DOCSLIB.ORG

Functors and Natural Transformations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Functors and Natural Transformations Functors and natural transformations functors ; category morphisms natural transformations ; functor morphisms Andrzej Tarlecki: Category Theory, 2018 - 90 - Functors A functor F: K ! K0 from a category K to a category K0 consists of: • a function F: jKj ! jK0j, and • for all A; B 2 jKj, a function F: K(A; B) ! K0(F(A); F(B)) such that: Make explicit categories in which we work at various places here • F preserves identities, i.e., F(idA) = idF(A) for all A 2 jKj, and • F preserves composition, i.e., F(f;g) = F(f);F(g) for all f : A ! B and g : B ! C in K. We really should differentiate between various components of F Andrzej Tarlecki: Category Theory, 2018 - 91 - Examples • identity functors: IdK : K ! K, for any category K 0 0 • inclusions: IK,!K0 : K ! K , for any subcategory K of K 0 0 0 • constant functors: CA : K ! K , for any categories K; K and A 2 jK j, with CA(f) = idA for all morphisms f in K • powerset functor: P: Set ! Set given by − P(X) = fY j Y ⊆ Xg, for all X 2 jSetj − P(f): P(X) ! P(X0) for all f : X ! X0 in Set, P(f)(Y ) = ff(y) j y 2 Y g for all Y ⊆ X op • contravariant powerset functor: P−1 : Set ! Set given by − P−1(X) = fY j Y ⊆ Xg, for all X 2 jSetj 0 0 − P−1(f): P(X ) ! P(X) for all f : X ! X in Set, 0 0 0 0 P−1(f)(Y ) = fx 2 X j f(x) 2 Y g for all Y ⊆ X Andrzej Tarlecki: Category Theory, 2018 - 92 - Examples, cont'd. 0 0 0 • projection functors: π1 : K × K ! K, π2 : K × K ! K • list functor: List: Set ! Monoid, where Monoid is the category of monoids (as objects) with monoid homomorphisms as morphisms: ∗ ∗ − List(X) = hX ; b; i, for all X 2 jSetj, where X is the set of all finite lists of elements from X, b is the list concatenation, and is the empty list. − List(f): List(X) ! List(X0) for f : X ! X0 in Set, List(f)(hx1; : : : ; xni) = hf(x1); : : : ; f(xn)i for all x1; : : : ; xn 2 X • totalisation functor: Tot: Pfn ! Set∗, where Set∗ is the subcategory of Set of sets with a distinguished element ∗ and ∗-preserving functions − Tot(X) = X ] f∗g Define Set∗ as the category of algebras 8 < f(x) if it is defined − Tot(f)(x) = : ∗ otherwise Andrzej Tarlecki: Category Theory, 2018 - 93 - Examples, cont'd. • carrier set functors: j j: Alg(Σ) ! SetS, for any algebraic signature Σ = hS; Ωi, yielding the algebra carriers and homomorphisms as functions between them 0 • reduct functors: σ : Alg(Σ ) ! Alg(Σ), for any signature morphism σ :Σ ! Σ0, as defined earlier • term algebra functors: TΣ : Set ! Alg(Σ) for all (single-sorted) algebraic signatures Σ 2 jAlgSigj Generalise to many-sorted signatures − TΣ(X) = TΣ(X) for all X 2 jSetj # 0 0 − TΣ(f) = f : TΣ(X) ! TΣ(X ) for all functions f : X ! X G G • diagonal functors: ∆K : K ! DiagK for any graph G with nodes N = jGjnodes and edges E = jGjedges , and category K G A A A − ∆K(A) = D , where D is the \constant" diagram, with Dn = A for all A n 2 N and De = idA for all e 2 E G f A B f − ∆K(f) = µ : D ! D , for all f : A ! B, where µn = f for all n 2 N Andrzej Tarlecki: Category Theory, 2018 - 94 - Hom-functors Given a locally small category K, define op HomK : K × K ! Set a binary hom-functor, contravariant on the first argument and covariant on the second argument, as follows: op • HomK(hA; Bi) = K(A; B), for all hA; Bi 2 jK × Kj, i.e., A; B 2 jKj 0 0 0 0 • HomK(hf; gi): K(A; B) ! K(A ;B ), for hf; gi: hA; Bi ! hA ;B i in Kop × K, i.e., f : A0 ! A and g : B ! B0 in K, as a function given by Hom (hf; gi)(h) = f;h;g. f K A 0 A K(A; B) h K(A0;B0) ? g ? B -B0 Also: HomK(A; ): K ! Set op HomK(f; g) 6 HomK( ;B): K ! Set Andrzej Tarlecki: Category Theory, 2018 - 95 - Functors preserve. • Check whether functors preserve: − monomorphisms − epimorphisms − (co)retractions − isomorphisms − (co)cones − (co)limits − ... • A functor is (finitely) continuous if it preserves all existing (finite) limits. Which of the above functors are (finitely) continuous? Dualise! Andrzej Tarlecki: Category Theory, 2018 - 96 - Functors compose. Given two functors F: K ! K0 and G: K0 ! K00, their composition F;G: K ! K00 is defined as expected: • (F;G)(A) = G(F(A)) for all A 2 jKj • (F;G)(f) = G(F(f)) for all f : A ! B in K Cat, the category of (sm)all categories − objects: (sm)all categories − morphisms: functors between them − composition: as above Characterise isomorphisms in Cat Define products, terminal objects, equalisers and pullback in Cat Try to define their duals Andrzej Tarlecki: Category Theory, 2018 - 97 - Comma categories Given two functors with a common target, F: K1 ! K and G: K2 ! K, define their comma category (F; G) − objects: triples hA1; f : F(A1) ! G(A2);A2i, where A1 2 jK1j, A2 2 jK2j, and f : F(A1) ! G(A2) in K − morphisms: a morphism in (F; G) is any pair hh1; h2i: hA1; f : F(A1) ! G(A2);A2i ! hB1; g : F(B1) ! G(B2);B2i, where h1 : A1 ! B1 in K1, h2 : A2 ! B2 in K2, and F(h1);g = f;G(h2) in K. K1: K: K2: f - − composition: component-wise A1 F(A1) G(A2) A2 h1 F(h1) G(h2) h2 ? ? g ? ? B1 F(B1) - G(B2) B2 Andrzej Tarlecki: Category Theory, 2018 - 98 - Examples • The category of graphs as a comma category: Graph = (IdSet; CP) where CP: Set ! Set is the (Cartesian) product functor (CP(X) = X × X and CP(f)(hx; x0i) = hf(x); f(x0)i). Hint: write objects of this category as hE; hsource; targeti: E ! N × N;Ni • The category of algebraic signatures as a comma category: + AlgSig = (IdSet; ( ) ) where ( )+ : Set ! Set is the non-empty list functor ((X)+ is the set of all + non-empty lists of elements from X, (f) (hx1; : : : ; xni) = hf(x1); : : : ; f(xn)i). Hint: write objects of this category as hΩ; harity; sorti:Ω ! S+;Si Define K!, K#A as comma categories. The same for Alg(Σ). Andrzej Tarlecki: Category Theory, 2018 - 99 - Cocompleteness of comma categories Fact: If K1 and K2 are (finitely) cocomplete categories, F: K1 ! K is a (finitely) cocontinuous functor, and G: K2 ! K is a functor then the comma category (F; G) is (finitely) cocomplete. Proof (idea): Construct coproducts and coequalisers in (F; G), using the corresponding constructions in K1 and K2, and cocontinuity of F. State and prove the dual fact, concerning completeness of comma categories Andrzej Tarlecki: Category Theory, 2018 - 100 - Coproducts: f A1 F(A1) - G(A2) A2 ιA1 ¡ A ¡ A ιA2 F(ιA1 ) G(ιA2 ) ¡ ¡ AAU ¡ ¡ AAU A1 + B1 F(A1 + B1) - G(A2 + B2) A2 + B2 AK ¡ AK ¡ A F(ιB )¡ AG(ιB ) ¡ ιB1 1 2 ιB2 A ¡ g A ¡ B1 F(B1) - G(B2) B2 Coequalisers: f A1 F(A1) - G(A2) A2 0 0 0 0 h1 h1 F(h1) F(h1) G(h2) G(h2) h2 h2 ?? ?? g ?? ?? B1 F(B1) - G(B2) B2 c1 F(c1) G(c2) c2 ? ? ? ? C1 F(C1) - G(C2) C2 Andrzej Tarlecki: Category Theory, 2018 - 101 - Indexed categories An indexed category is a functor C : Indop ! Cat . Standard example: Alg: AlgSigop ! Cat The Grothendieck construction: Given C : Indop ! Cat, define a category Flat(C): − objects: hi; Ai for all i 2 jIndj, A 2 jC(i)j − morphisms: a morphism from hi; Ai to hj; Bi, hσ; fi: hi; Ai ! hj; Bi, consists of a morphism σ : i ! j in Ind and a morphism f : A !C(σ)(B) in C(i) − composition: given hσ; fi: hi; Ai ! hi0;A0i and hσ0; f 0i: hi0;A0i ! hi00;A00i, their composition in Flat(C), hσ; fi;hσ0; f 0i: hi; Ai ! hi00;A00i, is given by hσ; fi;hσ0; f 0i = hσ;σ0; f;C(σ)(f 0)i Fact: If Ind is complete, C(i) are complete for all i 2 jIndj, and C(σ) are continuous for all σ : i ! j in Ind, then Flat(C) is complete. Try to formulate and prove a theorem concerning cocompleteness of Flat(C) Andrzej Tarlecki: Category Theory, 2018 - 102 - Natural transformations Given two parallel functors F; G: K ! K0, a natural transformation from F to G τ : F ! G 0 is a family τ = hτA : F(A) ! G(A)iA2jKj of K -morphisms such that for all f : A ! B in K (with A; B 2 jKj), τA;G(f) = F(f);τB K: K0: τ A F(A) A - G(A) f F(f) G(f) Then, τ is a natural isomorphism if for ? ? ? τB all A 2 jKj, τA is an isomorphism. B F(B) - G(B) Andrzej Tarlecki: Category Theory, 2018 - 103 - Examples 0 • identity transformations: idF : F ! F, where F: K ! K , for all objects A 2 jKj, (idF)A = idA : F(A) ! F(A) • singleton functions: sing : IdSet ! P (: Set ! Set), where for all X 2 jSetj, singX : X ! P(X) is a function defined by singX (x) = fxg for x 2 X List • singleton-list functions: sing : IdSet ! jListj (: Set ! Set), where jListj = List;j j: Set(! Monoid) ! Set, and for all X 2 jSetj, List ∗ List singX : X ! X is a function defined by singX (x) = hxi for x 2 X • append functions: append : jListj;CP ! jListj (: Set ! Set), where for all ∗ ∗ ∗ X 2 jSetj, append X :(X × X ) ! X is the usual append function (list concatenation) polymorphic functions between algebraic types Andrzej Tarlecki: Category Theory, 2018 - 104 - Polymorphic functions Work out the following generalisation of the last two examples: − for each algebraic type scheme 8α1 : : : αn · T , built in Standard ML using at least products and algebraic data types (no function types though), define the corresponding functor [[T ]]: Setn ! Set − argue that in a representative subset of Standard ML, for each polymorphic 0 expression E : 8α1 : : : αn · T ! T its semantics is a natural transformation [[E]]: [[T ]] ! [[T 0]] Theorems for free! (see Wadler 89) Andrzej Tarlecki: Category Theory, 2018 - 105 - Yoneda lemma Given a locally small category K, functor F: K ! Set and object A 2 jKj: ∼ Nat(HomK(A; ); F) = F(A) natural transformations from HomK(A; ) to F, between functors from K to Set, are given exactly by the elements of the set F(A) EXERCISES: • Dualise: for G: Kop ! Set, ∼ Nat(HomK( ;A); G) = G(A) .
Load more

Recommended publications
  • 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]
  • 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]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [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]
  • 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]
  • Sheaves with Values in a Category7
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology Vol. 3, pp. l-18, Pergamon Press. 196s. F’rinted in Great Britain SHEAVES WITH VALUES IN A CATEGORY7 JOHN W. GRAY (Received 3 September 1963) $4. IN’IXODUCTION LET T be a topology (i.e., a collection of open sets) on a set X. Consider T as a category in which the objects are the open sets and such that there is a unique ‘restriction’ morphism U -+ V if and only if V c U. For any category A, the functor category F(T, A) (i.e., objects are covariant functors from T to A and morphisms are natural transformations) is called the category ofpresheaces on X (really, on T) with values in A. A presheaf F is called a sheaf if for every open set UE T and for every strong open covering {U,) of U(strong means {U,} is closed under finite, non-empty intersections) we have F(U) = Llim F( U,) (Urn means ‘generalized’ inverse limit. See $4.) The category S(T, A) of sheares on X with values in A is the full subcategory (i.e., morphisms the same as in F(T, A)) determined by the sheaves. In this paper we shall demonstrate several properties of S(T, A): (i) S(T, A) is left closed in F(T, A). (Theorem l(i), $8). This says that if a left limit (generalized inverse limit) of sheaves exists as a presheaf then that presheaf is a sheaf.
    [Show full text]
  • Abelian Categories
    Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists.
    [Show full text]
  • Notes on Categorical Logic
    Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour.
    [Show full text]
  • Homological Algebra in Characteristic One Arxiv:1703.02325V1
    Homological algebra in characteristic one Alain Connes, Caterina Consani∗ Abstract This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B := f0, 1g is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on Bmod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category Bmods is simply Bmod in the topos of sets endowed with an involution and as such it shares with Bmod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category Bmods is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between Bmods and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of Bmods the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
    [Show full text]
  • Categories, Functors, Natural Transformations
    CHAPTERCHAPTER 1 1 Categories, Functors, Natural Transformations Frequently in modern mathematics there occur phenomena of “naturality”. Samuel Eilenberg and Saunders Mac Lane, “Natural isomorphisms in group theory” [EM42b] A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G E as a normal subgroup and a surjective homomorphism → E H that displays H as the quotient group E/G. This data is typically displayed in a diagram of group homomorphisms: 0 G E H 0.1 → → → → A pair of group extensions E and E of G and H are considered to be equivalent whenever there is an isomorphism E E that commutes with the inclusions of G and quotient maps to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian group extensions E of H by G defines an abelian group Ext(H, G). In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which 1 he computed for a prime p that Ext(Z[ p ]/Z, Z) Zp, the group of p-adic integers, where 1 Z[ p ]/Z is the Prüfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times inside the preceding torus. In teasing apart this connection, the pair of them discovered what is now known as the universal coefficient theorem in algebraic topology, which relates the homology H and cohomology groups H∗ ∗ associated to a space X via a group extension [ML05]: n (1.0.1) 0 Ext(Hn 1(X), G) H (X, G) Hom(Hn(X), G) 0 .
    [Show full text]
  • Math 395: Category Theory Northwestern University, Lecture Notes
    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
    [Show full text]