A Peter May Picture Book, Part 1
|
|
- Abraham Osborne
- 5 years ago
- Views:
Transcription
1 A Peter May Picture Book, Part 1 Steve Balady Auust 17, 2007 This is the beinnin o a larer project, a notebook o sorts intended to clariy, elucidate, and/or illustrate the principal ideas in A Concise Course in Alebraic Topoloy. Many o the worthwhile exercises suested in the text will be done in relatively complete detail, one-line proos expanded, constructions illustrated, and deinitions iven as assertions that a certain diaram commutes (which, when I try to explain to my riends, I am walked away rom) iven explanations anyway. The bulk o this paper ocuses on the cateorical intuition that the book assumes, with the concepts illustrated in the cateory o topoloical spaces. Followin that is a careul illustration o a proo that well illustrates the utility o such intuition, which I irst presented as part o a directed readin proram durin the previous academic year. This paper concludes with a list o currently intended illustrations or chapters 5-10, which will be added to a workin version o this document as they are done. p. 16, on limits and colimits in Top and Grp At this point, anyone who hasn t seen cateory theory beore will probably be utterly despondent. Some pictures will help. We ll do the product and coproduct, initial and terminal object, equalizer and coequalizer, pullback and pushout in the cateories o topoloical spaces and o roups these are the most important examples, and in practice the only ones that come up all that oten. Let s start with the product in Top. Remember, in this cateory the objects are topoloical spaces, and the arrows are continuous maps between spaces. Eectively, with the product we start with a collection o unrelated topoloical spaces. We want the limit o this collection, and assert that it is their Cartesian product endowed with the product topoloy, alon with the canonical projection maps onto each coordinate. 1
2 The ollowin is a diaram indicatin this or two spaces X and Y ; see also Fiure 1. X p X Z X p Y X Y Z Y Y Z To say that this diaram commutes is to say that the Cartesian product endowed with the product topoloy has two special properties: irst, it is itsel endowed with projection maps onto each coordinate; second, that any other space with such projections has a unique projection onto the Cartesian product. We shall check this assertion in eneral. Let I be an indexin set; we assert that the product o X i, i I is Π i I X i endowed with the product topoloy, alon with its projection maps p j : Π i I X i X j or j I. On the set-theoretic level, this is relatively clear: iven any other space Q with maps Q i : Q X i, we et a actorization throuh the Cartesian product. Speciically, deine the map : Q Π i I X i by q (Q i (q)) i I ; uniqueness up to bijection is orced by deinition. We need only impose a topoloy on the Cartesian product as a set; since the product topoloy is homeomorphic to the weak topoloy enerated by the projection maps, and the weak topoloy is the weakest topoloy relative to which the projection maps are continuous, since Q i are actually continuous maps, we are done. Dually, we assert that the coproduct o topoloical spaces is the disjoint union endowed with the topoloy enerated by its disjoint subspaces, alon with the standard inclusion maps. The diaram X Z X i X i Y X Y Z says that each coordinate embeds in the disjoint union, and also that the disjoint union embeds uniquely in any other space endowed with embeddin maps. See also Fiure 2. A terminal object is the speciic case o the product on the empty set; its dual, the coproduct on the empty set, is called an initial object. It should be clear now that or topoloical spaces, the terminal object is the one-point space with its only topoloy and the initial object is the empty set with its only topoloy. Z Y Y 2
3 Now we restrict ourselves to a pair o spaces X and Y, but we also have a pair o continuous maps, between them. We call the limit o this diaram the equalizer o and. As in the case o the product, its name is suestive in Top, it is the subspace o X on which and are equal. In the ollowin, we set L = {x X (x) = (x)} and l to be the standard inclusion map. l L X m M Y Dually, the coequalizer o X and Y with maps and will be a quotient space; speciically, the quotient by the smallest equivalence relation or which, or all x X, (x) (x). In the ollowin, we call it C, with q the quotient map. X Y q C d D The pullback is the limit o A C B. For topoloical spaces, it is the subspace o A B iven by {(x, y) x X, y Y, (x) = (y)} with its standard maps (projection to each actor ollowed by inclusion), and is usually written as A C B. D A C B A B C 3
4 The dual to the pullback is the pushout; it is the colimit o A C B. For a topoloical space, it is a quotient o A B, and as with the coequalizer it is the quotient induced by the weakest equivalence relation such that (x) (y). It is usually written as A C B. An illustration provides ar more intuition than paes o ormality; thus, see Fiure 3. C A B A C B In eneral, the intuition used or calculatin the limitin objects in Top carry throuh to Grp with minor modiications. We assert the ollowin or roups: the product is aain the Cartesian product with its natural roup structure, the coproduct is the ree product amalamated over the identity, an equalizer is the larest subroup contained in {a (a) = (a)}, a coequalizer is the quotient by the larest normal subroup contained in {a (a) = (a)}, and similarly or the pullback and pushout. We prove only the ollowin (which will be useul or the theorem on pae 36, which we later illustrate): i G is a roup and N is normal in G, the pushout o G i N 0 in the cateory o roups is isomorphic to G/N. D N i G k 0 G/N r j H We must show that any object makin the diaram commute actors uniquely throuh G/N and its associated maps. Assume that H, j : 0 H, r : G H does so, and consider n N; we trace all o its possible paths. Followin i irst, we see that n n, where the latter is in G. On the other hand, or jk to be a roup homomorphism, we must have that jk(n) = 0. Since the diaram commutes, jk(n) = ri(n) = 0, which immediately implies that ker(r) N. Identiyin H by the irst isomorphism theorem as G/ker(r), the above ives us a unique actorization throuh G/N, which veriies the universal property o the pushout. 4
5 p. 17, completeness is equivalent to the existence o products and equalizers I m relatively convinced that in this case, the proo is a worthwhile exercise should be interpreted to mean the proo is an awul diaram-chasin mess. The idea, however, is well worth seein. We present the basic idea o the standard proo, and reer the reader to Borceux or the orey details; eectively, we eneralize the idea above that a pullback is a subset o the cartesian product (a product and an equalizer). Thinkin dually, a pushout is a quotient o a disjoint union (a coproduct and a coequalizer), and by reversin all the arrows, the ollowin will also prove that cocompleteness is equivalent to the existence o coproducts and coequalizers. Let Cbe a cateory. I Cis complete, it contains in particular all products and equalizers, so that s ine. So assume that Chas all o its products and equalizers, let Dbe a small cateory (that is, a set o objects equipped with a collection o arrows between objects), and let F be a unctor rom Dto C. This is just a way o oranizin inormation. For example, the equalizer o, : A B written in this way would be the limit o F x, F y : F 1 F 2 where Dis the cateory with objects 1 and 2 and arrows x, y : 1 2, F 1 = A, F 2 = B, F x =, F y =. What we want is the equalizer o a pair o maps α, β. L l F D F (t()) D D D Here s() is the source (domain, corane) o, t() is its taret (rane, codomain), α((x D ) D D ) = (x t() ) D, β((x D ) D D ) = (F (x s() )) D, and L, l is the equalizer o α, β. We assert that L is the limit o F. In our example above, α(x A, x B ) = (x B, x B ), β(x A, x B ) = ((x A ), (x A )), and equalizin α, β is the assertion that (x A ) = (x A ); that is, L is the equalizer o,. From here the proo is airly straiht-orward: show that L, l ives a cone on F, and then that any other cone actors uniquely throuh it. I can oer no more intuition than that ained by starin at the diarams which prove this, so I leave the remainder to Borceux. p. 36, an illustration o the proo that every roup is π 1 o some space See Fiures 4 and 5. p. 39, on the adjunction o product and Hom in CGHTop See Fiure 6. 5
6 To be continued... p. 41, some intuition or the HEP p. 42, a mappin cylinder p. 42, a coibration is an inclusion with closed imae p. 43, illustratin the retraction which a coibration eects p. 47, on duality, intuition or the CHP p. 47, a mappin path space p. 49, a iber bundle is a ibration, with picture p. 55, S m S n = S m+n, with picture p. 56, [Σ 2 X, Y ] is an Abelian roup p. 58, an illustration that Ci Ci/CA is a homotopy equivalence p. 58, an illustration that up to homotopy equivalence, each pair o maps in our coiber sequence is the composite o a map and the inclusion o its taret in its coiber p. 59, a homotopy iber p. 61, illustration o unit and counit between loops and suspension p. 63, illustration o the maps in the lon exact sequence o pairs p. 64, a ew calculations actually calculated p. 65, the Hop map is a iber bundle p. 66, a word and picture on chane o basepoint p. 67, homotopy equivalence induces weak equivalence p. 68, the point o the technical lemma, with picture, explanation and condolances p. 73, how the lemma implies the HELP p. 74, the Whitehead theorem, and what it is not Reerences J. Peter May, A Concise Course in Alebraic Topoloy. University o Chicao Press, Chicao, may/ F. Borceux, Handbook o cateorical alebra 1. Cambride University Press, Cambride, Basic cateory theory. 6
UMS 7/2/14. Nawaz John Sultani. July 12, Abstract
UMS 7/2/14 Nawaz John Sultani July 12, 2014 Notes or July, 2 2014 UMS lecture Abstract 1 Quick Review o Universals Deinition 1.1. I S : D C is a unctor and c an object o C, a universal arrow rom c to S
More informationCategory Theory 2. Eugenia Cheng Department of Mathematics, University of Sheffield Draft: 31st October 2007
Cateory Theory 2 Euenia Chen Department o Mathematics, niversity o Sheield E-mail: echen@sheieldacuk Drat: 31st October 2007 Contents 2 Some basic universal properties 1 21 Isomorphisms 2 22 Terminal objects
More informationTHE GORENSTEIN DEFECT CATEGORY
THE GORENSTEIN DEFECT CATEGORY PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN Dedicated to Ranar-Ola Buchweitz on the occasion o his sixtieth birthday Abstract. We consider the homotopy cateory
More informationBECK'S THEOREM CHARACTERIZING ALGEBRAS
BEK'S THEOREM HARATERIZING ALGEBRAS SOFI GJING JOVANOVSKA Abstract. In this paper, I will construct a proo o Beck's Theorem characterizin T -alebras. Suppose we have an adjoint pair o unctors F and G between
More informationCLASS NOTES MATH 527 (SPRING 2011) WEEK 6
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6 BERTRAND GUILLOU 1. Mon, Feb. 21 Note that since we have C() = X A C (A) and the inclusion A C (A) at time 0 is a coibration, it ollows that the pushout map i
More informationCategorical Background (Lecture 2)
Cateorical Backround (Lecture 2) February 2, 2011 In the last lecture, we stated the main theorem o simply-connected surery (at least or maniolds o dimension 4m), which hihlihts the importance o the sinature
More informationLIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset
5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used,
More informationCLASSIFICATION OF GROUP EXTENSIONS AND H 2
CLASSIFICATION OF GROUP EXTENSIONS AND H 2 RAPHAEL HO Abstract. In this paper we will outline the oundations o homoloical alebra, startin with the theory o chain complexes which arose in alebraic topoloy.
More informationThe Category of Sets
The Cateory o Sets Hans Halvorson October 16, 2016 [[Note to students: this is a irst drat. Please report typos. A revised version will be posted within the next couple o weeks.]] 1 Introduction The aim
More informationGENERAL ABSTRACT NONSENSE
GENERAL ABSTRACT NONSENSE MARCELLO DELGADO Abstract. In this paper, we seek to understand limits, a uniying notion that brings together the ideas o pullbacks, products, and equalizers. To do this, we will
More informationTHE AXIOMS FOR TRIANGULATED CATEGORIES
THE AIOMS FOR TRIANGULATED CATEGORIES J. P. MA Contents 1. Trianulated cateories 1 2. Weak pusouts and weak pullbacks 4 3. How to prove Verdier s axiom 6 Reerences 9 Tis is an edited extract rom my paper
More informationINTERSECTION THEORY CLASSES 20 AND 21: BIVARIANT INTERSECTION THEORY
INTERSECTION THEORY CLASSES 20 AND 21: BIVARIANT INTERSECTION THEORY RAVI VAKIL CONTENTS 1. What we re doin this week 1 2. Precise statements 2 2.1. Basic operations and properties 4 3. Provin thins 6
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationELEMENTS IN MATHEMATICS FOR INFORMATION SCIENCE NO.14 CATEGORY THEORY. Tatsuya Hagino
1 ELEMENTS IN MTHEMTICS FOR INFORMTION SCIENCE NO.14 CTEGORY THEORY Tatsuya Haino haino@sc.keio.ac.jp 2 Set Theory Set Theory Foundation o Modern Mathematics a set a collection o elements with some property
More informationTHE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS
THE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS FINNUR LÁRUSSON Abstract. We give a detailed exposition o the homotopy theory o equivalence relations, perhaps the simplest nontrivial example o a model structure.
More informationCONVENIENT CATEGORIES OF SMOOTH SPACES
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 11, November 2011, Paes 5789 5825 S 0002-9947(2011)05107-X Article electronically published on June 6, 2011 CONVENIENT CATEGORIES OF
More informationCOARSE-GRAINING OPEN MARKOV PROCESSES. John C. Baez. Kenny Courser
COARE-GRAINING OPEN MARKOV PROCEE John C. Baez Department o Mathematics University o Caliornia Riverside CA, UA 95 and Centre or Quantum echnoloies National University o inapore inapore 7543 Kenny Courser
More informationCategories and Quantum Informatics: Monoidal categories
Cateories and Quantum Inormatics: Monoidal cateories Chris Heunen Sprin 2018 A monoidal cateory is a cateory equipped with extra data, describin how objects and morphisms can be combined in parallel. This
More informationReview of category theory
Review of category theory Proseminar on stable homotopy theory, University of Pittsburgh Friday 17 th January 2014 Friday 24 th January 2014 Clive Newstead Abstract This talk will be a review of the fundamentals
More information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationThe excess intersection formula for Grothendieck-Witt groups
manuscripta mathematica manuscript No. (will be inserted by the editor) Jean Fasel The excess intersection ormula or Grothendieck-Witt roups Received: date / Revised version: date Abstract. We prove the
More informationGENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationRELATIVE GOURSAT CATEGORIES
RELTIVE GOURST CTEGORIES JULI GOEDECKE ND TMR JNELIDZE bstract. We deine relative Goursat cateories and prove relative versions o the equivalent conditions deinin reular Goursat cateories. These include
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationJoseph Muscat Categories. 1 December 2012
Joseph Muscat 2015 1 Categories joseph.muscat@um.edu.mt 1 December 2012 1 Objects and Morphisms category is a class o objects with morphisms : (a way o comparing/substituting/mapping/processing to ) such
More informationDERIVED CATEGORIES AND THEIR APPLICATIONS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 48, Número 3, 2007, Páinas 1 26 DERIVED CATEGORIES AND THEIR APPLICATIONS Abstract In this notes we start with the basic deinitions o derived cateories,
More informationSPLITTING OF SHORT EXACT SEQUENCES FOR MODULES. 1. Introduction Let R be a commutative ring. A sequence of R-modules and R-linear maps.
SPLITTING OF SHORT EXACT SEQUENCES FOR MODULES KEITH CONRAD 1. Introduction Let R be a commutative rin. A sequence o R-modules and R-linear maps N M P is called exact at M i im = ker. For example, to say
More informationFrom Loop Groups To 2-Groups
From Loop Groups To 2-Groups John C. Baez Joint work with: Aaron Lauda Alissa Crans Danny Stevenson & Urs Schreiber 1 G f 1 f D 2 More details at: http://math.ucr.edu/home/baez/2roup/ 1 Hiher Gaue Theory
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationEpimorphisms and maximal covers in categories of compact spaces
@ Applied General Topoloy c Universidad Politécnica de Valencia Volume 14, no. 1, 2013 pp. 41-52 Epimorphisms and maximal covers in cateories o compact spaces B. Banaschewski and A. W. Haer Abstract The
More informationA crash course in homological algebra
Chapter 2 A crash course in homoloical alebra By the 194 s techniques from alebraic topoloy bean to be applied to pure alebra, ivin rise to a new subject. To bein with, recall that a cateory C consists
More informationTangent Categories. David M. Roberts, Urs Schreiber and Todd Trimble. September 5, 2007
Tangent Categories David M Roberts, Urs Schreiber and Todd Trimble September 5, 2007 Abstract For any n-category C we consider the sub-n-category T C C 2 o squares in C with pinned let boundary This resolves
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More informationPart 1. 1 Basic Theory of Categories. 1.1 Concrete Categories
Part 1 1 Basic Theory o Cateories 1.1 Concrete Cateories One reason or inventin the concept o cateory was to ormalize the notion o a set with additional structure, as it appears in several branches o mathematics.
More information( ) x y z. 3 Functions 36. SECTION D Composite Functions
3 Functions 36 SECTION D Composite Functions By the end o this section you will be able to understand what is meant by a composite unction ind composition o unctions combine unctions by addition, subtraction,
More informationPART I. Abstract algebraic categories
PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.
More informationHSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS
HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationWestern University. Imants Barušs King's University College, Robert Woodrow
Western University Scholarship@Western Psycholoy Psycholoy 2013 A reduction theorem or the Kripke-Joyal semantics: Forcin over an arbitrary cateory can always be replaced by orcin over a complete Heytin
More informationTOPOLOGY FROM THE CATEGORICAL VIEWPOINT
TOOLOGY FROM THE CATEGORICAL VIEWOINT KYLE ORMSBY One o the primary insihts o twentieth century mathematics is that obects should not be studied in isolation. Rather, to understand obects we must also
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More informationarxiv:math/ v1 [math.ct] 16 Jun 2006
arxiv:math/0606393v1 [math.ct] 16 Jun 2006 Strict 2-toposes Mark Weber bstract. 2-cateorical eneralisation o the notion o elementary topos is provided, and some o the properties o the yoneda structure
More informationINTERSECTION THEORY CLASS 17
INTERSECTION THEORY CLASS 17 RAVI VAKIL CONTENTS 1. Were we are 1 1.1. Reined Gysin omomorpisms i! 2 1.2. Excess intersection ormula 4 2. Local complete intersection morpisms 6 Were we re oin, by popular
More informationDUALITY AND SMALL FUNCTORS
DUALITY AND SMALL FUNCTORS GEORG BIEDERMANN AND BORIS CHORNY Abstract. The homotopy theory o small unctors is a useul tool or studying various questions in homotopy theory. In this paper, we develop the
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and
More informationThe Sectional Category of a Map
arxiv:math/0403387v1 [math.t] 23 Mar 2004 The Sectional Category o a Map M. rkowitz and J. Strom bstract We study a generalization o the Svarc genus o a iber map. For an arbitrary collection E o spaces
More informationarxiv: v3 [math.at] 28 Feb 2014
arxiv:1101.1025v3 [math.at] 28 Feb 2014 CROSS EFFECTS AND CALCULUS IN AN UNBASED SETTING (WITH AN APPENDIX BY ROSONA ELDRED) KRISTINE BAUER, BRENDA JOHNSON, AND RANDY MCCARTHY Abstract. We studyunctors
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationMADE-TO-ORDER WEAK FACTORIZATION SYSTEMS
MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS EMILY RIEHL The aim o this note is to briely summarize techniques or building weak actorization systems whose right class is characterized by a particular liting
More informationON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES
ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES TOBIAS BARTHEL AND EMIL RIEHL Abstract. We present general techniques or constructing unctorial actorizations appropriate or model
More informationOriented Bivariant Theories, I
ESI The Erwin Schrödiner International Boltzmannasse 9 Institute or Mathematical Physics A-1090 Wien, Austria Oriented Bivariant Theories, I Shoji Yokura Vienna, Preprint ESI 1911 2007 April 27, 2007 Supported
More informationSpan, Cospan, and Other Double Categories
ariv:1201.3789v1 [math.ct] 18 Jan 2012 Span, Cospan, and Other Double Categories Susan Nieield July 19, 2018 Abstract Given a double category D such that D 0 has pushouts, we characterize oplax/lax adjunctions
More informationIn the index (pages ), reduce all page numbers by 2.
Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by
More informationUniversity of Cape Town
The copyright o this thesis rests with the. No quotation rom it or inormation derived rom it is to be published without ull acknowledgement o the source. The thesis is to be used or private study or non-commercial
More informationON THE CONSTRUCTION OF LIMITS AND COLIMITS IN -CATEGORIES
ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES EMILY RIEHL ND DOMINIC VERITY bstract. In previous work, we introduce an axiomatic ramework within which to prove theorems about many varieties o
More informationCategory Theory (UMV/TK/07)
P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written
More informationSTRONGLY GORENSTEIN PROJECTIVE MODULES OVER UPPER TRIANGULAR MATRIX ARTIN ALGEBRAS. Shanghai , P. R. China. Shanghai , P. R.
STRONGLY GORENSTEIN PROJETIVE MODULES OVER UPPER TRINGULR MTRIX RTIN LGERS NN GO PU ZHNG Department o Mathematics Shanhai University Shanhai 2444 P R hina Department o Mathematics Shanhai Jiao Ton University
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationPROJECTIVE REPRESENTATIONS OF QUIVERS
IJMMS 31:2 22 97 11 II. S1611712218192 ttp://ijmms.indawi.com Hindawi ublisin Corp. ROJECTIVE RERESENTATIONS OF QUIVERS SANGWON ARK Received 3 Auust 21 We prove tat 2 is a projective representation o a
More informationThe inner automorphism 3-group of a strict 2-group
The inner automorphism 3-roup o a strict 2-roup Dav Roberts and Urs Schreiber July 25, 2007 Abstract or any roup G, there is a 2-roup o inner automorphisms, INN(G). This plays the role o the universal
More informationThe basics of frame theory
First version released on 30 June 2006 This version released on 30 June 2006 The basics o rame theory Harold Simmons The University o Manchester hsimmons@ manchester.ac.uk This is the irst part o a series
More informationarxiv: v1 [math.ct] 27 Oct 2017
arxiv:1710.10238v1 [math.ct] 27 Oct 2017 Notes on clans and tribes. Joyal October 30, 2017 bstract The purpose o these notes is to give a categorical presentation/analysis o homotopy type theory. The notes
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationTHE COALGEBRAIC STRUCTURE OF CELL COMPLEXES
Theory and pplications o Categories, Vol. 26, No. 11, 2012, pp. 304 330. THE COLGEBRIC STRUCTURE OF CELL COMPLEXES THOMS THORNE bstract. The relative cell complexes with respect to a generating set o coibrations
More informationFORMAL CATEGORY THEORY
FORML TEGORY THEORY OURSE HELD T MSRYK UNIVERSITY IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN bstract. These are the notes o a readin seminar on ormal cateory theory we are runnin at Masaryk
More informationRepresentation Theory of Hopf Algebroids. Atsushi Yamaguchi
Representation Theory o H Algebroids Atsushi Yamaguchi Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions
More informationAdjunctions! Everywhere!
Adjunctions! Everywhere! Carnegie Mellon University Thursday 19 th September 2013 Clive Newstead Abstract What do free groups, existential quantifiers and Stone-Čech compactifications all have in common?
More informationBasic properties of limits
Roberto s Notes on Dierential Calculus Chapter : Limits and continuity Section Basic properties o its What you need to know already: The basic concepts, notation and terminology related to its. What you
More informationTHE SNAIL LEMMA ENRICO M. VITALE
THE SNIL LEMM ENRICO M. VITLE STRCT. The classical snake lemma produces a six terms exact sequence starting rom a commutative square with one o the edge being a regular epimorphism. We establish a new
More informationNotes on Beilinson s How to glue perverse sheaves
Notes on Beilinson s How to glue perverse sheaves Ryan Reich June 4, 2009 In this paper I provide something o a skeleton key to A.A. Beilinson s How to glue perverse sheaves [1], which I ound hard to understand
More informationFunctions. Introduction
Functions,00 P,000 00 0 y 970 97 980 98 990 99 000 00 00 Fiure Standard and Poor s Inde with dividends reinvested (credit "bull": modiication o work by Prayitno Hadinata; credit "raph": modiication o work
More informationSJÄLVSTÄNDIGA ARBETEN I MATEMATIK
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Equivariant Sheaves on Topological Categories av Johan Lindberg 2015 - No 7 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET,
More informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
More informationCategory Theory. Travis Dirle. December 12, 2017
Category Theory 2 Category Theory Travis Dirle December 12, 2017 2 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives
More informationare well-formed, provided Φ ( X, x)
(October 27) 1 We deine an axiomatic system, called the First-Order Theory o Abstract Sets (FOTAS) Its syntax will be completely speciied Certain axioms will be iven; but these may be extended by additional
More informationAssume the left square is a pushout. Then the right square is a pushout if and only if the big rectangle is.
COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY VIVEK SHENDE Last time we learned about Yoneda s lemma, and various universal constructions initial and final objects, products and coproducts (which
More informationAdic spaces. Sophie Morel. March 5, 2019
Adic spaces Sophie Morel March 5, 2019 Conventions : - Every rin is commutative. - N is the set of nonneative inteers. 2 Contents I The valuation spectrum 5 I.1 Valuations......................................
More informationSECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES
SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there
More information3 3 Lemma and Protomodularity
Ž. Journal o Algebra 236, 778795 2001 doi:10.1006jabr.2000.8526, available online at http:www.idealibrary.com on 3 3 Lemma and Protomodularity Dominique Bourn Uniersite du Littoral, 220 a. de l Uniersite,
More informationLECTURE 6: FIBER BUNDLES
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationThe concept of limit
Roberto s Notes on Dierential Calculus Chapter 1: Limits and continuity Section 1 The concept o limit What you need to know already: All basic concepts about unctions. What you can learn here: What limits
More informationA CHARACTERIZATION OF CENTRAL EXTENSIONS IN THE VARIETY OF QUANDLES VALÉRIAN EVEN, MARINO GRAN AND ANDREA MONTOLI
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 15 12 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES VLÉRIN EVEN, MRINO GRN ND NDRE MONTOLI bstract: The
More information2 Coherent D-Modules. 2.1 Good filtrations
2 Coherent D-Modules As described in the introduction, any system o linear partial dierential equations can be considered as a coherent D-module. In this chapter we ocus our attention on coherent D-modules
More informationExercises for Algebraic Topology
Sheet 1, September 13, 2017 Definition. Let A be an abelian group and let M be a set. The A-linearization of M is the set A[M] = {f : M A f 1 (A \ {0}) is finite}. We view A[M] as an abelian group via
More informationSMSTC Geometry & Topology 1 Assignment 1 Matt Booth
SMSTC Geometry & Topology 1 Assignment 1 Matt Booth Question 1 i) Let be the space with one point. Suppose X is contractible. Then by definition we have maps f : X and g : X such that gf id X and fg id.
More informationHOMOTOPY THEORY ADAM KAYE
HOMOTOPY THEORY ADAM KAYE 1. CW Approximation The CW approximation theorem says that every space is weakly equivalent to a CW complex. Theorem 1.1 (CW Approximation). There exists a functor Γ from the
More informationLectures - XXIII and XXIV Coproducts and Pushouts
Lectures - XXIII and XXIV Coproducts and Pushouts We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationLecture notes for a class on perfectoid spaces
Lecture notes for a class on perfectoid spaces Bharav Bhatt April 23, 2017 Contents Preface 3 1 Conventions on non-archimedean fields 4 2 Perfections and tiltin 6 3 Perfectoid fields 9 3.1 Definition and
More informationBARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS
BARYCENTRIC SUBDIVISION AND ISOMORPHISMS OF GROUPOIDS JASHA SOMMER-SIMPSON Abstract Given roupoids G and H as well as an isomorpism Ψ : Sd G = Sd H between subdivisions, we construct an isomorpism P :
More informationTriangulated categories and localization
Trianulated cateories and localization Karin Marie Jacobsen Master o Science in Physics and Mathematics Submission date: Januar 212 Supervisor: Aslak Bakke Buan, MATH Norweian University o Science and
More informationThe Ordinary RO(C 2 )-graded Cohomology of a Point
The Ordinary RO(C 2 )-graded Cohomology of a Point Tiago uerreiro May 27, 2015 Abstract This paper consists of an extended abstract of the Master Thesis of the author. Here, we outline the most important
More informationPOLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS. Contents. Introduction
Theory and Applications o Categories, Vol. 18, No. 2, 2007, pp. 4 101. Contents POLARIZED CATEGORY THEORY, MODULES, AND GAME SEMANTICS J.R.B. COCKETT AND R.A.G. SEELY Abstract. Motivated by an analysis
More informationArrow-theoretic differential theory
Arrow-theoretic dierential theory Urs Schreiber August 17, 2007 Abstract We propose and study a notion o a tangent (n + 1)-bundle to an arbitrary n-category Despite its simplicity, this notion turns out
More informationTHE FUNDAMENTAL GROUP AND CW COMPLEXES
THE FUNDAMENTAL GROUP AND CW COMPLEXES JAE HYUNG SIM Abstract. This paper is a quick introduction to some basic concepts in Algebraic Topology. We start by defining homotopy and delving into the Fundamental
More informationBALANCED CATEGORY THEORY
Theory and Applications o Categories, Vol. 20, No. 6, 2008, pp. 85 5. BALANCED CATEGORY THEORY CLAUDIO PISANI Abstract. Some aspects o basic category theory are developed in a initely complete category
More information