Homotopy spectral sequences and obstructions homotopy. Computing simplicial representatives of homotopy group. Recently, broto, levi and oliver have taken the first steps towards merging the theory. Univalent foundations program, homotopy type theory. A central resource is the hott book, although it is hardgoing for the unexperienced. Simplicial homotopy theory, link homology and khovanov homology article pdf available in journal of knot theory and its ramifications. In contrast to previously encountered situations, model theoretic techniques are intrinsically insu. Pdf from simplicial homotopy to crossed module homotopy. 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 77 reads how we measure reads. Algebraic methods in unstable homotopy theory this is a comprehensive uptodate treatment of unstable homotopy. An elementary illustrated introduction to simplicial sets. The following was cited from simplicial homotopy theory by john f. This construction relies essentially on classical principles.
The notation catht 1,t 2 or t ht 1 2 denotes the homotopy theory of functors from the. The relations between spaces and simplicial sets 58 5. Thierry coquand has suggested a modi cation of this construction using semi simplicial sets. Instead, one assumes a space is a reasonable space. Clearly, as sets, jkj r, but not as topological spaces, e. As far as i understand, simplicial techniques are indispensible in modern topology. Then we have axiomatic modeltheoretic homotopy theory. Simplicial homology and singular homology 33 chapter 2. Simplicial sets and homotopy theories to give another example of a homotopy theory, we consider simplicial sets, which are combinatorial models for spaces. Being a new topic, homotopy type theory is still developing. 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 study of simplicial homology requires basic knowledge of some fundamental concepts from abstract algebra.
Pdf from simplicial homotopy to crossed module homotopy in. A triangulation of a topological space x is a simplicial complex k together with a homeomorphism jkj x. A simplicial set is a combinatorial model of a topological space formed by gluing simplices together along their faces. Then we have axiomatic modeltheoretic homotopy theory, stable homotopy theory, chromatic homotopy theory. Pdf simplicial homotopy theory, link homology and khovanov. Homology can be described elegantly in this theory. Thierry coquand has suggested a modi cation of this construction using semisimplicial sets. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. Since i began to work under norman steenrod as a graduate student at chicago in 1939 and received my ph.
Difference between two concepts of homotopy for simplicial. The origin of simplicial homotopy theory coincides with the beginning of alge braic topology almost a century ago. Contiguous implies topologically homotopic finding a family of curves, like you wrote, between the two maps topological homotopy of continuos maps between simplicial spaces implies homotopic equivalence in the chain complex, a bit harder to prove using simplicial approximations and baricentric operator. An introduction to simplicial sets mit opencourseware. The homotopy theory of simplical groups is well known 2, ch. Minimal fibrations and the organizing theorem of simplicial. A doctoral course on homotopy theory and homotopy type theory given by andrej bauer and jaka smrekar at the faculty of mathematics and physics, university of ljubljana, in the spring of 2019. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry e. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. Thus, goerssjardine appealed to topological methods for the verification. In mathematical logic and computer science, homotopy type theory hott h.
Ferry and weinberger 20 and in the theory of embeddings freedman and krushkal 2014. The notation tht 1 2 is very similar to a notation for homotopy. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing. Another motivation for representing homotopy classes by simplicial maps and complexity bounds for such algorithms is the connection to quantitative questions in homotopy theory gromov 1999. Instead, a rather intricate blend of model theory and classical homotopy theory is required. They form the rst four chapters of a book on simplicial homotopy theory. In this paper we give a new proof of this organizing.
The purpose of this introductory chapter is to introduce these concepts. Introduces many of the basic tools of modern homotopy theory. Homotopy theory and simplicial groupoids sciencedirect. Consider the following simplicial complex of the real line. Algebraic ktheory algebraic topology homological algebra homotopy k. Given a suitable measure of complexity for the maps in question, typical questions are.
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. One such way is to replace topological spaces by combinatorial objects, called simplicial sets which can be viewed as generalised simplicial complexes. Vi to be equivalent to the pointed homotopy theory of reduced i. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, postnikov towers, and bisimplicial sets. The starting point is the classical homotopy theory of topological spaces.
Simplicial functors and stable homotopy theory manos lydakis fakult. Introduction to the homotopy theory of homotopy theories to understand homotopy theories, and then the homotopy theory of them, we. This functor is left adjoint to the functor s which takes topological spaces and gives a simplicial set consisting of the singular maps. The starting point is the classical homotopy theory of. Difference between two concepts of homotopy for simplicial maps. Before we dive any deeper into the theory of simplicial sets let us recall a few. In these terms a deformation retract of x onto a subspace a is a homotopy. The homotopy theory of top is of course familiar, and it turns out that the homotopy theory ofch ris what is usually called homological algebra. Enriched functors and enriched natural transformations 34 3. The topology underlying this is well represented in the literature in the papers of adams, barratt unpublished, james. Cubical type theory is one attempt to give computational content to 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. The most important result is bott periodicity, which states that the kgroups are periodic with period 2. Covering simplicial theory in different ways are l, ml, and 181. The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of sn are trivial and that the third homotopy group of s2 is also. In this course we first overview the basics of classical homotopy theory. For example, we have simplicial homotopy theory, where one studies simplicial sets instead of topological spaces.
What is modern algebraic topologyhomotopy theory about. The theory of simplicial sets offers a model of homotopy theory without using topological spaces. Ktheory forms what is called an extraordinary cohomology theory. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by cohen, moore, and the author, on the exponents of homotopy groups. The homotopy theory of simplicial sets in this chapter we introduce simplicial sets and study their basic homotopy theory. Homotopy invariance while it is clear that homology groups are invariant under homeomorphism, it is also true that it is invariant under homotopy equivalence. I have tried to show aspects of homotopy theory from the simplicial point of view. The category of simplicial spaces with morphisms being naturaltransformationsisdenotedssp.
In homotopy theory as well as algebraic topology, one typically does not work with an arbitrary topological space to avoid pathologies in pointset topology. This approach allows to define the notion of homology for digraphs that is dual to the notion of cohomology of 2, 3, and 8. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Whitehead the subject of homotopy theory may be said to have begun in 1930 with the discovery of the hopf map. A pchain of k with coecients in g is a function cp from the oriented psimplices of k to g that vanishes on all but. Since z p is the set of all pchains that go to zero under the pth boundary homomorphism, z p is the kernel of. American mathematical society volume 8, number 1, january 1983 fifty years of homotopy theory by george w.
For a more detailed treatment of the subject, we refer the reader to a textbook on groups, rings and modules. Introduction the problem of constructing a nice smash product of spectra is an old and wellknown problem of algebraic topology. The homotopy interpretation of type theory uses kan fibrations in simplicial sets the simplicial set model. The homotopy spectral sequence of a cosimplicial space 390 2. Homotopy spectral sequences and obstructions homotopy limits. Comparison with the homotopy theory of simplicial sets in order to prove that the homotopy theory of simplicial groupoids of is equivalent to the usual homotopy theory of simplicial sets, we need the following generalizations of the loop group functor g and the classifying complex functor w 2, ch. Consequently, reading material and resources are a bit more fluid and scattered. Since the beginning of the modern era of algebraic topology, simplicial methods have. Contiguous implies topologically homotopic finding a family of curves, like you wrote, between the two maps topological homotopy of continuos maps between simplicial spaces implies homotopic equivalence in the chain complex, a bit harder to prove using. A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. Introduction to simplicial homology work in progress. Notes on homology theory mcgill university school of. A simplicial approach to stratified homotopy theory. We begin with directed simplicial complexes, which can either be drawn as spaces, or written as a diagram of sets with face maps between them.
By combining e with the degenerate edge sx from x to itself we. 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. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory the homotopy hypothesis. 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. They are filtered simplicial sets built from the homotopy groups of the different. 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. Simplicial homotopy groups are the basic invariants of simplicial setskan complexes in simplicial homotopy theory given that a kan complex is a special simplicial set that behaves like a combinatorial model for a topological space, the simplicial homotopy groups of a kan complex are accordingly the combinatorial analog of the homotopy groups of topological spaces.
Simplicial homotopy theory, link homology and khovanov. Thus for a simplicial space, we can think of it as a horizontal sequence. Comparing these two examples helps explain why quillen called the study of model categories \homotopical algebra and thought of it as a. Algebraic ktheory algebraic topology homological algebra homotopy ktheory. Purchase lectures on homotopy theory, volume 171 1st edition. 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. Lectures on homotopy theory, volume 171 1st edition. His proof is very difficult and uses even the classification theory of principal bundles. Simplicial functors and stable homotopy theory manos lydakis fakultat fur. Y are said to be homotopic maps, if there exists a homotopy ft connecting them and once writes f0 f1. In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them.
800 382 271 165 817 933 738 438 433 1451 1356 747 1279 1060 189 533 374 1554 525 1218 1339 1361 1030 847 736 1106 592 745 359 1232 487 1005 576 1228 314 160 931 1020 540