In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. The study of simplicial homology requires basic knowledge of some fundamental concepts from abstract algebra. For a more detailed treatment of the subject, we refer the reader to a textbook on groups, rings and modules. An elementary illustrated introduction to simplicial sets. Obstruction theory 417 chapter ix simplicial functors and homotopy coherence. A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. A quick tour of basic concepts in simplicial homotopy theory john baez september 24, 2018. In fact, this category was shown to have a model structure in 2004, making it a homotopy theory in the more rigorous sense 15. May 9780226511818 published on 19930115 by university of chicago press. The theory of simplicial sets offers a model of homotopy theory without using topological spaces. The course material is widely applicable, in fields including topology, geometry, number theory, mathematical pysics, and some forms of data analysis. Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets, but also the study of those properties of simplicial sets detectable by means of techniques adapted from topological homotopy theory.
Consequently, reading material and resources are a bit more fluid and scattered. Pdf an introduction to a1homotopy theory researchgate. Enriched functors and enriched natural transformations 34 3. What appears here as appendix a on quillen model structures will, in.
The purpose of this introductory chapter is to introduce these concepts. The advantage of working with simplicial sets is that no topology is really involved in any construction, and yet they allow to do homotopy theory. Homotopy theory department of mathematics faculty of. Lectures on homotopy theory the links below are to pdf files, which comprise my lecture notes for a first course on homotopy theory. Oct 22, 20 quillen showed that simplicial sets form a model category with appropriate choices of three classes of morphisms, which organized the homotopy theory of simplicial sets. Instead, one assumes a space is a reasonable space. In this paper we give a new proof of this organizing. Homotopy theory department of mathematics faculty of science. Back in the 1990s, james dolan got me interested in homotopy theory by explaining how it offers many important clues to ncategories. These are the notes from a minicourse in rational homotopy theory given at utrecht uni. This course will give a detailed account on how to construct the homotopy theory more precisely, the quillen model structure of spaces in the category of simplicial sets, and establish an equivalence of homotopy theories between it and the homotopy theory of topological spaces.
Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. Simplicial homotopy theory, link homology and khovanov homology. Homotopy theory is an important subfield of algebraic topology. Constructing a simplicial homotopy from some starting data. Simplicial homotopy theory find, read and cite all the research. From simplicial homotopy to crossed module homotopy in modified categories of interest article pdf available in georgian mathematical journal march 2016 with 104 reads how we measure reads. Simplicial functors and stable homotopy theory manos lydakis fakult.
Local homotopy theory department of mathematics faculty. Homotopy theory of presheaves of simplicial groupoids. Jan 17, 2017 simplicial homotopy theory, link homology and khovanov homology article pdf available in journal of knot theory and its ramifications january 2017 with 79 reads how we measure reads. Simplicial sets are discrete analogs of topological spaces. Chief among these are the homotopy groups of spaces, specifically those of spheres. Also in section 3, we introduce the fundamental category of a simplicial. Notice that the cech complex is equal to the nerve of the collection of balls. Univalent foundations program, homotopy type theory.
Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry e. A printed on demand paper copy of the book is also. 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. 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. In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. I last gave this course at the university of western ontario during the winter term of 2018. Lecture notes on simplicial homotopy theory springer. The most important result is bott periodicity, which states that the kgroups are periodic with period 2. His proof is very difficult and uses even the classification theory of principal bundles. 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. This is the homotopy theory of simplicial sheaves, simplicial presheaves and presheaves of spectra.
Pdf contents1 introduction 32 recollection on simplicial homotopy theory 52. Simplicial objects in algebraic topology pdf download. The links below are to pdf files for my lecture notes for a course on local homotopy theory. The notation catht 1,t 2 or t ht 1 2 denotes the homotopy theory of functors from the. Covering simplicial theory in different ways are l, ml, and 181. This has given rise to a new field, which has been christened homotopy type theory. Dec 05, 2009 discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. But avoid asking for help, clarification, or responding to other answers.
The interaction of category theory and homotopy theory a revised version of the 2001 article timothy porter february 12, 2010 abstract this article is an expanded version of notes for a series of lectures given at the corso estivo categorie e topologia organised by the gruppo nazionale di topologia del m. This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modu. Local homotopy theory university of western ontario. Thanks for contributing an answer to mathematics stack exchange. Minimal fibrations and the organizing theorem of simplicial. In homotopy theory as well as algebraic topology, one typically does not work with an arbitrary topological space to avoid pathologies in pointset topology. The main reference for the course is the goerssjardine book simplicial homotopy theory. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. A central resource is the hott book, although it is hardgoing for the unexperienced. Hence, we have a \ homotopy theory of homotopy theories, the topic of this workshop.
It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. The notation tht 1 2 is very similar to a notation for homotopy. Quillen showed that simplicial sets form a model category with appropriate choices of three classes of morphisms, which organized the homotopy theory of simplicial sets. The origin of simplicial homotopy theory coincides with the beginning of alge braic topology almost a century ago. Introduction the problem of constructing a nice smash product of spectra is an old and wellknown problem of algebraic topology. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. In addition to these notes, the basic source material for the course is the book local homotopy theory, by j. In this direction, vladimir voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property. There is also work by baues with ayala, marquez, and quintero which is in preprint form and uses a direct abstract homotopy theory approach to proper homotopy theory. The presentation of the homotopy theory and the account of duality in homology manifolds make the text ideal for a course on either homotopy or homology theory. Introduction to simplicial sets roughly speaking, simplicial sets are combinatorial models for topological spaces.
The homotopy spectral sequence of a cosimplicial space 390 2. The homotopy theory of simplicial sets in this chapter we introduce simplicial sets and study their basic homotopy theory. Algebraic k theory algebraic topology homological algebra homotopy k. The topology underlying this is well represented in the literature in the papers of adams, barratt unpublished, james. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. 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 links below are to pdf files, which comprise my lecture notes for a first course on homotopy theory. Thus, goerssjardine appealed to topological methods for the verification. We spent a bunch of time trying to learn this fascinating subject. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. K theory forms what is called an extraordinary cohomology theory. Simplicial homotopy theory department of mathematics. To provide possible answers to this question, we start by listing some examples. The notes contained in this booklet were printed directly from files supplied by the authors before the course.
Simplicial homotopy theory request pdf researchgate. Download free ebook of simplicial objects in algebraic topology in pdf format or read online by j. Sset has a cylinder functor given by cartesian product with the standard 1simplex, i. Simplicial homotopy theory, which appears in the references file below. Since trying to explain something is often the best way to learn it, i wrote a quick tour of basic. Jan 11, 2019 goerss jardine simplicial homotopy theory pdf as the commenters already argued, i would not regard this book as a self contained introduction. Download simplicial objects in algebraic topology pdf free. This approach allows to define the notion of homology for digraphs that is dual to the notion of cohomology of 2, 3, and 8. In a recent paper 7 the authors developed the theory of homology of path complexes, that can be considered as a natural generalization of a simplicial homology theory see, for example, 9, 11, and 12. Pdf from simplicial homotopy to crossed module homotopy in. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
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