Simplicial objects in algebraic topology chicago lectures in. Both the algebraic ktheory and the andrequillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set. 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. Simplicial homotopy theory request pdf researchgate. Model categories and simplicial methods 3 and gis unique up to chain homotopy. The book can be highly recommended to anybody who wants to learn and to apply simplicial techniques andor the theory of simplicial closed model categories.
Instead, one assumes a space is a reasonable space. Homotopy theory of simplicial abelian hopf algebras, james. Since the beginning of the modern era of algebraic topology, simplicial methods have. Homotopy theory of simplicial abelian hopf algebras deepdyve. Topological modular forms after hopkins, miller, and lurie. The homotopy theory of simplicial presheaves on a grothendieck site c is determined by the given topology on the site c see 8. 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.
Jan 11, 2019 for instance, from a brief browse through the. Introduces many of the basic tools of modern homotopy theory. Algebraic ktheory algebraic topology homological algebra homotopy ktheory algebra colimit. The main reference for the course is the goerssjardine book simplicial homotopy theory. Bibliography simplicial homotopy theory, by paul g. This is false, because there could be several surjective ordinal number maps n k such that. Four classical expositions of simplicial homotopy theory appeared between 1967 and 1971. This was also the point of view of lurie in his thesis lur1.
The starting point is the classical homotopy theory of topological spaces. In homotopy theory as well as algebraic topology, one typically does not work with an arbitrary topological space to avoid pathologies in pointset topology. Since it was first published in 1967, simplicial objects in algebraic topology has been the standard reference for the theory of simplicial sets and their relationship to. Paul goerss, rick jardine, simplicial homotopy theory, progress in mathematics, birkhauser 1996 andre joyal, myles tierney an introduction to simplicial homotopy theory, 2005 chapter i, more notes pdf zhen lin low, notes on homotopical algebra. In part ii, riehl further examines this topic, separating categorical arguments from homotopical ones. The homotopy theory of simplicial sets in this chapter we introduce simplicial sets and study their basic homotopy theory. Homotopy spectral sequences and obstructions homotopy limits. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. Paul goerss, rick jardine, simplicial homotopy theory, progress in. Homotopy theory of simplicial abelian hopf algebras homotopy theory of simplicial abelian hopf algebras goerss, paul. Topological modular forms after hopkins, miller, and. J f jardine discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. Now is one reduced to intuition and experimentation when trying to define homotopy groups group objects. We examine the homotopy coniveau tower for a general cohomology theory on smooth kschemes and give a new proof that the layers of this tower for ktheory agree with motivic cohomology.
Of course, not everyone prefers the austere joys of algebra to the earthy pleasures of geometry. Rational homotopy theory lecture 7 3 where p is an acyclic bration, and where the dotted arrow exists since f is a co bration since the compositions u. In addition, we show that the homotopy coniveau tower agrees with voevodskys slice tower for s 1 spectra, giving a proof of a connectedness conjecture. The origin of simplicial homotopy theory coincides with the beginning of alge braic topology almost a century ago. The notation catht 1,t 2 or t ht 1 2 denotes the homotopy theory of functors from the.
Obstruction theory 417 chapter ix simplicial functors and homotopy coherence. Discussed here are the homotopy theory of simplicial sets, and other basictopics such as simplicial groups, postnikov towers, and bisimplicial more. Courses on homotopy theory mathematics stack exchange. Since it was first published in 1967, simplicial objects in algebraic topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. In the 1960s quillen organized the homotopy theory of simplicial sets in the framework of model categories which is a. A good i my view from the modern language the best reference is simplicial homotopy theory by goerss and jardine. 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. With the development of quillens concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe nonabelian homological algebra and to address homotopytheoretical issues in a variety of fields, including algebraic ktheory. Also in section 3, we introduce the fundamental category of a simplicial.
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. Jardine, simplicial homotopy theory, birkhauser, 1999. A printed on demand paper copy of the book is also. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing. Goerss jardine simplicial homotopy theory pdf as the commenters already argued, i would not regard this book as a self contained introduction.
Simplicial homotopy theory find, read and cite all the research. Algebraic k theory algebraic topology homological algebra homotopy k theory algebra colimit homology homotopy theory. The homotopy spectral sequence of a cosimplicial space 390 2. Porter, abstract homotopy and simple homotopy theory, world scientific publishing co. 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. Finally, i provided a brief summary of the extremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme interactions between homotopy theory and algebra. However, formatting rules can vary widely between applications and fields of interest or study. Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets. The course material is widely applicable, in fields including topology, geometry, number theory, mathematical pysics, and some forms of data analysis. It turns out that we can solve a much more general lifting problem than in 1. Homotopy theory of simplicial abelian hopf algebras paul goerss1 and james turner january 22, 1997 abstract we examine the homotopy theory of simplicial graded abelian hopf algebras over a prime eld fp, p0, proving that two very di erent notions of weak equivalence yield the same homotopy category. Repeat is to be understood in the naivest possible sense here. His work made homotopy theory independent of general topology.
This fact is then used to prove the uniqueness of projective resolutions up to chain homotopy. Goerss, jardine, simplicial homotopy theory cylinder based homotopy is also discussed extensively in k. Peter may, simplicial objects in algebraic topology, van nostrand, princeton, 1968. Algebraic k theory algebraic topology homological algebra homotopy k theory algebra colimit. Homotopy colimits are explained to be a special case of weighted colimits, a foundational topic in enriched category theory. The starting point is the classical homotopy theory of. Simplicial homotopy theory department of mathematics. Simplicial homotopy theory modern birkhauser classics.
I last gave this course at the university of western ontario during the winter term of 2018. Homotopy theory of simplicial abelian hopf algebras paul goerss a, james turner, adepartment of mathematics, university of washington, seattle, wa 98195, usa bdepartment of mathematics, university of virginia, charlottsville, va 22903, usa. Part iii treats the most ubiquitous axiomatic framework for homotopy theory quillens model categories. The links below are to pdf files, which comprise my lecture notes for a first course on homotopy theory. The book is an excellent account of simplicial homotopy theory from a modern point of view the book is well written. The notation tht 1 2 is very similar to a notation for homotopy. The main reference for the course is the goerss jardine book simplicial homotopy theory. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. They form the rst four chapters of a book on simplicial homotopy theory. Homotopy spectral sequences and obstructions homotopy. Edward curtis, simplicial homotopy theory, advances in math. Jardine, simplicial homotopy theory, progress in math. However, our goal will be to develop a homotopy theory for simplicial sets independently of topological spaces and only make the comparison at the very end.
Introduction to the homotopy theory of homotopy theories to understand homotopy theories, and then the homotopy theory of them, we. As the commenters already argued, i would not regard this book as a self contained introduction. Homotopy theory of simplicial abelian hopf algebras. Pdf minimal fibrations and the organizing theorem of. Is there a highconcept explanation for why simplicial. Simplicial methods are often useful when one wants to prove that a space is a loop space. The organizing theorem of simplicial homotopy theory asserts that simplicial sets form a model category.
The homotopy theory of simplicial sets was developed mostly by kan in the 1950s. We will verify m4 before m3, as we use factorizations in the proof. Paul goerss, rick jardine, simplicial homotopy theory, progress in mathematics, birkhauser. The nth homotopy sheaf nx is the sheaf associated to the nth homotopy presheaf. 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. Simplicial homotopy theory find, read and cite all the research you need on researchgate. Homotopy theory department of mathematics faculty of. When studying the relationship to topological spaces now it depends on how much homotopy theory you already know. An elementary illustrated introduction to simplicial sets. However, the role of pointset topology in homotopy theory is contingent, and there are several ways to model the homotopy theory of spaces without reference to the notion of topology. One such way is to replace topological spaces by combinatorial objects, called simplicial sets which can be viewed as generalised simplicial complexes.
1174 1145 949 1504 672 1503 17 1040 1510 1516 140 1383 502 929 885 280 1650 399 999 759 1110 1154 785 945 821 709 863 1555 822 23 333 615 1362 522 200 416 1185 1447 811 249