notes for a secondyear graduate course in advanced topology at mit, designed to introduce the student to some of the important concepts of homotopy theory. In particular, if you calculate something like a homotopy group of a sphere in the type theory, then its also a true statement about the homotopy groups of spheres in classical algebraic topology. As voevodskys work became integrated with the community of other researchers working on homotopy type theory, univalent foundations was sometimes used interchangeably with homotopy type theory, and other times to refer only to its use as a foundational system excluding, for example, the study of modelcategorical semantics or.
Jeffrey stroms big book over 800 pages on the subject of homotopy theory, more precisely, modern classical homotopy theory, sports a useful blurb on its backcover, which goes a long way to explaining what at first glance seems like a contradiction, or at best a stretch, viz. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Let k be an algebraically closed field of characteristic zero. A classical 1groupoid is a structured collection ofmorphisms. Historical transition from classical homotopy to modern. This allows us to rethink all of stable homotopy theory. In homotopy type theory, this is just its tower of truncations.
The set of all such classes can be given an algebraic structure called a group, the fundamental group of the region, whose structure varies according to the type of region. Jeffrey strom, modern classical homotopy theory, prerequesites. Here global refers to simultaneous and compatible actions of compact lie groups. Modern classical homotopy theory jeffrey strom american mathematical society providence, rhode island graduate studies in mathematics volume 127.
Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed. Classical, neoclassical and modern organisation theory. Both ideas are impossible to capture directly in classical settheoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of logic of homotopy. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model. In particular, for an ideal i and multiplicatively closed. The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The course offers an introduction to algebraic topology centered around the theory of higher homotopy groups of a topological space. The exposition is guided by the principle that it is generally preferable to prove topological results using topology rather than algebra. A comparison of motivic and classical homotopy theories. Relative homotopy groups, homotopy fiber, long exact sequence in homotopy, whitehead theorem. Point 1 is just about using the model in the other direction.
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. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. In order to prove this result, we make use of an intermediate category. 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. What is the role of homotopy theory in the classical. Classical homotopy theory let x and y be topological spaces, each with a speci. Consider the full subcategory secat of the category of simplicial spaces whose objects are simplicial spaces with a discrete simplicial set in degree zero. 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. At further section, by looking at the perspectives of the 29 pioneering people, main principles of the classical to modern organization theory are presented one by one. This book introduces a context for global homotopy 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. Jeffrey strom, modern classical homotopy theory, prerequesites and recommended knowledge. Homology groups were originally defined in algebraic topology.
Modern classical homotopy theory, jeffrey strom graduate studies in mathematics, 127. Homotopy type theory is a new branch of mathematics that combines aspects of several different. In this setting, we consider topological spaces up to homotopy equivalence, or up to weak homotopy equivalence. Let shk denote the motivic stable homotopy category of tspectra over k and sh the classical stable homotopy category. The classical writers viewed organisation as a machine and human beings as components of that machine. Homotopy theory of higher categories hal archive ouverte. We survey axioms for a system of modal operators that may naturally be added to homotopy type theory. The core of classical homotopy is a body of ideas and theorems that. For much of what will follow, we will deal with arbitrary topological spaces, which may, for example, not be hausdor recall the quotient space r 0 r tra. Various ways to provide a home for global stable homotopy types have previously. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, lie algebras, galois theory, and algebraic. Browns representability theorems show that homology and cohomology are also contained in classical homotopy theory. Cannon and conner developed the theory of big fundamental groups.
82 1104 1065 927 795 1517 511 1312 1618 311 683 1022 1165 32 643 834 594 1621 498 417 439 547 731 1288 528 1430 707 1127 756 1666 977 1409 483 755 826 1434 75 892 692 824 461 509