It would be quite helpful for a student to have a background in basic algebraic topology andor homological algebra prior to working through this course. The homotopy theory of simplicial sets was developed mostly by kan in the 1950s. John baez and james dolan, higherdimensional algebra and topological quantum field theory. Homotopy theory there is a homotopy theory for sheaves of groupoids joyaltierney, 1990. Simplicial homotopy theory modern birkhauser classics. Lecture notes on local homotopy theory local homotopy. An elementary illustrated introduction to simplicial sets. The homotopy theory of simplicial sets in this chapter we introduce simplicial sets and study their basic homotopy theory. The origin of simplicial homotopy theory coincides with the beginning of alge braic topology almost a century ago. Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets. Paul goerss, rick jardine, simplicial homotopy theory, progress in.
Homotopy theories of dynamical systems western university. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. Since the beginning of the modern era of algebraic topology, simplicial. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by apresheaves in the sense that there is a corresponding model structure for apresheaves with an equivalent homotopy category. If your institution has the right kind of springerlink subscription as does western, you can download a pdf file for the book free of charge from the springerlink site, for example at this link. Jardine december 11, 2002 introduction the purpose of this paper is to display a di erent approach to the construction of the homotopy theory of simplicial sets and the corresponding equivalence with the homotopy theory of topological spaces.
Goerss, jardine, simplicial homotopy theory cylinder based homotopy is also discussed extensively in k. Simplicial homotopy theory, volume 174 of progress in mathematics. Algebraic ktheory algebraic topology homological algebra homotopy ktheory algebra colimit homology homotopy theory. The thread of ideas started with the work of poincar. Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. Model structures for prosimplicial presheaves volume 7 issue 3 j. Model structures for prosimplicial presheaves journal. This book presents formal descriptions of the structures comprising these theories, and the links between them. In the 1960s quillen organized the homotopy theory of simplicial sets in the framework of model categories which is a modern foundation of homotopy theory. Simplicial homotopy theory find, read and cite all the research you need on researchgate. This is the homotopy theory of simplicial sheaves, simplicial presheaves and presheaves of spectra. In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site e.
Simplicial homotopy theory bookopen free ebooks for all. Porter, abstract homotopy and simple homotopy theory, world scientific publishing co. A printed on demand paper copy of the book is also. This homotopy theory is a modification of the traditional category of cofibrant objects structure, and is effectively a calculus of controlled equivalences. John baez and michael shulman, lectures on ncategories and cohomology. Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, nonabelian cohomology, stacks, and local stable homotopy theory. Homotopy theory department of mathematics faculty of. Friedlander received 5 october 1985 revised january 1986 istroduction the central organizational theorem of simplicial homotopy theory asserts that the. Discussed here are the homotopy theory of simplicial sets, and other basictopics such as simplicial groups, postnikov towers, and bisimplicial more. Homotopy spectral sequences and obstructions homotopy. Goerss jardine simplicial homotopy theory pdf as the commenters already argued, i would not regard this book as a self contained introduction. A stack is a sheaf or presheaf of groupoids which satis es descent in this homotopy theory. The idea is to use the theory of minimal fibrations to show that every. Local homotopy theory springer monographs in mathematics.
The main components of the theory are the local homotopy theories of simplicial presheaves and simplicial sheaves, local stable homotopy theories, derived categories, and nonabelian cohomology theory. We will endow this category with a model category structure such that the. 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 theory specializes, for example, to the homotopy theories of cubical sets and cubical presheaves, and. Local homotopy theory university of western ontario. More detail on topics covered here can be found in the goerss jardine book simplicial homotopy theory, which appears in the references. Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, nonabelian cohomology. The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. Rational homotopy theory lecture 17 benjamin antieau 1. Simplicial methods are often useful when one wants to prove that a space is a loop space. Relations with algebraic geometry, group cohomology, and algebraic ktheory.
His work made homotopy theory independent of general topology. Local homotopy theory springer monographs in mathematics 2015th edition. The main reference for the course is the goerss jardine book simplicial homotopy theory. This paper displays an approach to the construction of the homotopy theory of simplicial sets and the corresponding equivalence with the homotopy theory of topological spaces which is based on simplicial approximation.
Using the language of that paper, we prove theorem 1. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. Obstruction theory 417 chapter ix simplicial functors and homotopy coherence. Architectural theory mallgrave pdf mallgrave has integrated and consolidated an enormous body of literature, thought and imagery on the emergence of modern architectural theory. The category of systems of spaces admits a homotopy theory that is based on rweak equivalences suitably defined that are the output of homotopy stability results.
Rational homotopy theory lecture 11 benjamin antieau 1. Jardine s model category sprecl is quillen equivalent to the universal homotopy theory ucs constructed by. The homotopy theory of simplicial presheaves is well prepared. Jardine mathematics department, university of western ontario, london, ontario n6a 5b7, canada communicated by e. Jardine, simplicial homotopy theory, progress in math.
Introduces many of the basic tools of modern homotopy theory. It should mean something in the homotopy theory of dynamical systems if the parameter space. 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. In addition to these notes, the basic source material for the course is the book local homotopy theory, by j. The more advanced material includes homotopy limits and colimits,localization with respect to a map and with respect to a homology theory,cosimplicial spaces, and homotopy coherence. The homotopy spectral sequence of a cosimplicial space 390 2. For a gentle introduction to ncategories and the homotopy hypothesis, try these. Simplicial homotopy theory request pdf researchgate. Paul goerss, rick jardine, simplicial homotopy theory, progress in mathematics, birkhauser. They form the rst four chapters of a book on simplicial homotopy theory. Lemma 14 is a variant of a classical result from simplicial homotopy theory 3, i. Jardine, local homotopy theory, springer monographs in mathematics. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing.
The homotopy theory of cosimplicial spaces we will allow spaces to mean either topological spaces or simplicial sets, and we will write spc for the category of spaces. The model category on rational cdgas throughout this section, ch ch 0 q denotes the category of nonnegatively graded rational cochain complexes, and cdga cdga 0 q is the category of commutative algebra objects in ch 0 q. The links below are to pdf files for my lecture notes for a course on local homotopy theory. Simplicial sets topological spaces are horrible, so for the purposes of proving our main theorems in rational homotopy theory, we will use a homotopically equivalent category, the category of simplicial sets. The closed model structure for simplicial sets is much easier to derive from this point of view, and. Algebraic k theory algebraic topology homological algebra homotopy k theory algebra colimit homology homotopy theory. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. Discussed here are the homotopy theory of simplicial sets, and other basictopics such as simplicial groups, postnikov towers, and bisimplicial sets. The central organizational theorem of simplicial homotopy theory asserts that the category s of.