The general construction of homotopy that we will use allows us to talk about homotopy theories in many other settings though. The topics covered are separation axioms, connectivity, mappings, and function spaces. But, to quote a slogan from a tshirt worn by one of my students. Universal covering spaces of homotopy equivalent spaces are homotopy equivalent. For prelude classical homotopy theory a concise and selfcontained rewrite of the proof of the classical model structure on topological spaces is in. This paper is a sequal to homotopy associativity of h spaces. A base of neighborhoods of a point x is a collection b of open neighborhoods of x such that an y neighborhood of x contains an element of b. Universal covering spaces of homotopy equivalent spaces are. Readers will only have to know that a cellular inclusion is the main example of a co bration, and that a cwcomplex is. A map of homotopy associative and homotopy commutative hspaces f. Philip hirschhorn, the quillen model category of topological spaces arxiv.
The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Convenient categories of topological spaces for homotopy. Top topological spaces, we have to choose a precise notion of directed space. Two continuous functions from one topological space to another are called homotopic if one can be \continuously deformed into the other, such a deformation being called a homotopy between the two functions. In the standard model structures on topological spaces, simplicial sets, or chain complexes, the left homotopy relation correspond to the usual notion of strong homotopy using an intervala simplicial intervala chain homotopy. Homotopy theory of topological spaces and simplicial sets. Given two spaces x and y, we say they are homotopy equivalent, or of the same homotopy type, if there exist continuous maps f. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Convenient categories of topological spaces for homotopy theory. Every homeomorphism is a homotopy equivalence, but the converse is. Im working on an exercise from hatchers algebraic topology exercise 1. Most of the material here only depends on part 1, but every once in a while we need to use something more advanced. For a higher category theorist, the following examples of model categories are perhaps the most useful to keep in mind.
Basically it is given by declaring which subsets are open sets. Homotopy colimits of topological spaces 202 part iv. In homotopy theory, spaces are glued together fromdiscs. Slides of a talk on finite topological spaces selected papers and a thesis about finite topological spaces. Something about the definition of homotopy in algebraic topology and in particular in the study of the fundamental group always puzzled me. A category renriched over topological spaces is an ordinary category.
Let x,y be two topological spaces, and i the closed unit interval 0,1. Classical homotopy theory let x and y be topological spaces, each with a speci. Pdf homotopy theory of topological spaces and simplicial. In particular, we characterize pairs of spaces x,y such that the compactopen topology on cx,y is alexandro. A classical 1groupoid is a structured collection ofmorphisms. This paper is a sequal to homotopy associativity of hspaces. In addition to superb coverage of the fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory, and other essentials, elementary topology gives added perspective as the author demonstrates how abstract topological notions developed from classical mathematics. The goal of this paper is to provide a thorough explication of mccords results and prove a new extension of his main theorem. Recall that two spaces are homeomorphic if and only if there exists a homeomorphism, that is, an open continuous bijection, between them. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopy theoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9.
Notation and some standard spaces and constructions1 1. The homotopy hypothesis generalizing to weak ngroupoids. What is modern algebraic topologyhomotopy theory about. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopytheoretic version of sheaf theory developed. Most readers will be able to understand the basic ideas without having read parts 2 and 3 rst, but will occasionally have to ip back for. Universal covering spaces of homotopy equivalent spaces. Algebraic topology is the branch of topology where algebraic methods are used to solve topological problems. We will use, in this we will use, in this sense, a topological space x equipped with a set dx of directed paths 0. Then we have axiomatic modeltheoretic homotopy theory. The older definition of the homotopy category htop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. Pdf an illustrated introduction to topology and homotopy. I 28, hereafter referred to as hah i, in that it continues the study of the associative law from the point of view of homotopy theory, but knowledge of hah i is assumed only in a few places. A primer on homotopy colimits 3 to conclude the paper we have part 4, concerning examples.
A primer on homotopy colimits university of oregon. For our purposes the \ homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. For topological spaces, quillens fundamental theorem then recovers whiteheads theorem. Sections 7 and 8 describe in detail two basic examples of model categories, namely the category top of topological spaces and the category ch rof nonnegative chain. The category of topological spaces top has objects the topological spaces and morphisms the continuous maps between them. I to y is called homotopy relative to a if for each a in a the map fa,t is constant independent of t. Intuitively, the second argument can be viewed as time, and then the homotopy describes a continuous. Intuitively, we can think of a homotopy between two functions as a kind of continuous mapping between the two. A primer on homotopy colimits daniel dugger contents 1. It is shown in 4 that the category of topological spaces is a closed model category ff one puts f serre fibrations and iv weak homotopy equivalences, and takes c to be the class of all maps having a certain lifting property. This note contains comments to chapter 0 in allan hatchers book 5. Let x,y be two topological spaces, and a a subspace of x. Vogt, convenient categories of topological spaces for algebraic topology.
The idea of algebraic topology, at a first moment, is to distinguish topological spaces modulo homeomorphism by topological invariants and the focus is. Assign to a space x an algebraic object a group or ring and to any map of spaces x y an appropriate homomorphism. A base of neighborhoods of a point x is a collection b of open neighborhoods of x such that an. A homotopy between f and g is a continuous function hx,r from the set x.
The category of topological spaces and continuous maps3 2. Topologyhomotopy wikibooks, open books for an open world. Nov 26, 2010 on topological homotopy groups of nha waiian like spa ces 5 now to prove the main result of this section, theorem 1. Introduction to higher homotopy groups and obstruction theory. In addition to superb coverage of the fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory, and other essentials, elementary topology gives added perspective as the author demonstrates how abstract topological notions. One can likewise form the pointed homotopy category htop. Homotopy type theory is a new conception of foundations of mathematics, with intrinsic homotopical content, and an. Most books on the fundamental group often begin with the basic notion of a homotopy of curves or more generally, continuous functions between topological spaces and describe it intuitively as a. The definition of homotopy in algebraic topology mathoverflow. Such topological spaces are often called second countable. The homotopy category htop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. For our purposes the \homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. First, lets recall the fundamental problem of topology. The idea of homotopy can be turned into a formal category of category theory.
The main example of a model category that we will discuss in this article is the example on topological spaces. Being homotopic is an equivalence relation, so we have equivalence classes. Altogether, these additional topics amount to nearly half. Most topological spaces considered in analysis and geometry but not in algebraic geometry ha ve a countable base. For general model category theory a decent concise account is in. For example, we have simplicial homotopy theory, where one studies simplicial sets instead of topological spaces. Let x and y be topological spaces, and let fx and gx be continuous functions from x to y. The category of pointed topological spaces top is a coslice category over top. Topological space, in mathematics, generalization of euclidean spaces in which the idea of closeness, or limits, is described in terms of relationships between sets rather than in terms of distance. As far as i understand, simplicial techniques are indispensible in modern topology. We will see in x3 how to remove the \based condition.
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