Sheafification Johan M. Commelin, October 15, 2013
|
|
- Gervais Hugh Goodwin
- 5 years ago
- Views:
Transcription
1 heaiication Johan M. Commelin, October 15, 2013 Introduction Let C be a categor, equipped with a Grothendieck topolog J. Let 2 Ph(C) be a preshea. The purpose o these notes is to assign to a morphism to a shea! #, through which an other morphism to a shea! G actors. In doing so, a crucial r^ole is plaed b the plus construction, dened below. We rst some preliminar conventions and notation. Notation. Let U 2 C be an object. The contravariant Hom-unctor Hom( ; U ) will be denoted h U. A morphism : V! U in C will be identied with the induced morphism h V! h U. (Indeed, to avoid conusion, in the tet below we will introduce morphisms in C as morphisms between presheaves.) In the same spirit, we identi elements o (U ) with morphisms h U!. Remark. Let U 2 C be some object. A sieve on U is a subunctor o h U. Nevertheless, we will use the notation T, to denote the ``intersection'' o two sieves on U. ormall, this means z hu T. where Denition. The plus construction is given b: ( ) + : Ph(C)! Ph(C) 7! + ; + : C op! et U 7! colim 2J (U )Hom Ph(C)(; ): The plus construction is a well-deined unctor Remark. The colimit in the denition o + (U ) eists, since it is a colimit in et. Notation. Given some covering sieve on some U 2 C, and an element 2 Hom(; ), we write or the image o in + (U ) = Hom(h U ; + ). Lemma. The plus construction applied to gives a preshea +. Proo. Given a morphism : h V! h U rom C, there is an obvious candidate or + ( ). Ater all, given a covering sieve on U, we have a covering sieve { on V. There is a natural morphism {! b composition with. Consequentl we get a morphism Hom Ph(C)(; )! Hom Ph(C)( { ; ). This gives a natural transormation rom the diagram o + (U ) to the diagram o + (V ) {, and thereore an induced morphism between the colimits. In practice, this means that or :!, it maps to = j {. 1
2 h V h U + { We leave it to the reader to veri that + preserves identit and composition. Corollar. There is a natural morphism :! +, b mapping 2 (U ) to. Indeed, viewing as morphism h U!, we have =. + h U Corollar. or a covering sieve on U, and a morphism :!, we have = j. h U + Proo. We veri the identit b ``probing'' it with representables. Let : h V! be some morphism. We ma then identi h V with {. B the preceding lemma, we have = j { = : The preceding corollar gives, =. ince is arbitrar, we conclude that j =. Lemma. The plus construction is a unctor. Proo. Given a morphism o presheaves :! G, one obtains a natural transormation o the diagram dening + (U ) to the diagram dening G + (U ) just b composing with. This induces morphisms on the colimits, which are compatible with restriction morphisms. Thus we have a morphism +! G +, and this construction evidentl preserves identities and composition. eparatedness o + Lemma. The preshea + is separated. Proo. Let U 2 C be some object, and a covering sieve on U. We have to show that an natural transormation :! + etends to at most one natural transormation h U! +. Assume such an etension eists. Let and be two such etensions, represented b :!, and :!, where and are covering sieves on U. 2
3 h U + Put T =. Then we have jt = jt = jt : urther, jt represents, and jt represents. It ollows that we ma replace our setup with the ollowing: h U + T or an V 2 C, and 2 (V ), we have = = = = : Consequentl, and agree on some common renement T o h V. We use these T and the transitivit aiom o Grothendieck topologies to create a covering sieve o U, on which and agree. Recall the transitivit aiom or Grothendieck topologies: Let R be a sieve on U. I there is a covering sieve T on U, such that or all V 2 C and 2 T (V ), the sieve { R covers V, then R covers U. Let R be the sieve V; 2T (V ) ( T ). In other words, ever morphism in R(W ), when viewed as morphism h W! h U, actors via some : T! h U. Note that T { R, and thereore, b the transitivit aiom o Grothendieck topologies, R is a covering sieve on U. Observe that b construction R is a subsieve o T, and thereore we ma replace our setup with the ollowing: h U + R 3
4 To prove that =, it now suces to prove that =. Let W 2 C be arbitrar, and k : h W! R be an element o R(W ). B denition o R, there eists some V 2 C, and 2 T (V ), such that the composition k : h W! R! T equals h W! T! h V! T. ince jt = jt we also see that k = k. As W and k are arbitrar, we conclude that =, which implies =. I is separated, + is a shea We continue the notation o the previous section. Lemma. Assume that is separated. The + is a shea. Proo. Now we have to show that etends (and uniqueness will ollow rom the previous section). or an V 2 C, and an 2 (V ), the composition gives an element o + (V ). Represent this element b some morphism :!. These representing morphisms are compatible, in the sense that the are, up to renement, unctorial in V and. More precise: or some W 2 C, and g 2 (W ), the morphisms g and g might not be equal, but the do agree on some W -covering subsieve ;g o g. g h V + g h W g g ;g Let R be the sieve V; 2(V ) ( ). In other words, ever morphism in R(W ), when viewed as morphism h W! h U, actors via some :! h U. Note that { R, and thereore, b the transitivit aiom o Grothendieck topologies, R is a covering sieve on U. Giving a morphism ~ : R! boils down to giving its composition (an element o (W )) with ever morphism k : h W! R. B denition o R, such a morphism k actors as k = g, with 2 (V ), g 2 (W ). Now put ~ k = g. This does not depend on the actorisation k = g, since g etends k : k!, and is separated. 4
5 h V g ~ h W k R We are done, i we check that ~ etends. This is done b a doublelaered ``probing'' with representables. or an : h V!, we have to show that equals ~ j. We ma test this equalit on, because + is separated. But or an g 2 (W ), we have k = g 2 R(W ), and or such k we have just proven the etension. ~ h U + h V g ~ h W k R Idempotence, adjunction, and eactness The idempotence ollows immediatel rom the denition: I is a shea, then the colimit dening + (U ) has terms Hom Ph(C)(; ) = (U ), because o the shea propert, and the Yoneda lemma. It is then clear that + (U ) = (U ), which grants the idempotence. Let be a preshea, and G a shea. We have to show that We reduce this to proving that Hom(; G) = Hom( # ; G): Hom(; G) = Hom( + ; G): This is actuall not ver hard. B unctorialit o the plus construction, we have a map + : +! G associated to ever map :! G. I 2 + (U ) is represented b :!. Then + () = (), showing that + is the unique etension o along. + h U + G G id 5
6 Now that we have the adjunction established, it is a ormal consequence that ( ) # preserves all colimits. To show that it also preserves nite limits, again, we reduce this to showing that ( ) + preserves nite limits. Now it boils down to the observation that the colimit in the denition o + (U ) is ltered, and thereore commutes with nite limits. 6
Descent 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 informationHomotopy Colimits of Relative Categories (Preliminary Version)
Homotopy Colimits of Relative Categories (Preliminary Version) Lennart Meier October 28, 2014 Grothendieck constructions of model categories have recently received some attention as in [HP14]. We will
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 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 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 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 informationEtale cohomology of fields by Johan M. Commelin, December 5, 2013
Etale cohomology of fields by Johan M. Commelin, December 5, 2013 Etale cohomology The canonical topology on a Grothendieck topos Let E be a Grothendieck topos. The canonical topology T on E is given in
More informationLecture 9: Sheaves. February 11, 2018
Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with
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 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 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 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 information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
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 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 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 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 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 informationarxiv: v1 [math.at] 1 Aug 2016
Kan Complees, Homotop and Cohomolog 1 KAN COMPLEXES, HOMOTOPY AND COHOMOLOGY b JAN STEINEBRUNNER ABSTRACT This article shows several new methods for proofs on Kan complees while using them to give a compact
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 informationGrothendieck construction for bicategories
Grothendieck construction or bicategories Igor Baković Rudjer Bošković Institute Abstract In this article, we give the generalization o the Grothendieck construction or pseudo unctors given in [5], which
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 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 informationStabilization as a CW approximation
Journal of Pure and Applied Algebra 140 (1999) 23 32 Stabilization as a CW approximation A.D. Elmendorf Department of Mathematics, Purdue University Calumet, Hammond, IN 46323, USA Communicated by E.M.
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 informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
More informationGALOIS THEORY GROTHENDIECK
Faculty of Science and Technology Master Degree in Mathematics GALOIS THEORY OF GROTHENDIECK Supervisor: Prof. Luca BARBIERI VIALE Thesis by: Alessandro MONTRESOR Student ID: 808469 Academic year 2013-2014
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 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 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 informationLECTURE NOTES IN EQUIVARIANT ALGEBRAIC GEOMETRY. Spec k = (G G) G G (G G) G G G G i 1 G e
LECTURE NOTES IN EQUIVARIANT ALEBRAIC EOMETRY 8/4/5 Let k be field, not necessaril algebraicall closed. Definition: An algebraic group is a k-scheme together with morphisms (µ, i, e), k µ, i, Spec k, which
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 informationTensor products in Riesz space theory
Tensor products in Riesz space theor Jan van Waaij Master thesis defended on Jul 16, 2013 Thesis advisors dr. O.W. van Gaans dr. M.F.E. de Jeu Mathematical Institute, Universit of Leiden CONTENTS 2 Contents
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 informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 6
Ma 4: Introduction to Lebesgue Integration Solutions to Homework Assignment 6 Pro. Wickerhauser Due Thursda, April 5th, 3 Please return our solutions to the instructor b the end o class on the due date.
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 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 informationarxiv: v1 [math.kt] 9 Jul 2018
MOTIVIC TAMBARA FUNCTORS TOM BACHMANN arxiv:1807.02981v1 [math.kt] 9 Jul 2018 Abstract. Let k be a ield and denote by SH(k) the motivic stable homotopy category. Recall its ull subcategory SH(k) e [2].
More informationElementary (ha-ha) Aspects of Topos Theory
Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................
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 informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More information{1X } if X = Y otherwise.
C T E G O R Y T H E O R Y Dr E. L. Cheng e.cheng@dpmms.cam.ac.uk http://www.dpmms.cam.ac.uk/ elgc2 Michaelmas 2002 1 Categories, unctors and natural transormations 1.1 Categories DEFINITION 1.1.1 category
More informationThe Morita-equivalence between MV-algebras and abelian l-groups with strong unit
The Morita-equivalence between MV-algebras and abelian l-groups with strong unit Olivia Caramello and Anna Carla Russo December 4, 2013 Abstract We show that the theory of MV-algebras is Morita-equivalent
More informationarxiv: v1 [math.ct] 12 Nov 2015
double-dimensional approach to ormal category theory Seerp Roald Koudenburg arxiv:1511.04070v1 [math.t] 12 Nov 2015 Drat version as o November 13, 2015 bstract Whereas ormal category theory is classically
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 informationCategories and Modules
Categories and odules Takahiro Kato arch 2, 205 BSTRCT odules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract
More informationCategory Theory. Course by Dr. Arthur Hughes, Typset by Cathal Ormond
Category Theory Course by Dr. Arthur Hughes, 2010 Typset by Cathal Ormond Contents 1 Types, Composition and Identities 3 1.1 Programs..................................... 3 1.2 Functional Laws.................................
More informationThis is a repository copy of The homotopy theory of Khovanov homology.
This is a repository copy o The homotopy theory o Khovanov homology. White Rose Research Online URL or this paper: http://eprints.whiterose.ac.uk/90638/ Version: Published Version Article: Everitt, Brent
More informationPerverse Sheaves. Bhargav Bhatt. Fall The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications.
Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal o this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see
More informationNATURAL WEAK FACTORIZATION SYSTEMS
NATURAL WEAK FACTORIZATION SYSTEMS MARCO GRANDIS AND WALTER THOLEN Abstract. In order to acilitate a natural choice or morphisms created by the (let or right) liting property as used in the deinition o
More informationUNSTABLE MODULES OVER THE STEENROD ALGEBRA REVISITED
UNSTABLE MODULES OVER THE STEENROD ALGEBRA REVISITED GEOREY M.L. POWELL Abstract. A new and natural description of the category of unstable modules over the Steenrod algebra as a category of comodules
More informationON KAN EXTENSION OF HOMOLOGY AND ADAMS COCOMPLETION
Bulletin o the Institute o Mathematics Academia Sinica (New Series) Vol 4 (2009), No 1, pp 47-66 ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION B AKRUR BEHERA AND RADHESHAM OTA Abstract Under a set
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
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 informationBasic Category Theory
BRICS LS-95-1 J. van Oosten: Basic Category Theory BRICS Basic Research in Computer Science Basic Category Theory Jaap van Oosten BRICS Lecture Series LS-95-1 ISSN 1395-2048 January 1995 Copyright c 1995,
More informationIndCoh Seminar: Ind-coherent sheaves I
IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of
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 informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information9.1 The Square Root Function
Section 9.1 The Square Root Function 869 9.1 The Square Root Function In this section we turn our attention to the square root unction, the unction deined b the equation () =. (1) We begin the section
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 informationCOMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY
COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY VIVEK SHENDE A ring is a set R with two binary operations, an addition + and a multiplication. Always there should be an identity 0 for addition, an
More informationHomotopy, Quasi-Isomorphism, and Coinvariants
LECTURE 10 Homotopy, Quasi-Isomorphism, an Coinvariants Please note that proos o many o the claims in this lecture are let to Problem Set 5. Recall that a sequence o abelian groups with ierential is a
More informationUMS 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 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 informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
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 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 informationAn Introduction to Topos Theory
An Introduction to Topos Theory Ryszard Paweł Kostecki Institute o Theoretical Physics, University o Warsaw, Hoża 69, 00-681 Warszawa, Poland email: ryszard.kostecki % uw.edu.pl June 26, 2011 Abstract
More informationRepresentable presheaves
Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom
More informationMicro-support of sheaves
Micro-support of sheaves Vincent Humilière 17/01/14 The microlocal theory of sheaves and in particular the denition of the micro-support is due to Kashiwara and Schapira (the main reference is their book
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 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 informationModules over a Ringed Space
Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More informationSome glances at topos theory. Francis Borceux
Some glances at topos theory Francis Borceux Como, 2018 2 Francis Borceux francis.borceux@uclouvain.be Contents 1 Localic toposes 7 1.1 Sheaves on a topological space.................... 7 1.2 Sheaves
More informationThe Kervaire Invariant One Problem, Lecture 9, Independent University of Moscow, Fall semester 2016
The Kervaire Invariant One Problem, Lecture 9, Independent University of Moscow, Fall semester 2016 November 8, 2016 1 C-brations. Notation. We will denote by S the (, 1)-category of spaces and by Cat
More informationChapter 6. Self-Adjusting Data Structures
Chapter 6 Self-Adjusting Data Structures Chapter 5 describes a data structure that is able to achieve an epected quer time that is proportional to the entrop of the quer distribution. The most interesting
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 informationAn introduction to Yoneda structures
An introduction to Yoneda structures Paul-André Melliès CNRS, Université Paris Denis Diderot Groupe de travail Catégories supérieures, polygraphes et homotopie Paris 21 May 2010 1 Bibliography Ross Street
More informationTRANSFINITE LIMITS IN TOPOS THEORY
Theory and Applications of Categories, Vol. 31, No. 7, 2016, pp. 175200. TRANSFINITE LIMITS IN TOPOS THEORY MORITZ KERZ Abstract. For a coherent site we construct a canonically associated enlarged coherent
More informationarxiv: v3 [math.kt] 20 Oct 2008
THE MOTHER OF ALL ISOMORPHISM CONJECTURES VIA DG CATEGORIES AND DERIVATORS arxiv:0810.2099v3 [math.kt] 20 Oct 2008 PAUL BALMER AND GONÇALO TABUADA Abstract. We describe a undamental additive unctor E und
More informationAbstract structure of unitary oracles for quantum algorithms
Abstract structure o unitary oracles or quantum algorithms William Zeng 1 Jamie Vicary 2 1 Department o Computer Science University o Oxord 2 Centre or Quantum Technologies, University o Singapore and
More informationRepresentation of monoids in the category of monoid acts. 1. Introduction and preliminaries
Quasigroups and Related Systems 25 (2017), 251 259 Representation of monoids in the category of monoid acts Abolghasem Karimi Feizabadi, Hamid Rasouli and Mahdieh Haddadi To Bernhard Banaschewski on his
More information= s j Ui U j. i, j, then s F(U) with s Ui F(U) G(U) F(V ) G(V )
1 Lecture 2 Recap Last tme we talked about presheaves and sheaves. Preshea: F on a topologcal space X, wth groups (resp. rngs, sets, etc.) F(U) or each open set U X, wth restrcton homs ρ UV : F(U) F(V
More information9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
More informationThe fundamental theorem for the algebraic K-theory of spaces: II the canonical involution
Journal of Pure and Applied Algebra 167 (2002) 53 82 www.elsevier.com/locate/jpaa The fundamental theorem for the algebraic K-theory of spaces: II the canonical involution Thomas Huttemann a, John R. Klein
More information2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction
. ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties
More informationChapter 5. Localization. 5.1 Localization of categories
Chaper 5 Localizaion Consider a caegory C and a amily o morphisms in C. The aim o localizaion is o ind a new caegory C and a uncor Q : C C which sends he morphisms belonging o o isomorphisms in C, (Q,
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationCategorical Properties of Topological and Dierentiable Stacks
Categorical Properties of Topological and Dierentiable Stacks ISBN 978-90-5335-441-4 Categorical Properties of Topological and Dierentiable Stacks Categorische Eigenschappen van Topologische en Dierentieerbare
More informationGabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base
Under consideration or publication in J. Functional Programming 1 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base STEPHEN LACK School o Computing and Mathematics, University o Western
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More informationUniversity of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor
Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.
More informationThe Uniformity Principle on Traced Monoidal Categories
Electronic Notes in Theoretical Computer Science 69 (2003) URL: http://www.elsevier.nl/locate/entcs/volume69.html 19 pages The Uniormity Principle on Traced Monoidal Categories Masahito Hasegawa Research
More informationA calculus of fractions for the homotopy category of a Brown cofibration category
A calculus o ractions or the homotopy category o a Brown coibration category Sebastian Thomas Dissertation August 2012 Rheinisch-Westälisch Technische Hochschule Aachen Lehrstuhl D ür Mathematik ii Version:
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 informationarxiv: v2 [math.ct] 19 Feb 2008
Understanding the small object argument arxiv:0712.0724v2 [math.ct] 19 Feb 2008 Richard Garner Department o Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden October 14, 2011 Abstract
More informationSYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION
SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO Abstract. We give an operadic deinition o a genuine symmetric monoidal G-category, and we prove
More informationInvariants and semi-direct products for nite group actions on tensor categories
J. Math. Soc. Japan Vol. 53, No. 2, 2001 Invariants and semi-direct products for nite group actions on tensor categories B Daisuke Tambara (Received Nov. 15, 1999) Abstract. Suppose a group G acts on a
More informationarxiv:math/ v1 [math.at] 6 Oct 2004
arxiv:math/0410162v1 [math.at] 6 Oct 2004 EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL Abstract. We construct hyper-homology spectral sequences
More information