Higher Categorical Groups and the Classification of Topological Defects and Textures
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Crossed Modules and the Homotopy 2-Type of a Free Loop Space
Crossed modules and the homotopy 2-type of a free loop space Ronald Brown∗ November 5, 2018 Bangor University Maths Preprint 10.01 Abstract The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We show how this depends on the homotopy 2-type of X by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX; that is we describe crossed modules representing the 2-type of each component of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.1 1 Introduction It is well known that for a connected CW -complex X with fundamental group G the set of components of the free loop space LX of X is bijective with the set of conjugacy classes of the group G, and that the fundamental groups of LX fit into a family of exact sequences derived arXiv:1003.5617v3 [math.AT] 23 May 2010 from the fibration LX → X obtained by evaluation at the base point. Our aim is to describe the homotopy 2-type of LX, the free loop space on X, when X is a connected CW -complex, in terms of the 2-type of X. Weak homotopy 2-types are described by crossed modules (over groupoids), defined in [BH81b] as follows. A crossed module M is a morphism δ : M → P of groupoids which is the identity on objects such that M is just a disjoint union of groups M(x), x ∈ P0, together with an action of P on ∗School of Computer Science, University of Bangor, LL57 1UT, Wales 1MSClass:18D15,55Q05,55Q52; Keywords: free loop space, crossed module, crossed complex, closed category, classifying space, higher homotopies. -
Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension
LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K. -
A Comonadic Interpretation of Baues–Ellis Homology of Crossed Modules
A COMONADIC INTERPRETATION OF BAUES–ELLIS HOMOLOGY OF CROSSED MODULES GURAM DONADZE AND TIM VAN DER LINDEN Abstract. We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr–Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions. 1. Introduction The concept of a crossed module originates in the work of Whitehead [36]. It was introduced as an algebraic model for path-connected CW spaces whose homotopy groups are trivial in dimensions strictly above 2. The role of crossed modules and their importance in algebraic topology and homological algebra are nicely demon- strated in the articles [6, 8, 9, 28, 31, 36], just to mention a few. The study of crossed modules as algebraic objects in their own right has been the subject of many invest- igations, amongst which the Ph.D. thesis of Norrie [33] is prominent. She showed that some group-theoretical concepts and results have suitable counterparts in the category of crossed modules. In [2], Baues defined a (co)homology theory of crossed modules via classifying spaces, which was studied using algebraic methods by Ellis in [15] and Datuashvili– Pirashvili in [13]. A different approach to (co)homology of crossed modules is due to Carrasco, Cegarra and R.-Grandjeán. -
A Primer on Homotopy Colimits
A PRIMER ON HOMOTOPY COLIMITS DANIEL DUGGER Contents 1. Introduction2 Part 1. Getting started 4 2. First examples4 3. Simplicial spaces9 4. Construction of homotopy colimits 16 5. Homotopy limits and some useful adjunctions 21 6. Changing the indexing category 25 7. A few examples 29 Part 2. A closer look 30 8. Brief review of model categories 31 9. The derived functor perspective 34 10. More on changing the indexing category 40 11. The two-sided bar construction 44 12. Function spaces and the two-sided cobar construction 49 Part 3. The homotopy theory of diagrams 52 13. Model structures on diagram categories 53 14. Cofibrant diagrams 60 15. Diagrams in the homotopy category 66 16. Homotopy coherent diagrams 69 Part 4. Other useful tools 76 17. Homology and cohomology of categories 77 18. Spectral sequences for holims and hocolims 85 19. Homotopy limits and colimits in other model categories 90 20. Various results concerning simplicial objects 94 Part 5. Examples 96 21. Homotopy initial and terminal functors 96 22. Homotopical decompositions of spaces 103 23. A survey of other applications 108 Appendix A. The simplicial cone construction 108 References 108 1 2 DANIEL DUGGER 1. Introduction This is an expository paper on homotopy colimits and homotopy limits. These are constructions which should arguably be in the toolkit of every modern algebraic topologist, yet there does not seem to be a place in the literature where a graduate student can easily read about them. Certainly there are many fine sources: [BK], [DwS], [H], [HV], [V1], [V2], [CS], [S], among others. -
Theory of Covering Spaces
THEORY OF COVERING SPACES BY SAUL LUBKIN Introduction. In this paper I study covering spaces in which the base space is an arbitrary topological space. No use is made of arcs and no assumptions of a global or local nature are required. In consequence, the fundamental group ir(X, Xo) which we obtain fre- quently reflects local properties which are missed by the usual arcwise group iri(X, Xo). We obtain a perfect Galois theory of covering spaces over an arbitrary topological space. One runs into difficulties in the non-locally connected case unless one introduces the concept of space [§l], a common generalization of topological and uniform spaces. The theory of covering spaces (as well as other branches of algebraic topology) is better done in the more general do- main of spaces than in that of topological spaces. The Poincaré (or deck translation) group plays the role in the theory of covering spaces analogous to the role of the Galois group in Galois theory. However, in order to obtain a perfect Galois theory in the non-locally con- nected case one must define the Poincaré filtered group [§§6, 7, 8] of a cover- ing space. In §10 we prove that every filtered group is isomorphic to the Poincaré filtered group of some regular covering space of a connected topo- logical space [Theorem 2]. We use this result to translate topological ques- tions on covering spaces into purely algebraic questions on filtered groups. In consequence we construct examples and counterexamples answering various questions about covering spaces [§11 ]. In §11 we also -
Crossed Modules, Double Group-Groupoids and Crossed Squares
Filomat 34:6 (2020), 1755–1769 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL2006755T University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Crossed Modules, Double Group-Groupoids and Crossed Squares Sedat Temela, Tunc¸ar S¸ahanb, Osman Mucukc aDepartment of Mathematics, Recep Tayyip Erdogan University, Rize, Turkey bDepartment of Mathematics, Aksaray University, Aksaray, Turkey cDepartment of Mathematics, Erciyes University, Kayseri, Turkey Abstract. The purpose of this paper is to obtain the notion of crossed module over group-groupoids con- sidering split extensions and prove a categorical equivalence between these types of crossed modules and double group-groupoids. This equivalence enables us to produce various examples of double groupoids. We also prove that crossed modules over group-groupoids are equivalent to crossed squares. 1. Introduction In this paper we are interested in crossed modules of group-groupoids associated with the split exten- sions and producing new examples of double groupoids in which the sets of squares, edges and points are group-groupoids. The idea of crossed module over groups was initially introduced by Whitehead in [29, 30] during the investigation of the properties of second relative homotopy groups for topological spaces. The categori- cal equivalence between crossed modules over groups and group-groupoids which are widely called in literature 2-groups [3], -groupoids or group objects in the category of groupoids [9], was proved by Brown and Spencer in [9, TheoremG 1]. Following this equivalence normal and quotient objects in these two cate- gories have been recently compared and associated objects in the category of group-groupoids have been characterized in [22]. -
Appendix a Topological Groups and Lie Groups
Appendix A Topological Groups and Lie Groups This appendix studies topological groups, and also Lie groups which are special topological groups as well as manifolds with some compatibility conditions. The concept of a topological group arose through the work of Felix Klein (1849–1925) and Marius Sophus Lie (1842–1899). One of the concrete concepts of the the- ory of topological groups is the concept of Lie groups named after Sophus Lie. The concept of Lie groups arose in mathematics through the study of continuous transformations, which constitute in a natural way topological manifolds. Topo- logical groups occupy a vast territory in topology and geometry. The theory of topological groups first arose in the theory of Lie groups which carry differential structures and they form the most important class of topological groups. For exam- ple, GL (n, R), GL (n, C), GL (n, H), SL (n, R), SL (n, C), O(n, R), U(n, C), SL (n, H) are some important classical Lie Groups. Sophus Lie first systematically investigated groups of transformations and developed his theory of transformation groups to solve his integration problems. David Hilbert (1862–1943) presented to the International Congress of Mathe- maticians, 1900 (ICM 1900) in Paris a series of 23 research projects. He stated in this lecture that his Fifth Problem is linked to Sophus Lie theory of transformation groups, i.e., Lie groups act as groups of transformations on manifolds. A translation of Hilbert’s fifth problem says “It is well-known that Lie with the aid of the concept of continuous groups of transformations, had set up a system of geometrical axioms and, from the standpoint of his theory of groups has proved that this system of axioms suffices for geometry”. -
Elementary Homotopy Theory II: Pointed Homotopy
Elementary Homotopy Theory II: Pointed Homotopy Tyrone Cutler June 14, 2020 Contents 1 Pointed Homotopy 1 1.1 Cones and Paths . 4 1.2 Suspensions and Loops . 8 1.3 Understanding Maps Out of a Suspension . 11 1 Pointed Homotopy Definition 1 Let X; Y be based spaces. A based, or pointed, homotopy between based maps f; g : X ! Y is a continuous function H : X × I ! Y satisfying 1) H(x; 0) = f(x), 8x 2 X. 2) H(x; 1) = g(x), 8x 2 X. 3) H(∗; t) = ∗ , 8t 2 I. Thus a based homotopy is just a homotopy through pointed maps. Clearly H factors to produce a pointed map ∼ He : X × I= ∗ ×I = X ^ I+ ! Y (1.1) which satisfies the first two listed properties. Conversely, any pointed map X ^ I+ ! Y satisfying these properties also defines a pointed homotopy. As expected, pointed homotopy is an equivalence relation on T op∗(X; Y ), and is compatible with composition. We use the same notation f ' g as in the unpointed case to indicate that f; g are based homotopic. No confusion should arise, and if we need to be clear we often describe unpointed maps and homotopies as free. As in the unbased case, we can also take the adjoint of a homotopy H : X ^ I+ ! Y to view it as a based map # ∼ I H : X ! C∗(I+;Y ) = Y (1.2) or, if X is locally compact, an unbased map He : I ! C∗(X; Y ): (1.3) 1 All the notions and terminology introduced for the unpointed category are also available in the pointed category. -
ON CATEGORICAL CROSSED MODULES Introduction
Theory and Applications of Categories, Vol. 16, No. 22, 2006, pp. 585–618. ON CATEGORICAL CROSSED MODULES P. CARRASCO, A.R. GARZON´ AND E.M. VITALE Abstract. The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernel- cokernel lemma for a diagram of categorical crossed modules. We then study derivations with coefficients in categorical crossed modules and show the existence of a categorical crossed module given by inner derivations. This allows us to define the low-dimensional cohomology categorical groups and, finally, these invariants are connected by a six-term 2-exact sequence obtained by using the kernel-cokernel lemma. Introduction Crossed modules of groups were introduced by J.H.C. Whitehead [42]; they have been shown to be relevant both in topological and algebraic contexts. They provided algebraic models for connected 2-types [28, 30] and allowed to develop, as adequate coefficients, a non-abelian cohomology of groups [13, 27]. It is convenient and sensible to regard crossed modules of groups as 2-dimensional versions of groups (c.f. [33]). They correspond, in fact, to strict categorical groups, that is, strict monoidal groupoids where each object is invertible up to isomorphism. Categorical groups have been studied in the last twenty- five years by several authors from different points of view (algebraic models for homotopy types [4, 6], cohomology [8, 39], extensions [1, 2, 34], derivations [19, 20], ring theory [16, 40]). -
HOMOTOPY THEORY for BEGINNERS Contents 1. Notation
HOMOTOPY THEORY FOR BEGINNERS JESPER M. MØLLER Abstract. This note contains comments to Chapter 0 in Allan Hatcher's book [5]. Contents 1. Notation and some standard spaces and constructions1 1.1. Standard topological spaces1 1.2. The quotient topology 2 1.3. The category of topological spaces and continuous maps3 2. Homotopy 4 2.1. Relative homotopy 5 2.2. Retracts and deformation retracts5 3. Constructions on topological spaces6 4. CW-complexes 9 4.1. Topological properties of CW-complexes 11 4.2. Subcomplexes 12 4.3. Products of CW-complexes 12 5. The Homotopy Extension Property 14 5.1. What is the HEP good for? 14 5.2. Are there any pairs of spaces that have the HEP? 16 References 21 1. Notation and some standard spaces and constructions In this section we fix some notation and recollect some standard facts from general topology. 1.1. Standard topological spaces. We will often refer to these standard spaces: • R is the real line and Rn = R × · · · × R is the n-dimensional real vector space • C is the field of complex numbers and Cn = C × · · · × C is the n-dimensional complex vector space • H is the (skew-)field of quaternions and Hn = H × · · · × H is the n-dimensional quaternion vector space • Sn = fx 2 Rn+1 j jxj = 1g is the unit n-sphere in Rn+1 • Dn = fx 2 Rn j jxj ≤ 1g is the unit n-disc in Rn • I = [0; 1] ⊂ R is the unit interval • RP n, CP n, HP n is the topological space of 1-dimensional linear subspaces of Rn+1, Cn+1, Hn+1. -
The Infinite Symmetric Product and Homology Theory
THE INFINITE SYMMETRIC PRODUCT AND HOMOLOGY THEORY ANDREW VILLADSEN Abstract. Following the work of Aguilar, Gitler, and Prieto, I define the infi- nite symmetric product of a pointed topological space. The infinite symmetric product allows the construction of a reduced homology theory on CW com- plexes in terms of the homotopy groups. This relationship between homology and homotopy gives a means to convert Moore spaces that are CW complexes into Eilenberg-Mac Lane spaces, and I give an explicit construction for any finitely generated Abelian groups of CW complexes which are Moore spaces. Contents 1. Introduction 1 2. The infinite symmetric product 2 3. The homology groups 7 4. Moore spaces and Eilenberg-Mac Lane spaces 11 Acknowledgments 13 References 13 1. Introduction The basic elements of the field of algebraic topology are divided generally into two categories: homotopy theory, which considers the homotopy groups of a topo- logical space; and homology theory, which considers instead the homology groups. Each of these approaches has its particular benefits and drawbacks. The homotopy groups are relatively easy to define, and are in some sense a more powerful tool than homology, but are in general not easily computed. The homology groups, which up to natural isomorphism can be constructed in a number of seemingly disparate ways, while perhaps a less powerful tool are much more easily computable, giving them considerable practical value. In this paper I follow the program of Aguilar, Gitler, and Prieto given in [1], sup- plemented by the original work of Dold and Thom in [2], which relate homology theory to homotopy theory by means of the infinite symmetric product construc- tion, a functor from the category of pointed spaces to itself. -
Coalgebras in Symmetric Monoidal Categories of Spectra
Homology, Homotopy and Applications, vol. 21(1), 2019, pp.1–18 COALGEBRAS IN SYMMETRIC MONOIDAL CATEGORIES OF SPECTRA MAXIMILIEN PEROUX´ and BROOKE SHIPLEY (communicated by J.P.C. Greenlees) Abstract We show that all coalgebras over the sphere spectrum are cocommutative in the category of symmetric spectra, orthog- onal spectra, Γ-spaces, W-spaces and EKMM S-modules. Our result only applies to these strict monoidal categories of spectra and does not apply to the ∞-category setting. 1. Introduction It is well known that the diagonal map of a set, or a space, gives it the struc- ture of a comonoid. In fact, the only possible (counital) comonoidal structure on an object in a Cartesian symmetric monoidal category is given by the diagonal (see [AM10, Example 1.19]). Thus all comonoids are forced to be cocommutative in these settings. We prove that this rigidity is inherited by all of the strict monoidal cate- gories of spectra that have been developed over the last 20 years, including symmetric spectra (see [HSS00]), orthogonal spectra (see [MMSS01, MM02]), Γ-spaces (see [Seg74, BF78]), W-spaces (see [And74]) and S-modules (see [EKMM97]), which we call EKMM-spectra here. That is, S-coalgebra spectra in any of these categories are cocommutative, where S denotes the sphere spectrum. Theorem 1.1. Let (C, ∆,ε) be an S-coalgebra in symmetric spectra, orthogonal spec- tra, Γ-spaces, W-spaces or EKMM-spectra. Then C is a cocommutative S-coalgebra. Furthermore, we prove that all R-coalgebras are cocommutative whenever R is 0 a commutative S-algebra with R0 homeomorphic to S ; see Theorems 3.4 and 4.1.