Groups up to Congruence Relation and from Categorical Groups to C
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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. -
Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
8.6 Modular Arithmetic
“mcs” — 2015/5/18 — 1:43 — page 263 — #271 8.6 Modular Arithmetic On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae, Gauss introduced the notion of “congruence.” Now, Gauss is another guy who managed to cough up a half-decent idea every now and then, so let’s take a look at this one. Gauss said that a is congruent to b modulo n iff n .a b/. This is j written a b.mod n/: ⌘ For example: 29 15 .mod 7/ because 7 .29 15/: ⌘ j It’s not useful to allow a modulus n 1, and so we will assume from now on that moduli are greater than 1. There is a close connection between congruences and remainders: Lemma 8.6.1 (Remainder). a b.mod n/ iff rem.a; n/ rem.b; n/: ⌘ D Proof. By the Division Theorem 8.1.4, there exist unique pairs of integers q1;r1 and q2;r2 such that: a q1n r1 D C b q2n r2; D C “mcs” — 2015/5/18 — 1:43 — page 264 — #272 264 Chapter 8 Number Theory where r1;r2 Œ0::n/. Subtracting the second equation from the first gives: 2 a b .q1 q2/n .r1 r2/; D C where r1 r2 is in the interval . n; n/. Now a b.mod n/ if and only if n ⌘ divides the left side of this equation. This is true if and only if n divides the right side, which holds if and only if r1 r2 is a multiple of n. -
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. -
Math 327, Solutions to Hour Exam 2
Math 327, Solutions to Hour Exam 2 1. Prove that the function f : N × N ! N defined by f(m; n) = 5m10n is one-to-one. Solution: Begin by observing that we may rewrite the formula for f as f(m; n) = 5m+n2n. Suppose f(m1; n1) = f(m2; n2), then 5m1+n1 2n1 = 5m2+n2 2n2 = k for some common integer k. By the fundamental theorem of arithmetic, the exponents in the prime decomposition of k are unique and hence m1 + n1 = m2 + n2 and n1 = n2: Substituting the second equality into the first we have m1 +n1 = m2 +n1 and hence m1 = m2. Therefore, (m1; n1) = (m2; n2), which shows that the function is one-to-one. 2a. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 5x ≡ 2 (mod 10). If no such x exists, explain why not. Solution: If the relation has an integer solution x, then there exists an integer k such that 5x − 2 = 10k, which can be rewritten as 2 = 5x − 10k = 5(x − 2k). But since x − 2k is an integer, this shows that 5 2, which is false. This contradiction shows that there are no solutions to the relation. 2b. Find all integers x, 0 ≤ x < 10 satisfying the congruence relation 7x ≡ 6 (mod 10). If no such x exists, explain why not. Solution: By Proposition 4.4.9 in the book, since 7 and 10 are relatively prime, there exists a unique solution to the relation. To find it, we use the Euclidean algorithm to see that 21 = 7(3) + 10(−2), which means that 7(3) ≡ 1(mod 10). -
Congruence Relations and Fundamental Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 75, 44545 1 (1982) Congruence Relations and Fundamental Groups YASUTAKA IHARA Department of Mathematics, Faculty of Science, University of Tok.vo, Bunkyo-ku, Tokyo 113, Japar? Communicated by J. Dieudonne’ Received April 10, 1981 1 In the arithmetic study of fundamental groups of algebraic curves over finite fields, the liftings of the Frobeniuses play an essential role. Let C be a smooth proper irreducible algebraic curve over a finite field F;I and let II (resp. n’) be the graphs on C x C of the qth (resp. q-‘th) power correspon- dences of C. In several interesting cases, C admits a lifting to characteristic 0 together with its correspondence T = 17 U P. It will be shown that in such a case the quotient of the algebraic fundamental group xi(C) modulo its normal subgroup generated by the Frobenius elements of “special points” (see below) is determined by the homomorphisms between the topological fundamental groups of the liftings of C and of T. The main result of this paper, which generalizes [2a, b] (Sect. 4), is formulated roughly as follows. Suppose that there exist two compact Riemann surfaces 31, 93’ that Yift” C, and another one, go, equipped with two finite morphisms p: 91°+ 93, o’: 93’ -+ 3’ such that 9 x @: ‘3’ --P 3 X 3’ is generically injective and (o X 4~‘) (‘3”) lifts T. Call an F,,rational point x of C special if the point (x, x9) of T (which lies on the crossing of 17 and n’) does not lift to a singular point of (u, x rp’)(l#‘). -
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]. -
Higher Categorical Groups and the Classification of Topological Defects and Textures
YITP-SB-18-29 Higher categorical groups and the classification of topological defects and textures J. P. Anga;b;∗ and Abhishodh Prakasha;b;c;y aDepartment of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA bC. N. Yang Institute for Theoretical Physics, Stony Brook, NY 11794-3840, USA cInternational Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Bangalore 560089, India Abstract Sigma models effectively describe ordered phases of systems with spontaneously broken sym- metries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context, these solitons are known as textures or defect sectors. In this paper, we address the problem of enumerating and describing the solitonic sectors of sigma models. We approach this problem via an algebraic topological method { com- binatorial homotopy, in which one models both spacetime and the target space with algebraic arXiv:1810.12965v1 [math-ph] 30 Oct 2018 objects which are higher categorical generalizations of fundamental groups, and then counts the homomorphisms between them. We give a self-contained discussion with plenty of examples and a discussion on how our work fits in with the existing literature on higher groups in physics. ∗[email protected] (corresponding author) [email protected] 1 Higher groups, defects and textures 1 Introduction Methods of algebraic topology have now been successfully used in physics for over half a cen- tury. One of the earliest uses of homotopy theory in physics was by Skyrme [1], who attempted to model the nucleon using solitons, or topologically stable field configurations, which are now called skyrmions. -
Bisimple Semigroups
/-BISIMPLE SEMIGROUPS BY R. J. WARNE Let S be a semigroup and let Es denote the set of idempotents of S. As usual Es is partially ordered in the following fashion: if e,feEs, efíf if and only if ef=fe = e. Let /denote the set of all integers and let 1° denote the set of nonnegative integers. A bisimple semigroup Sis called an 7-bisimple semigroup if and only if Es is order isomorphic to 7 under the reverse of the usual order. We show that S is an 7-bisimple semigroup if and only if S^Gx Ixl, where G is a group, under the multiplication (g, a, b)(h, c, d) = (gfb-}c.chab-cfb-c.d, a,b + d-c) if b ^ c, = (fc~-\,ag<xc~''fc-b,bh,a+c-b, d) if c ^ b, where a is an endomorphism of G, a0 denoting the identity automorphism of G, and for me Io, ne I, /o,n=e> the identity of G while if m>0, fim.n = un + i"m~1un + 2am-2- ■ -un + (m.X)aun + m, where {un : ne/} is a collection of elements of G with un = e, the identity of G, if n > 0. If we let G = {e}, the one element group, in the above multiplication we obtain S=IxI under the multiplication (a, b)(c, d) = (a + c —r, b + d—r). We will denote S under this multiplication by C*, and we will call C* the extended bicyclic semigroup. C* is the union of the chain I of bicyclic semigroups C. -
UNIVERSAL ALGEBRA Abstract Approved _____MMEMIIIMI Norman Franzen
AN ABSTRACT OF THE THESIS OF JUNPEI SEKINO for the MASTER OF SCIENCE (Name) (Degree) ,, in MATHEMATICS presented on () t ,{, i >_: ¡ ``¡2 (Major) (Date) Title: UNIVERSAL ALGEBRA Abstract approved _____MMEMIIIMI Norman Franzen In this paper, we are concerned with the very general notion of a universal algebra. A universal algebra essentially consists of a set A together with a possibly infinite set of finitary operations on. A. Generally, these operations are related by means of equations, yield- ing different algebraic structures such as groups, groups with oper- ators, modules, rings, lattices, etc. This theory is concerned with those theorems which are common to all these various algebraic sys- tems. In particular, the fundamental isomorphism and homomorphism theorems are covered, as well as, the Jordan- Holder theorem and the Zassenhaus lemma. Furthermore, new existence proofs are given for sums and free algebras in any primitive class of universal algebras. The last part treats the theory of groups with multi- operators in a manner essentially different from that of P. J. Higgins. The ap- proach taken here generalizes the theorems on groups with operators as found in Jacobson's "Lectures in Abstract Algebra, " vol. I. The basic language of category theory is used whenever conven- ient. Universal Algebra by Junpei Sekino A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science June 1969 APPROVED: Assistant Professor of Mathematics in charge of major Acting Chairman of Department of Mathematics Dean of ra uate c ool Date thesis is presented ( ;( Typed by Clover Redfern for JUNPEI SEKINO ACKNOWLEDGMENT I wish to acknlowledge my gratitude to Mr. -
Structures and Congruences
Structures and Congruences Oliver Kullmann Mathematical structures Equivalence Structures and Congruences relations Subspaces and quotients Oliver Kullmann Isomorphisms theorems Computer Science Department Congruences for Swansea University groups and rings MRes Seminar Swansea, November 10, 2008 Structures and Introduction Congruences Oliver Kullmann This lecture tries to convey some key concepts from Mathematical mathematics in general, relevant for our investigations structures Equivalence into boolean algebra and Stone duality: relations First we reflect upon “mathematical structures”. Subspaces and quotients Of fundamental importance is the concept of an Isomorphisms equivalence relation “compatible” with the structure, theorems Congruences for which enables us to derive a “quotient”. groups and rings We discuss this concept in general, and especially for the algebraic structures here. Basically no proofs are given, but emphasise is put on the definitions and properties, leaving the proofs as relatively straightforward exercises. You must fill the gaps yourself!! Structures and Overview Congruences Oliver Kullmann Mathematical structures 1 Mathematical structures Equivalence relations Subspaces and quotients 2 Equivalence relations Isomorphisms theorems Congruences for 3 Subspaces and quotients groups and rings 4 Isomorphisms theorems 5 Congruences for groups and rings Structures and Three major mathematical structures Congruences Oliver Kullmann Bourbaki introduced “structuralism” to mathematics (and Mathematical the world), distinguishing -
A Study of Universal Algebras in Fuzzy Set Theory
RHODES UNIVERSITY DEPARTMENT OF MATHEMATICS A STUDY OF UNIVERSAL ALGEBRAS IN FUZZY SET THEORY by V. MURAU A thesis submitted in fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics. February 1987 ( i ) ABSTRACT This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [IX,F] is constructed by extending operations on X to that on IX, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in IX. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived . Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting.