DOCSLIB.ORG

Appendix. Foundations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix. Foundations Appendix. Foundations We have described a category in terms of sets, as a set of objects and a set of arrows. However, categories can be described directly-and they can then be used as a possible foundation for all of mathematics, thus replacing the use in such a foundation of the usual Zermelo-Fraenkel axioms for set theory. Here is the direct description: 1. Objects and arrows. A category consists of objects a, b, c, ... and arrows f, g, h. Sets form a category with sets as the objects and functions as the arrows. 2. Domain. Each arrow f has an object a as its "domain" and an object b as its "codomain"; we then write f : a --+ b. 3. Composition. Given f : a --+ band g : b --+ c, their composite is an arrow g 0 f : a --+ c. 4. Associativity. If also h: c --+ d, then the triple composition is associative: h 0 (g 0 f) = (h 0 g) 0 f : a --+ d . 5. Identities. Each object b has an identity arrow Ib: b --+ b. If also f : a --+ b, then Ib 0 f = f. If also g : b --+ c, then go Ib = g. An elementary topos is a category with a certain additional structure: terminal object, pullbacks, truth, a subobject classifier, and power objects (sets of subsets). The axioms for this addtional structure are as follows: 6. Terminal object. There is a terminal object 1 such that every ob­ ject a has exactly one arrow a --+ 1. 7. Pullbacks. Every pair of arrows f: a --+ b +- c : g with a common codomain b has a pullback, defined as in § 111.4: (1) a ----+ b. f In particular, (take b = 1), any two objects a and c have a product a x c. 8. Truth. There is an object Q (the object of truth values) and a 289 290 Appendix. Foundations monomorphism t: 1 -+ Q called truth; to any monomorphism m : a -+ b, there is a unique arrow ljJ : b -+ Q such that the fol­ lowing square is a pullback: a -----+ 1 mi It (2) b -----+ Q. '" 9. Power objects. To each object b, there is an associated object P b and an arrow eb : b x P b -+ Q such that for every arrow f : b x a -+ Q there is a unique arrow g : a -+ P b for which the following diagram commutes: f bxa -----+ Q (3) lxgI II b x Pb -----+ Q. tb To understand these axioms, we observe how they apply to the usual category of all sets. There, any set with just one element can serve as a terminal object 1, because each set a has a unique function a -+ 1 to 1. For two sets a and b, the pullback of two arrows a -+ 1 +-- b is then the usual set-theoretic product, with its projections to the given factors a and b. For truth values, take the object Q to be any set 2 consisting of two objects, 1 and 0, while the monomorphism t : 1 -+ 2 is just the usual in­ clusion of 1 in 2. Then a monomorphism m : a -+ b, as in Axiom 8, is a subset a of b. This subset has a well-known characteristic function ljJ : b -+ 2 with ljJ(y) = 1 or 0 according as the element of y of b is or is not in the subset a. This produces the pullback (2) above. Axiom 9 describes P b, the set of all subsets s of b, often called the "power set" Pb. Indeed, one can then set eb(x, s) = 0 if the element x of b is in the subset s and equal to 1 otherwise. This does give a pullback, as in (3) above. These axioms for a topos then hold for the category of sets. They have a number of strong consequences. For example, they give all finite cate­ gorical products and pullbacks, as well as all finite coproducts, including an initial object 0, the empty set. For example, they provide a right adjoint to the product a x b as a functor of a; this is the exponential ch with (see § IV.6) hom(a x b, c) ~ (a, cb ) . A category of sets can now be described as an elementary topos, defined as above, with three additional properties: Appendix. Foundations 291 (a) it is well-pointed, (b) it has the axiom of choice (AC), (c) it has a natural-numbers object (NNO). In describing these properties, it is useful to think of the objects as sets and the arrows as functions. Well-pointed requires that if two arrows f, g : a -+ b have (f"# g) then there must exist an arrow p : 1 -+ a for whichfp "# gpo The intention is that when the functions f and g differ they must differ at some "point" p, that is, at some element p of the set a. The axiom of choice (AC) requires that every subjectionf : a -+ b has a right inverse 'l" : b -+ a for which f 0 'l" = b. This right inverse picks out to each point p : 1 -+ b of b a point of a, to wit, the composite 'l"P which is mapped by f onto p. The natural-numbers object (NNO) can be described as a set N with an initial object 0 : 1 -+ N and a successor function s : N -+ N in terms of which functions f : N -+ X on N can be defined by recursion, by specifying f(O) and the composite fos. In other words, an NNO N in a category is a diagram 1 !!..,N':"'N consisting of a point 0 of N and a map s such that, given any arrows 1 !!... b .!:.. b, there is a unique arrow f : N -+ b which makes the following diagram commute: 1~ II 1~ In the usual functional notation, this states that fO=h, fs=kf; that is,fis defined by givingf( 0) and thenf(n + 1) in terms off(n). Thus, the category of sets may be described as a well-pointed topos with the AC in which there is an NNO. This set of axioms for set theory is weakly consistent with a version of the Zermelo axioms (the so-called "bounded" Zermelo; see Mac Lane and Moerdijk [1992]). They are originally due to Lawvere [1964], who called them the "elementary theory of the category of sets." Among the other examples of an elementary topos are the sheaves on a topological space; see Mac Lane and Moerdijk [1992]. In these axioms, it is often assumed that the subject classifier Q has just two elements. This makes it a Boolean algebra. Table of Standard Categories: Objects and Arrows Ab Abelian groups Adj Small categories, adjunctions, p. 104 Alg <, E)-algebras Bool Boolean algebras CAb Compact topological abelian groups CAT Categories, functors CG Haus Compactly generated Hausdorff spaces p. 185 Comp Bool Complete Boolean algebras Comp Haus Compact Hausdorff spaces, p. 125 CRng Commutatative rings, homomorphisms Ensv Sets and functions, within a universe V, p. 11 Euclid Euclidean vector spaces, orthogonal transformations Fin Skeletal category of finite sets Finord Finite ordinals, all set functions, p. 12 Grp Groups and homomorphisms Grpb Directed graphs and morphisms Haus Hausdorff Spaces, continuous maps Lconn Locally connected topological spaces K-Mod K-Modules and their morphisms Mod-R Right R-modules, R a ring R-Mod Left R-modules and morphisms MatrK Natural numbers, morphisms rectangular matrices, p. 11 Mon Monoids and morphisms of monoids, p. 12 Moncat Monoidal categories and strict morphisms, p. 160 Ord Ordered sets, order-preserving maps, p. 123 Rng Rings and homomorphisms Ses-A Short exact sequences of A -modules Set All small sets and functions Set. Sets with base point Smgrp Semigroups and morphisms Top Topological spaces, continuous maps, p. 122 Topb Topological spaces, homotopy classes of maps, p. 12 Vet Vector spaces, linear transformations o Empty category, p. 10 1 One-object category, p. 10 2 Two objects, p. 10 3 Three objects, p. 11 <Q,E) Universal algebras, type r, p. 120, 152 b 1 c Objects of C under b, p. 46 c 1 a Objects of C over a, p. 46 T1S Comma category, p. 46 293 Table of Terminology This Book Elsewhere (for abbreviations, see below) arrow map (E & M), morphism (Gr) domain source (Ehr) codomain target (Ehr) graph precategory, diagram scheme (Mit) natural transformation morphism of functors (Gr), functorial map (G-Z) natural isomorphism natural equivalence (E & M; now obsolete) mOlllc monomorphism epi epimorphism, epic idempotent projector (Gr) opposite dual coproduct sum equalizer kernel, difference kernel pullback fibered product (Gr), cartesian square pushout cocartesian square, comeet universal arrow left liberty map (G-Z) limit exists limit is representable (Gr) limit projective limit, inverse limit colimit inductive limit, direct limit cone to a functor projective cone, inverse cone (G-Z) cone from a functor inductive cone, co-cone left adjoint coadjoint (Mit), adjoint right adjoint adjoint (Mit), coadjoint unit of adjunction adjunction morphism (G-Z) triangular identities 8 quasi-inverse to IJ (G-Z) monad triple biproduct direct sum (in Ab-categories) Ab-category preadditive category (old) Gr = Grothendieck Ehr = Ehresmann Mit = Mitchell E & M = Eilenberg & Mac Lane G-Z = Gabriel-Zisman 295 Bibliography Adamek, J., Herrlich, H., Strecker, G.B. [1990]: Abstract and concrete cate­ gories. New York: John Wiley & Sons 1990. Andre, Michel [1967]: Methode simpliciale en algebre homologique et algebre commutative IV. Lecture Notes in Mathematics, Vol. 32. Berlin-Heidelberg­ New York: Springer 1967. - [1970]: Homology of simplicial objects. Proceeding of the AMS symposium in Pure Mathematics on Applications of Categorical Algebra, pp. 15-36.
Load more

Recommended publications
  • A Very Short Note on Homotopy Λ-Calculus
    A very short note on homotopy λ-calculus Vladimir Voevodsky September 27, 2006, October 10, 2009 The homotopy λ-calculus is a hypothetical (at the moment) type system. To some extent one may say that Hλ is an attempt to bridge the gap between the "classical" type systems such as the ones of PVS or HOL Light and polymorphic type systems such as the one of Coq. The main problem with the polymorphic type systems lies in the properties of the equality types. As soon as we have a universe U of which P rop is a member we are in trouble. In the Boolean case, P rop has an automorphism of order 2 (the negation) and it is clear that this automorphism should correspond to a member of Eq(U; P rop; P rop). However, as far as I understand there is no way to produce such a member in, say, Coq. A related problem looks as follows. Suppose T;T 0 : U are two type expressions and there exists an isomorphism T ! T 0 (the later notion of course requires the notion of equality for members of T and T 0). Clearly, any proposition which is true for T should be true for T 0 i.e. for all functions P : U ! P rop one should have P (T ) = P (T 0). Again as far as I understand this can not be proved in Coq no matter what notion of equality for members of T and T 0 we use. Here is the general picture as I understand it at the moment.
    [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]
  • Topological Properties of the Real Numbers Object in a Topos Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 17, No 3 (1976), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES LAWRENCE NEFF STOUT Topological properties of the real numbers object in a topos Cahiers de topologie et géométrie différentielle catégoriques, tome 17, no 3 (1976), p. 295-326 <http://www.numdam.org/item?id=CTGDC_1976__17_3_295_0> © Andrée C. Ehresmann et les auteurs, 1976, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE Vol. XVIl-3 (1976) ET GEOMETRIE DIFFERENTIELLE TOPOLOGICAL PROPERTIES OF THE REAL NUMBERS OBJECT IN A TOPOS * by Lawrence Neff STOUT In his presentation at the categories Session at Oberwolfach in 1973, Tierney defined the continuous reals for a topos with a natural numbers ob- ject (he called them Dedekind reals). Mulvey studied the algebraic proper- ties of the object of continuous reals and proved that the construction gave the sheaf of germs of continuous functions from X to R in the spatial topos Sh(X). This paper presents the results of the study of the topological prop- erties of the continuous reals with an emphasis on similarities with classi- cal mathematics and applications to familiar concepts rephrased in topos terms. The notations used for the constructions in the internal logic of a topos conform to that of Osius [11].
    [Show full text]
  • The Petit Topos of Globular Sets
    Journal of Pure and Applied Algebra 154 (2000) 299–315 www.elsevier.com/locate/jpaa View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector The petit topos of globular sets Ross Street ∗ Macquarie University, N. S. W. 2109, Australia Communicated by M. Tierney Dedicated to Bill Lawvere Abstract There are now several deÿnitions of weak !-category [1,2,5,19]. What is pleasing is that they are not achieved by ad hoc combinatorics. In particular, the theory of higher operads which underlies Michael Batanin’s deÿnition is based on globular sets. The purpose of this paper is to show that many of the concepts of [2] (also see [17]) arise in the natural development of category theory internal to the petit 1 topos Glob of globular sets. For example, higher spans turn out to be internal sets, and, in a sense, trees turn out to be internal natural numbers. c 2000 Elsevier Science B.V. All rights reserved. MSC: 18D05 1. Globular objects and !-categories A globular set is an inÿnite-dimensional graph. To formalize this, let G denote the category whose objects are natural numbers and whose only non-identity arrows are m;m : m → n for all m¡n ∗ Tel.: +61-2-9850-8921; fax: 61-2-9850-8114. E-mail address: [email protected] (R. Street). 1 The distinction between toposes that are “space like” (or petit) and those which are “category-of-space like” (or gros) was investigated by Lawvere [9,10]. The gros topos of re exive globular sets has been studied extensively by Michael Roy [12].
    [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]
  • The Sierpinski Object in the Scott Realizability Topos
    Logical Methods in Computer Science Volume 16, Issue 3, 2020, pp. 12:1–12:16 Submitted May 01, 2019 https://lmcs.episciences.org/ Published Aug. 20, 2020 THE SIERPINSKI OBJECT IN THE SCOTT REALIZABILITY TOPOS TOM DE JONG AND JAAP VAN OOSTEN School of Computer Science, University of Birmingham e-mail address: [email protected] Department of Mathematics, Utrecht University e-mail address: [email protected] Abstract. We study the Sierpinski object Σ in the realizability topos based on Scott's graph model of the λ-calculus. Our starting observation is that the object of realizers in this topos is the exponential ΣN , where N is the natural numbers object. We define order-discrete objects by orthogonality to Σ. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider Σ as a dominance; we explicitly construct the lift functor and characterize Σ-subobjects. Contrary to our expectations the dominance Σ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths. 1. Introduction In this paper, we aim to revive interest in what we call the Sierpinski object in the Scott realizability topos. We show that it is of fundamental importance in studying the subcategory of order-discrete objects (section 3), arithmetic in the topos (section 4), the Sierpinski object as a dominance (section 5) and a notion of homotopy based on it (section 6).
    [Show full text]
  • Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension
    LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K.
    [Show full text]
  • Logic and Categories As Tools for Building Theories
    Logic and Categories As Tools For Building Theories Samson Abramsky Oxford University Computing Laboratory 1 Introduction My aim in this short article is to provide an impression of some of the ideas emerging at the interface of logic and computer science, in a form which I hope will be accessible to philosophers. Why is this even a good idea? Because there has been a huge interaction of logic and computer science over the past half-century which has not only played an important r^olein shaping Computer Science, but has also greatly broadened the scope and enriched the content of logic itself.1 This huge effect of Computer Science on Logic over the past five decades has several aspects: new ways of using logic, new attitudes to logic, new questions and methods. These lead to new perspectives on the question: What logic is | and should be! Our main concern is with method and attitude rather than matter; nevertheless, we shall base the general points we wish to make on a case study: Category theory. Many other examples could have been used to illustrate our theme, but this will serve to illustrate some of the points we wish to make. 2 Category Theory Category theory is a vast subject. It has enormous potential for any serious version of `formal philosophy' | and yet this has hardly been realized. We shall begin with introduction to some basic elements of category theory, focussing on the fascinating conceptual issues which arise even at the most elementary level of the subject, and then discuss some its consequences and philosophical ramifications.
    [Show full text]
  • Arxiv:0906.4931V2 [Math.CT] 6 Mar 2010
    POLYNOMIAL FUNCTORS AND POLYNOMIAL MONADS NICOLA GAMBINO AND JOACHIM KOCK Abstract. We study polynomial functors over locally cartesian closed cat- egories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored. Introduction Background. Notions of polynomial functor have proved useful in many areas of mathematics, ranging from algebra [41, 34] and topology [10, 50] to mathe- matical logic [17, 45] and theoretical computer science [24, 2, 20]. The present paper deals with the notion of polynomial functor over locally cartesian closed categories. Before outlining our results, let us briefly motivate this level of abstraction. Among the devices used to organise and manipulate numbers, polynomials are ubiquitous. While formally a polynomial is a sequence of coefficients, it can be viewed also as a function, and the fact that many operations on polynomial functions, including composition, can be performed in terms of the coefficients alone is a crucial feature. The idea of polynomial functor is to lift the machinery of polynomials and polynomial functions to the categorical level. An obvious notion results from letting the category of finite sets take the place of the semiring of natural numbers, and defining polynomial functors to be functors obtained by finite combinations of disjoint union and cartesian product. It is interesting and fruitful to allow infinite sets. One reason is the interplay between inductively defined sets and polynomial functors. For example, the set of natural numbers can be characterised as the least solution to the polynomial equation of sets X =∼ 1+ X , while the set of finite planar trees appears as least solution to the equation X =∼ 1+ Xn.
    [Show full text]
  • Absolute Algebra and Segal's Γ-Rings
    Absolute algebra and Segal’s Γ-rings au dessous de Spec (Z) ∗ y Alain CONNES and Caterina CONSANI Abstract We show that the basic categorical concept of an S-algebra as derived from the theory of Segal’s Γ-sets provides a unifying description of several construc- tions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoïds, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an S-algebra to a monad on pointed sets. The notion of an S-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homol- ogy. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories. Contents 1 Introduction 2 2 S-algebras and Segal’s Γ-sets 3 2.1 Segal’s Γ-sets . .4 2.2 A basic construction of Γ-sets . .4 2.3 S-algebras . .5 3 Basic constructions of S-algebras 6 3.1 S-algebras and monoïds . .6 3.2 From semirings to -algebras . .7 arXiv:1502.05585v2 [math.AG] 13 Dec 2015 S 4 Smash products 9 4.1 The S-algebra HB ......................................9 4.2 The set (HB ^ HB)(k+) ...................................9 4.3 k-relations . 10 ∗Collège de France, 3 rue d’Ulm, Paris F-75005 France. I.H.E.S.
    [Show full text]
  • Combinatorial Categorical Equivalences of Dold-Kan Type 3
    COMBINATORIAL CATEGORICAL EQUIVALENCES OF DOLD-KAN TYPE STEPHEN LACK AND ROSS STREET Abstract. We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X ; (b) the observation of Church, Ellenberg and Farb [9] that X -valued species are equivalent to X -valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of “Dold-Kan type”; and so on. When X is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourn. Contents 1. Introduction 1 2. The setting 4 3. Basic examples 6 4. The kernel module 8 5. Reduction of the problem 11 6. The case P = K 12 7. Examples of Theorem 6.7 14 8. When X is semiabelian 16 Appendix A. A general result from enriched category theory 19 Appendix B. Remarks on idempotents 21 References 22 2010 Mathematics Subject Classification: 18E05; 20C30; 18A32; 18A25; 18G35; 18G30 arXiv:1402.7151v5 [math.CT] 29 Mar 2019 Key words and phrases: additive category; Dold-Kan theorem; partial map; semi-abelian category; comonadic; Joyal species. 1. Introduction The intention of this paper is to prove a class of equivalences of categories that seem of interest in enumerative combinatorics as per [22], representation theory as per [9], and homotopy theory as per [2].
    [Show full text]
  • Cartesian Closed Categories for the Logic of Proofs
    City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2009 TR-2009002: Cartesian Closed Categories for the Logic of Proofs Florian Lengyel How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/323 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Cartesian Closed Categories for the Logic of Proofs Florian Lengyel, CUNY Graduate Center April 25, 2009 Abstract A generalization of the Curry-Howard-Lambek isomorphism for carte- sian closed categories and typed lambda calculi is given for the LP cate- gories with weak natural numbers object, which correspond to the positive conjunction fragment of the intuitionistic Logic of Proofs LP of Artemov, and LP-typed lambda calculi with natural numbers type. 1 Introduction The Logic of Proofs LP of Artemov is a Hilbert-style logical system extending classical and intuitionistic propositional logic, with additional propositions of the form (t : A), read as \term t is justification for A" [Art01a, Art01b]. LP and the related broader class of justification logics are, in a precise sense, refine- ments of epistemic and modal logics such as K, K4, K45, KD45, T, S4 and S5 [Art08]. Semantics for these systems include the Kripke{Fitting models, the Mkrtychev models, and arithmetical provability semantics [Fit05, Mkr97, Art01a]. In addition to its applications in epistemic logic, modal logic and proof theory, LP has been proposed as a logic for certified mobile computation [BF09].
    [Show full text]