Mar 28, 2021 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. Friedlander received 5 october 1985 revised january 1986 istroduction the central organizational theorem of simplicial homotopy theory asserts that the. Quillen, homotopical algebra, lecture notes in mathematics 4. This fact is shown to have strong implications for the homotopy theory of this category. Jan 01, 1984 the aim of this note is to show that similarly, the homotopy theory of simplical groupoids is equivalent to the unpointed homotopy theory of all simplical sets. The homotopy theory of quadratic forms, sfb lecture, university of regensburg, february 2021 new perspectives in hermitian k theory three talks, new perspectives on k and l theory, university of muenster, september 2020. In addition to these notes, the basic source material for the course is the book local homotopy theory, by j. I have tried to show aspects of homotopy theory from the simplicial point of view.
Jardine january 21, 2001 introduction this paper grew from a personal need for a collection of technical statements about homotopy theoretic objects related to stacks in connection with a project on transfer maps for presheaves of spectra. Lecture notes on local homotopy theory local homotopy theory. The origin of simplicial homotopy theory coincides with the beginning of alge braic topology almost a. Enriched functors and enriched natural transformations 34 3. Citeseerx document details isaac councill, lee giles, pradeep teregowda. A simplicial complex of dimension n is a dependent type that contains for each i n a list of faces of dimension i.
For our purposes the \ homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. Squares represented by a product of three ternary quadratic forms, and a homogeneous variant of a method of swinnertondyer, rational points on irrational varieties. In september 2006 he posted to the types mailing list a very short note on homotopy lambda calculus, which sketched the outlines of a type theory with dependent products, sums and universes and of a model of this type theory in kan simplicial sets. Simplicial objects in algebra topology, van nostrand, princeton, 1%7.
An elementary illustrated introduction to simplicial sets. Covering simplicial theory in different ways are l, ml, and 181. A key tool is the classical model structure on simplicial sets. Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets, but also the study of those properties of simplicial sets. Lecture notes on local homotopy theory local homotopy.
It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Modeling homotopy theories american mathematical society. Various aspects of homotopy theory in the category of minimal simplicial spaces are studied. Towards simplicial complexes in homotopy type theory. In these notes, whenever we refer to a topological space we mean a compactly generated topological space or kelley space. They form the rst four chapters of a book on simplicial homotopy theory. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely alge braic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic ktheory. Lecture notes in algebraic topology pdf 392p this note covers the following topics. In particular for us the category of topological spaces will be cartesian closed. We will now explain simplicial model categories since we will need to discuss mapping spaces. Homotopy theory is a subdomain of topology where, instead of considering the category. As background material, we recommend the lectures of dundas dun and levine lev in this volume. Homotopy spectral sequences and obstructions homotopy limits.
We have been using homotopy theory of spectra, so we need a way to add homotopy theory to these categories. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. Lectures on homotopy theory homotopy theories springer. On cellularization for simplicial presheaves and motivic homotopy. Mapping spaces this section is borrowed from my forthcoming notes on motivic homotopy theory with elden elmanto. This topological space, called the geometric realization of the simplicial set, is defined. The course material is widely applicable, in fields including topology, geometry, number theory, mathematical. Andre joyal, myles tierney notes on simplicial homotopy theory, lecture at advanced course on simplicial methods in higher categories, crm 2008 pdf. For example, hott seems very appropriate to formulate algorithms in. Pdf contents1 introduction 32 recollection on simplicial homotopy theory 52. Jardine mathematics department, university of western ontario, london, ontario n6a 5b7, canada communicated by e. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. On simplicial homotopy theory, the classical model structure on simplicial sets and the classical model structure on topological spaces. Simply put, the a 1 homotopy category, or rather the canonical functor, is the universal functor from the category of smooth schemes towards an infinity category which satisfies nisnevich descent, such that the affine line a 1 becomes contractible.
In this paper we study a particular model for a homotopy theory, called a com. Categorical homotopy theory emily riehl johns hopkins university. I last gave this course at the university of western ontario during the winter term of 2018. Homotopy theory for truncated weak equivalences of.
The simplicial category is a category having the nite totally ordered. Homotopy theory and simplicial groupoids sciencedirect. We will only describe one of these setting, namely the case of simplicial sets. Simplicial functors and stable homotopy theory manos lydakis fakult. Homotopy theory of topological spaces and simplicial sets. Towards simplicial complexes in homotopy type theory maximilian dor e and samson abramsky university of oxford, united kingdom fmaximilian. Model categories and simplicial methods department of. A face of dimension i is a vector of i natural numbers, which stand for. Note that the set v is not necessarily nite, nor v s. Chapter 1 homotopy theory of categories 1 basics of simplicial sets 1. As named before we will be using category theory to make this generalization. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use. 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. These are studied in detail elsewhere in these notes.
The topology underlying this is well represented in the literature in the papers of adams, barratt unpublished, james, mahowald, toda, whitehead, and others. The course material is widely applicable, in fields including topology, geometry, number theory, mathematical pysics, and some forms of data analysis. Introduction the problem of constructing a nice smash product of spectra is an old and wellknown problem of algebraic topology. Other examples can for example be found in hov99 and ds95. The links below are to pdf files for my lecture notes for a course on local homotopy theory. Introduction to combinatorial homotopy theory institut fourier. We construct cellular homotopy theories for categories of simplicial presheaves on. Hence one can simply write the boundary of a psimplex is the addition of all its p. Abstract homotopy theory 32715 in the last lecture we constructed various categories of spectra.
Spaces, null spaces and homotopy localization, lecture notes in math. Woolf, transversal homotopy theory, theory and applications of categories, vol 24, issue 7, pp 148178, 2010. Aside from this relationship with topological spaces, simplicial sets. Notes on the course algebraic topology download book.
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. Important examples of topological spaces, constructions, homotopy and homotopy equivalence, cw complexes and homotopy, fundamental group, covering spaces, higher homotopy groups, fiber bundles, suspension theorem and whitehead product, homotopy groups of cw complexes, homology groups, homology groups of cw. Simplicial homotopy and simplicial model categories. This is the homotopy theory of simplicial sheaves, simplicial presheaves and presheaves of spectra. Simplicial homotopy theory modern birkhauser classics. 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. It is shown that the usual results of homotopy theory hold in this category, and necessary and sufficient conditions are given under which a simplicial space has the homotopy type of a minimal simplicial space. Roughly speaking, you decide not to distinguish two.
In this lecture, we discuss two standard ways to bring homotopy theory into category theory. In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. 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. Vi to be equivalent to the pointed homotopy theory of reduced i. These notes will not require much in the way of simplicial methods simply because its hard to reconcile simplicial methods with nondiscrete lie groups, but you will want to know the bar construction. An introductory reference to motivic homotopy theory is voevodskys icm address voe98. Chain complexes, homology, and cohomology, homological algebra, products, fiber bundles, homology with local coefficient, fibrations, cofibrations and homotopy groups, obstruction theory and eilenbergmaclane spaces, bordism, spectra, and generalized homology. Lectures on homotopy theory the links below are to pdf files, which comprise my lecture notes for a first course on homotopy theory. There is a more recent corrected version published by springer. The following notes grew out of my own difficulties in attempting to learn. Local homotopy theory university of western ontario. Pdf notes on simplicial homotopy theory semantic scholar. Simplicial homotopy theory find, read and cite all the research you need on researchgate. The links below are to pdf files, which comprise my lecture notes for a first course on homotopy theory.
X is a weak homotopy equivalence, and proposition 1. Interested readers are referred to this excellent text for a comprehensive introduction. Such functor categories are of signi cant practical interest. The general construction of homotopy that we will use allows us to talk about homotopy theories in many other settings though. Stacks and the homotopy theory of simplicial sheaves. I livet e xed them using, and as such there may be typos. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry e. To prove the third one, one notes that for any fibrant on. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing.
Minimal fibrations and the organizing theorem of simplicial. For more indepth references, there are mays simplicial objects in algebraic topology, and simplicial homotopy theory by goerssjardine. The homotopy theory of simplicial presheaves on a grothendieck site c is. Homotopy theory of categories 1 basics of simplicial sets. Pdf an introduction to a1homotopy theory researchgate. Homotopy theory for truncated weak equivalences of simplicial groups volume 121 issue 1. I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019. The links below are to pdf files, which comprise my lecture notes for a first course. 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. Rational homotopy theory arun debray october 6, 2015 these notes were taken in ut austins math 392 c rational homotopy theory class in fall 2015, taught by jonathan campbell. Wtypes in homotopy type theory mathematical structures. Woolf, the fundamental category of a stratified space, journal of homotopy and related structures, vol 4, issue 1 pp 359387, 2009. We provide a short introduction to the various concepts of homology theory in algebraic topology.
A 1 homotopy theory is founded on a category called the a 1 homotopy category. The authors acknowledge the financial support given by the dgi. Elements x, 0, which generate the homotopy groups of spheres in the category of simplicial lie algebras are shown to have hopf invariant one. 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 origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. Homotopy spectral sequences and obstructions homotopy.
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