The Freyd's Adjoint Functor Theorem

Total Page:16

File Type:pdf, Size:1020Kb

The Freyd's Adjoint Functor Theorem Universit´edu Qu´ebec `aMontr´eal D´epartement de math´ematiques Marco A. P´erezB. [email protected] THE FREYD'S ADJOINT FUNCTOR THEOREM January 2011 Contents Introduction i 1 Universal Arrows 1 1.1 Comma categories and universal arrows . .1 1.2 Limits and colimits . .3 2 Limits and Adjunction 11 2.1 Adjoint functors . 11 2.2 Preservation of limits . 17 2.3 The Freyd's Adjoint Functor Theroem . 20 Bibliography 25 Introduction Adjoint functors are pairs of functors, G : D −! C and F : C −! D, which stand in a particular relationship with one another, called an adjunction. Specifically, (F; G) is an adjunction if there exists a natural isomorphism θ : HomD(F (·); ·) −! HomC(·;G(·)). Under these conditions, G is said to have a left adjoint, namely F . Of course, not every functor admits a left adjoint. For instance, if X is a set having two or more elements, then the functor X × − : Set −! Set does not have a left adjoint. A natural question that comes to us is under which conditions it is possible to determine the existence of a left adjoint F for a given functor G. Peter Freyd answered that question, for the particular case where the category D is complete. If D is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem: Theorem. Given a small and complete category D, a functor G : D −! C has a left adjoint if and only if it preserves all limits and satisfies the following Solution Set Condition. For each object C 2 Ob(C) there is a set I and an I-indexed family of arrows fi : C −! G(Di) such that every arrow h : C −! G(D) can be written as a composite h = G(t) ◦ fi for some index i and some t : Di −! D. In these notes we shall give a prove of this result due to Saunders Mac Lane. First, we study universal arrows and characterize them as initial objects in certain comma categories. Then we study the concept of a limit for a functor F : J −! C. Limits give rise to a special type of categories known as complete categories. In these categories there exists a lot of universal constructions, such as products, equalizers and pullbacks. We begin the last chapter recalling the notion of adjunction. Adjoint functors have a deep relation with limits, in fact, every functor having a left adjoint preserves limits. Before proving the Freyd's theorem, we first study the case of the existence of an initial object in a category and then use the fact that each universal arrow defined by the unit of a left adjoint is an initial object in a suitable comma category. i ii Chapter 1 Universal Arrows 1.1 Comma categories and universal arrows A comma category is a construction in category theory, introduced in 1963 by F. W. Lawvere, which provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. We shall see there are certain guarantees about the existence of limits and colimits in the context of comma categories. Let F : C −! D be a functor and D 2 Ob(D). We define (D # F ) the comma category of objects F -under D as follows: (1) the objects of (D # F ) are all pairs (u; C) where C 2 Ob(C) and u : D −! F (C) is an arrow of D; (2) the arrows of (D # F ), h :(u1;C1) −! (u2;C2), are arrows h : C1 −! C2 of C making the following triangle commute: D E yy EE u1 yy EE u2 yy EE yy EE |yy E" F (C1) / F (C2) F (h) Similarly, the comma category (F # D) of objects F -over D is defined as follows: (1) the objects of (F # D) are all pairs (u; C) where C 2 Ob(C) and u : F (C) −! D is an arrow of D; 1 (2) the arrows of (F # D), h :(u1;C1) −! (u2;C2), are arrows h : C1 −! C2 of C such that the triangle D < bE yy EE u1 yy EE u2 yy EE yy EE yy E F (C1) / F (C2) F (h) commutes. Example 1.1.1. Let U : Grp −! Set be the forgetful functor and X a set. An object of (X # U) is a function X −! U(G) from X into the underlying set of G, where G is a group. If G : D −! C is a functor and C 2 Ob(C), a universal arrow from C to G is a pair (D; µ) consisting of an object D of D and an arrow µ : C −! G(D) of C, such that for every arrow f : C −! G(D0) of C there is a unique arrow h : D −! D0 of D such that the diagram µ D C / G(D) C CC CC CC 9!h CC G(h) f CC CC C! D0 G(D0) commutes. The arrow µ is also called G-free. Similarly, a universal arrow from G to C is a pair (D; ν) where D is an object of D and ν : G(D) −! C is an arrow of C, such that for every arrow f : G(D0) −! C of C there exists a unique arrow h : D0 −! D of D such that the diagram ν D G(D) / C O = O {{ {{ {{ 9!h G(h) {{ {{ f {{ {{ D0 G(D0) commutes. The arrow ν is also called G-cofree. Proposition 1.1.1. (1) If (D1; µ1) and (D2; µ2) are universal arrows from C to G, then D1 and D2 are isomor- phic. (2) If (D1; ν1) and (D2; ν2) are universal arrows from G to C, then D1 and D2 are isomor- phic. 2 Proof: We only prove (1). Part (2) can be proven in a similar way. Since (D1; µ1) is a universal arrow from C to G, there exists a unique arrow h : D1 −! D2 0 of D such that G(h) ◦ µ1 = µ2. Similarly, there exists a unique arrow h : D2 −! D1 such 0 that G(h ) ◦ µ2 = µ1. So we get 0 G(h ) ◦ (G(h) ◦ µ1) = µ1 0 (G(h ) ◦ (G(h)) ◦ µ1 = µ1 0 (G(h ◦ h)) ◦ µ1 = µ1: On the other hand, idD1 is the only arrow of D satisfying G(idD1 ) ◦ µ1 = µ1. Hence 0 0 h ◦ h = idD1 . Similarly, h ◦ h = idD2 . Therefore, h : D1 −! D2 is an isomorphism. From the definitions of comma categories and universal arrows, the following proposition is immediate. Proposition 1.1.2. (1) µ : C −! G(D) is a universal arrow from C to G if and only if (D; µ) is an initial object in the comma category (C # G). (2) ν : G(D) −! C is a universal arrow from G to C if and only if (D; ν) is a terminal object in the comma category (G # C). 1.2 Limits and colimits The abstract notion of a limit captures the essential properties of universal constructions such as products, equalizers and pullbacks. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts and pushouts. Let J be a small category, i.e., Ob(J ) and Hom(J ) are sets. Let C be a category. Consider the category [J ; C] whose objects are all functors F : J −! C and whose arrows are all natural transformations α : F −! G. Define a functor ∆ : C −! [J ; C] as follows: (1) ∆(C): J −! C is the constant functor ∆(i) = C for every i 2 Ob(J ), and ∆(a) = idC for every arrow a : i −! j of J ; 3 (2) for every arrow f : C1 −! C2 of C, ∆(f) : ∆(C1) −! ∆(C2) is the natural transforma- tion defined as the J -indexed set ∆(f) = f∆(f)i = f : C1 −! C2gi2Ob(J ): The functor ∆ is known as the diagonal functor.A limit for a functor F : J −! C is a universal arrow (L; ν) from ∆ to F . In this situation, ν : ∆(L) −! F is a natural transformation, i.e., a J -indexed family of arrows νi : L −! F (i) of C, with i 2 Ob(J ), such that the triangle νi L / F (i) ?? ?? ?? ?? F (a) νj ?? ?? ? F (j) commutes, for every arrow a : i −! j of J . Moreover, if β : ∆(L0) −! F is another natural 0 transformation, i.e., a J -indexed family of arrows (βi : L −! F (i))i2Ob(J ) of C satisfying 0 F (a) ◦ βi = βj for every arrow a : i −! j, then there exists a unique arrow h : L −! L of C such that β = ν ◦ ∆(h), i.e., βi = νi ◦ h for every i 2 Ob(J ). In pictures, we have the commutative diagram L0 9!h β βi L j < ÓÓ << ÓÓ << νi ÓÓ << νj ÓÓ << ÓÓ << ÓÓ << ÑÓÓ < F (i) / F (j) F (a) By Proposition 1.1.1, the object L is unique up to isomorphisms. Example 1.2.1. (1) Terminal objects: Let J be the empty category. Then any F : J −! C is the empty functor. Thus, every pair (L; ν : ∆(L) −! F ) is just the object L. If the previous pair is the limit for F , then for any other pair (L0; β : ∆(L0) −! F ), i.e., any other object L0, there exists a unique arrow h : L0 −! L of C. Hence, L is a terminal object in C. (2) Products: Let J = I be a discrete small category, i.e., a small category whose only arrows are the identity arrows. A functor F : I −! C is just a I-indexed set of objects fCi = F (i)gi2I .
Recommended publications
  • Congruences Between Derivatives of Geometric L-Functions
    1 Congruences between Derivatives of Geometric L-Functions David Burns with an appendix by David Burns, King Fai Lai and Ki-Seng Tan Abstract. We prove a natural equivariant re¯nement of a theorem of Licht- enbaum describing the leading terms of Zeta functions of curves over ¯nite ¯elds in terms of Weil-¶etalecohomology. We then use this result to prove the validity of Chinburg's ­(3)-Conjecture for all abelian extensions of global function ¯elds, to prove natural re¯nements and generalisations of the re- ¯ned Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-¶etalecohomology groups. In an appendix coau- thored with K. F. Lai and K-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules. 1991 Mathematics Subject Classi¯cation: Primary 11G40; Secondary 11R65; 19A31; 19B28. Keywords and Phrases: Geometric L-functions, leading terms, congruences, Iwasawa theory 2 David Burns 1. Introduction The main result of the present article is the following Theorem 1.1. The central conjecture of [6] is valid for all global function ¯elds. (For a more explicit statement of this result see Theorem 3.1.) Theorem 1.1 is a natural equivariant re¯nement of the leading term formula proved by Lichtenbaum in [27] and also implies an extensive new family of integral congruence relations between the leading terms of L-functions associated to abelian characters of global functions ¯elds (see Remark 3.2).
    [Show full text]
  • Op → Sset in Simplicial Sets, Or Equivalently a Functor X : ∆Op × ∆Op → Set
    Contents 21 Bisimplicial sets 1 22 Homotopy colimits and limits (revisited) 10 23 Applications, Quillen’s Theorem B 23 21 Bisimplicial sets A bisimplicial set X is a simplicial object X : Dop ! sSet in simplicial sets, or equivalently a functor X : Dop × Dop ! Set: I write Xm;n = X(m;n) for the set of bisimplices in bidgree (m;n) and Xm = Xm;∗ for the vertical simplicial set in horiz. degree m. Morphisms X ! Y of bisimplicial sets are natural transformations. s2Set is the category of bisimplicial sets. Examples: 1) Dp;q is the contravariant representable functor Dp;q = hom( ;(p;q)) 1 on D × D. p;q G q Dm = D : m!p p;q The maps D ! X classify bisimplices in Xp;q. The bisimplex category (D × D)=X has the bisim- plices of X as objects, with morphisms the inci- dence relations Dp;q ' 7 X Dr;s 2) Suppose K and L are simplicial sets. The bisimplicial set Kט L has bisimplices (Kט L)p;q = Kp × Lq: The object Kט L is the external product of K and L. There is a natural isomorphism Dp;q =∼ Dpט Dq: 3) Suppose I is a small category and that X : I ! sSet is an I-diagram in simplicial sets. Recall (Lecture 04) that there is a bisimplicial set −−−!holim IX (“the” homotopy colimit) with vertical sim- 2 plicial sets G X(i0) i0→···→in in horizontal degrees n. The transformation X ! ∗ induces a bisimplicial set map G G p : X(i0) ! ∗ = BIn; i0→···→in i0→···→in where the set BIn has been identified with the dis- crete simplicial set K(BIn;0) in each horizontal de- gree.
    [Show full text]
  • Category Theory for Autonomous and Networked Dynamical Systems
    Discussion Category Theory for Autonomous and Networked Dynamical Systems Jean-Charles Delvenne Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) and Center for Operations Research and Econometrics (CORE), Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium; [email protected] Received: 7 February 2019; Accepted: 18 March 2019; Published: 20 March 2019 Abstract: In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way. Keywords: ergodic theory; topological dynamics; control theory 1. Introduction The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics (studying continuous maps on topological spaces) and ergodic theory (studying probability measures invariant under the map) as two prominent examples. These theories have followed parallel tracks and built multiple bridges. For example, both have grown successful tools from the interaction with information theory soon after its emergence, around the key invariants, namely the topological entropy and metric entropy, which are themselves linked by a variational theorem. The situation is more complex and heterogeneous in the field of what we here call open dynamical systems, or controlled dynamical systems.
    [Show full text]
  • A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
    A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience.
    [Show full text]
  • Category Theory
    Michael Paluch Category Theory April 29, 2014 Preface These note are based in part on the the book [2] by Saunders Mac Lane and on the book [3] by Saunders Mac Lane and Ieke Moerdijk. v Contents 1 Foundations ....................................................... 1 1.1 Extensionality and comprehension . .1 1.2 Zermelo Frankel set theory . .3 1.3 Universes.....................................................5 1.4 Classes and Gödel-Bernays . .5 1.5 Categories....................................................6 1.6 Functors .....................................................7 1.7 Natural Transformations. .8 1.8 Basic terminology . 10 2 Constructions on Categories ....................................... 11 2.1 Contravariance and Opposites . 11 2.2 Products of Categories . 13 2.3 Functor Categories . 15 2.4 The category of all categories . 16 2.5 Comma categories . 17 3 Universals and Limits .............................................. 19 3.1 Universal Morphisms. 19 3.2 Products, Coproducts, Limits and Colimits . 20 3.3 YonedaLemma ............................................... 24 3.4 Free cocompletion . 28 4 Adjoints ........................................................... 31 4.1 Adjoint functors and universal morphisms . 31 4.2 Freyd’s adjoint functor theorem . 38 5 Topos Theory ...................................................... 43 5.1 Subobject classifier . 43 5.2 Sieves........................................................ 45 5.3 Exponentials . 47 vii viii Contents Index .................................................................. 53 Acronyms List of categories. Ab The category of small abelian groups and group homomorphisms. AlgA The category of commutative A-algebras. Cb The category Func(Cop,Sets). Cat The category of small categories and functors. CRings The category of commutative ring with an identity and ring homomor- phisms which preserve identities. Grp The category of small groups and group homomorphisms. Sets Category of small set and functions. Sets Category of small pointed set and pointed functions.
    [Show full text]
  • Notes and Solutions to Exercises for Mac Lane's Categories for The
    Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits .
    [Show full text]
  • Derived Functors and Homological Dimension (Pdf)
    DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic.
    [Show full text]
  • N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
    N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y).
    [Show full text]
  • Coreflective Subcategories
    transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1).
    [Show full text]
  • Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0.
    [Show full text]
  • A Few Points in Topos Theory
    A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces.
    [Show full text]
  • Basic Category Theory and Topos Theory
    Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object .
    [Show full text]