DOCSLIB.ORG

Nonsymmetric Operads in Combinatorics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nonsymmetric Operads in Combinatorics Nonsymmetric operads in combinatorics Samuele Giraudo Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEM, F-77454, Marne-la-Vallée, France Email address [email protected] : arXiv:2104.12398v1 [math.CO] 26 Apr 2021 Mathematics Subject Classification. 2010 05-00, 05C05, 05E15, 16T05, 18D50. Key words and phrases. binatorics; Algebraic combinatorics; Computer science; Com Tree; Formal power series; Rewrite system; Operad; Bialgebra; Hopf bialgebra; Pre-Lie algebra; Dendriform algebra. Operads are algebraic devices offering a formalization of the Abstract. concept of operations with several inputs and one output. Such opera- tions can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompany- ing it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide con- nections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their proper- ties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this the- ory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combina- torics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmet- ric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics. April 27, 2021 Contents Introduction 1 Chapter 1. Enriched collections 9 1. Collections 9 1.1. Structured collections 9 1.2. Operations over collections 12 1.3. Examples 19 2. Posets 23 2.1. Posets on collections 23 2.2. Examples 24 3. Rewrite systems 25 3.1. Rewrite systems on collections 26 3.2. Examples 28 Chapter 2. Treelike structures 33 1. Planar rooted trees 33 1.1. Collection of planar rooted trees 33 1.2. Subcollections of planar rooted trees 37 2. Syntax trees 40 2.1. Collections of syntax trees 40 2.2. Grafting operations 42 2.3. Patterns and rewrite systems 44 3. Treelike structures 48 3.1. Rooted trees 49 3.2. Colored syntax trees 49 Chapter 3. Algebraic structures 53 1. Polynomials spaces 53 1.1. Polynomials on collections 53 1.2. Operations over polynomial spaces 56 iii iv CONTENTS 1.3. Changes of basis and posets 59 2. Bialgebras 61 2.1. Biproducts on polynomial spaces 61 2.2. Products on polynomial spaces 63 2.3. Polynomial bialgebras 66 3. Types of polynomial bialgebras 69 3.1. Associative and coassociative algebras 69 3.2. Dendriform algebras 71 3.3. Pre-Lie algebras 72 3.4. Hopf bialgebras 73 Chapter 4. Nonsymmetric operads 81 1. Operads as polynomial spaces 81 1.1. Composition maps 81 1.2. Operads 87 1.3. Algebras over operads 89 2. Free operads, presentations, and Koszulity 92 2.1. Free operads 92 2.2. Presentations by generators and relations 93 2.3. Koszulity 95 3. Main operads 97 3.1. Operads of words 97 3.2. Operads of trees 102 3.3. Operads of graphs 107 Chapter 5. Applications and generalizations 117 1. Series on operads 117 1.1. Series on algebraic structures 117 1.2. Generalizing series multiplication 119 1.3. Generalizing series composition 121 2. Enriched operads 123 2.1. Colored operads 123 2.2. Cyclic operads 126 2.3. Symmetric operads 128 3. Product categories 132 3.1. Abstract bioperators 132 CONTENTS v 3.2. Pros 134 3.3. Main pros 136 Bibliography 141 Introduction Combinatorics, algebra, and programming Let us start this text by considering two fields of mathematics: combinatorics and algebra, and a field of computer science: programming. These topics have far more in common than it appears at first glance and we shall explain why. Combinatorics. One of the most common activities in combinatorics consists in studying families of combinatorial objects such as permutations, words, trees, or graphs. One of the most common activities about such families is to provide formulas to enu- merate them according to a decent size notion. A large number of tools exists for this purpose such as manipulation of generating series, use of bijections, and use of decom- positions rules to break objects into elementary pieces. This last tool is crucial and relies on the fact that if one knows a way to compose objects to build a bigger one, then with- out much effort one knows a way to decompose them. Let us, for instance, consider the celebrated set of Motzkin paths. Recall that a Motzkin path is a walk in the quarter plane connecting the origin to a point on the x-axis by means of steps , , and . Let us denote by u ; u the binary composition rule consisting in substituting u (resp. u ) 1 2) 1 2 with the first( (resp. second) point of , and by u ; u ; u the ternary composition 1 2 3 rule consisting in substituting u (resp. u , u ) with the( first (resp.) second, third) point of 1 2 3 . By using these two operations, one can, for instance, provide the decomposition ; ; ; ; ; (0.0.1) = ( ) of the Motzkin path appearing as left member. An easy result says that a nonempty Motzkin path decomposes in a unique way either as a step followed by a Motzkin path, or as a step followed by a Motzkin path, a step , and another Motzkin path. This leads to the expression M { } ⊔ { ; u u ∈ M} ⊔ { ; u ; u u ; u ∈ M} (0.0.2) = ( 2) : 2 ( 2 3) : 2 3 describing the set M of all Motzkin paths. From (0.0.2), one harvests combinatorial data as a formula to count Motzkin paths of a given size (with respect to an adequate notion of size), bijections between Motzkin paths and other families of combinatorial objects admitting similar decompositions (like Motzkin trees or Motzkin configurations 1 2 INTRODUCTION of chords in polygons), or even the definition of generalizations of these objects (by adding colors on the steps, or by adding other possible sorts of steps). Algebra. On another side, in algebra, the study of algebraic structures consisting in a set of operations and the relations they satisfy is very habitual. To specify such an algebraic structure, one provides the nontrivial relations satisfied by its operations. For instance, any binary operation ⋆ satisfying for any inputs x , x , and x the relation 1 2 3 x ⋆x ⋆x − x ⋆ x ⋆x x ⋆x ⋆x − x ⋆ x ⋆x (0.0.3) ( 1 2) 3 1 ( 2 3) = ( 1 3) 2 1 ( 3 2) specifies the class of the so-called pre-Lie algebras. Classical examples include among others monoids, groups, lattices, associative algebras, commutative algebras, and dendri- form algebras. An obvious but important fact is that expressions and relations can be composed: the variables in a relation refer to any term of the algebraic structure. For instance, in (0.0.3), all the occurrences of x can be replaced by a term x4 ⋆ x5, leading to 2 the relation x ⋆ x4 ⋆ x5 ⋆x − x ⋆ x4 ⋆ x5 ⋆x x ⋆x ⋆ x4 ⋆ x5 − x ⋆ x ⋆ x4 ⋆ x5 (0.0.4) ( 1 ( )) 3 1 (( ) 3) = ( 1 3) ( ) 1 ( 3 ( )) that still holds. Furthermore, by seeing the binary operation ⋆ as a device ⋆ (0.0.5) x x 1 2 having two inputs x and x and one output x ⋆x , Relation (0.0.3) reformulates as 1 2 1 2 ⋆ ⋆ ⋆ ⋆ − − ; (0.0.6) ⋆ x x ⋆ ⋆ x x ⋆ 3 1 = 2 1 x x x x x x x x 1 2 2 3 1 3 3 2 and Relation (0.0.4) as ⋆ ⋆ ⋆ ⋆ ⋆ x x ⋆ x ⋆ 3 − 1 ⋆ ⋆ − 1 : (0.0.7) x ⋆ ⋆ x = x ⋆ 1 3 3 x x x x x x x4 x5 x x 4 5 4 5 1 3 4 5 Therefore, one observes that algebraic expressions translates as syntax trees, and that composition and substitution of expressions translates as grafting of trees. Programming. In programming, and more precisely in the context of functional programming, one encounters expressions of the form v1; v2 ; g2 v3 ; v4 (0.0.8) f(g1( ) ( ) ) where f, g1, and g2 are function names, and v1, v2, v3, and v4 are value names. This func- tion call can be considered as a program, and the returned value is the value computed by the execution of the program. When our programming language satisfies referential NONSYMMETRIC OPERADS AS A MEETING POINT 3 transparency (that is, any expression can be substituted by its value without changing the overall computation), the call (0.0.8) is equivalent to both the calls f w1; g2 v3 ; v4 and f g1 v1; v2 ; w2; v4 (0.0.9) ( ( ) ) ( ( ) ) where w1 is the value of g1 v1; v2 and w2 is the value of g2 v3 . As a consequence, this ( ) ( ) leads to the fact that the order of evaluation of the sub-expressions g1 v1; v2 and g2 v3 does not influence the computation of (0.0.8). Furthermore, by seeing( the functions) ( f1,) g1, and g2 as black boxes ; ; (0.0.10) f g1 g2 where inputs are depicted below and outputs above the boxes, the program (0.0.8) is made of compositions of such black boxes where inputs are completed with values (that are arguments of the function calls).
Load more

Recommended publications
  • BIPRODUCTS WITHOUT POINTEDNESS 1. Introduction
    BIPRODUCTS WITHOUT POINTEDNESS MARTTI KARVONEN Abstract. We show how to define biproducts up to isomorphism in an ar- bitrary category without assuming any enrichment. The resulting notion co- incides with the usual definitions whenever all binary biproducts exist or the category is suitably enriched, resulting in a modest yet strict generalization otherwise. We also characterize when a category has all binary biproducts in terms of an ambidextrous adjunction. Finally, we give some new examples of biproducts that our definition recognizes. 1. Introduction Given two objects A and B living in some category C, their biproduct { according to a standard definition [4] { consists of an object A ⊕ B together with maps p i A A A ⊕ B B B iA pB such that pAiA = idA pBiB = idB pBiA = 0A;B pAiB = 0B;A and idA⊕B = iApA + iBpB. For us to be able to make sense of the equations, we must assume that C is enriched in commutative monoids. One can get a slightly more general definition that only requires zero morphisms but no addition { that is, enrichment in pointed sets { by replacing the last equation with the condition that (A ⊕ B; pA; pB) is a product of A and B and that (A ⊕ B; iA; iB) is their coproduct. We will call biproducts in the first sense additive biproducts and in the second sense pointed biproducts in order to contrast these definitions with our central object of study { a pointless generalization of biproducts that can be applied in any category C, with no assumptions concerning enrichment. This is achieved by replacing the equations referring to zero with the single equation (1.1) iApAiBpB = iBpBiApA, which states that the two canonical idempotents on A ⊕ B commute with one another.
    [Show full text]
  • The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras
    Algebras and Representation Theory 3: 19–42, 2000. 19 © 2000 Kluwer Academic Publishers. Printed in the Netherlands. The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras S. CAENEPEEL1, BOGDAN ION2, G. MILITARU3;? and SHENGLIN ZHU4;?? 1University of Brussels, VUB, Faculty of Applied Sciences, Pleinlaan 2, B-1050 Brussels, Belgium 2Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, U.S.A. 3University of Bucharest, Faculty of Mathematics, Str. Academiei 14, RO-70109 Bucharest 1, Romania 4Institute of Mathematics, Fudan University, Shanghai 200433, China (Received: July 1998) Presented by A. Verschoren Abstract. We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R: H ⊗ L ! L ⊗ H and W: L ⊗ H ! H ⊗ L. We introduce a product K D L W FG R H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dascˇ alescuˇ and Grünenfelder appear as special cases. Mathematics Subject Classification (2000): 16W30. Key words: Hopf algebra, smash product, factorization structure. Introduction The factorization problem for a ‘structure’ (group, algebra, coalgebra, bialgebra) can be roughly stated as follows: under which conditions can an object X be written as a product of two subobjects A and B which have minimal intersection (for example A \ B Df1Xg in the group case). A related problem is that of the construction of a new object (let us denote it by AB) out of the objects A and B.In the constructions of this type existing in the literature ([13, 20, 27]), the object AB factorizes into A and B.
    [Show full text]
  • Arxiv:1908.01212V3 [Math.CT] 3 Nov 2020 Step, One Needs to Apply a For-Loop to Divide a Matrix Into Blocks
    Typing Tensor Calculus in 2-Categories Fatimah Ahmadi Department of Computer Science University of Oxford November 4, 2020 Abstract We introduce semiadditive 2-categories, 2-categories with binary 2- biproducts and a zero object, as a suitable framework for typing tensor calculus. Tensors are the generalization of matrices, whose components have more than two indices. 1 Introduction Linear algebra is the primary mathematical toolbox for quantum physicists. Categorical quantum mechanics re-evaluates this toolbox by expressing each tool in the categorical language and leveraging the power of graphical calculus accompanying monoidal categories to gain an alternative insight into quantum features. In the categorical description of quantum mechanics, everything is typed in FHilb; the category whose objects are finite dimensional Hilbert spaces and morphisms are linear maps/matrices. In this category, Hilbert spaces associated with physical systems are typed as objects, and processes between systems as morphisms. MacLane [7] proposed the idea of typing matrices as morphisms while intro- ducing semiadditive categories(categories with well-behaved additions between objects). This line of research, further, has been explored by Macedo and Oliveira in the pursuit of avoiding the cumbersome indexed-based operations of matrices [6]. They address the quest for shifting the traditional perspective of formal methods in software development[10]. To observe why index-level operations do not offer an optimal approach, take the divide-and-conquer algorithm to multiplicate two matrices. In each arXiv:1908.01212v3 [math.CT] 3 Nov 2020 step, one needs to apply a for-loop to divide a matrix into blocks. While the division happens automatically if one takes matrices whose sources or targets are biproducts of objects.
    [Show full text]
  • UNIVERSITY of CALIFORNIA RIVERSIDE the Grothendieck
    UNIVERSITY OF CALIFORNIA RIVERSIDE The Grothendieck Construction in Categorical Network Theory A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Joseph Patrick Moeller December 2020 Dissertation Committee: Dr. John C. Baez, Chairperson Dr. Wee Liang Gan Dr. Carl Mautner Copyright by Joseph Patrick Moeller 2020 The Dissertation of Joseph Patrick Moeller is approved: Committee Chairperson University of California, Riverside Acknowledgments First of all, I owe all of my achievements to my wife, Paola. I couldn't have gotten here without my parents: Daniel, Andrea, Tonie, Maria, and Luis, or my siblings: Danielle, Anthony, Samantha, David, and Luis. I would like to thank my advisor, John Baez, for his support, dedication, and his unique and brilliant style of advising. I could not have become the researcher I am under another's instruction. I would also like to thank Christina Vasilakopoulou, whose kindness, energy, and expertise cultivated a deeper appreciation of category theory in me. My expe- rience was also greatly enriched by my academic siblings: Daniel Cicala, Kenny Courser, Brandon Coya, Jason Erbele, Jade Master, Franciscus Rebro, and Christian Williams, and by my cohort: Justin Davis, Ethan Kowalenko, Derek Lowenberg, Michel Manrique, and Michael Pierce. I would like to thank the UCR math department. Professors from whom I learned a ton of algebra, topology, and category theory include Julie Bergner, Vyjayanthi Chari, Wee-Liang Gan, Jos´eGonzalez, Jacob Greenstein, Carl Mautner, Reinhard Schultz, and Steffano Vidussi. Special thanks goes to the department chair Yat-Sun Poon, as well as Margarita Roman, Randy Morgan, and James Marberry, and many others who keep the whole thing together.
    [Show full text]
  • An Aop Approach to Typed Linear Algebra
    An AoP approach to typed linear algebra J.N. Oliveira (joint work with Hugo Macedo) Dept. Inform´atica, Universidade do Minho Braga, Portugal IFIP WG2.1 meeting #65 27th January 2010 Braga, Portugal Motivation Matrices = arrows Abelian category Abide laws Divide & conquer Vectorization References Context and Motivation • The advent of on-chip parallelism poses many challenges to current programming languages. • Traditional approaches (compiler + hand-coded optimization are giving place to trendy DSL-based generative techniques. • In areas such as scientific computing, image/video processing, the bulk of the work performed by so-called kernel functions. • Examples of kernels are matrix-matrix multiplication (MMM), the discrete Fourier transform (DFT), etc. • Kernel optimization is becoming very difficult due to the complexity of current computing platforms. Motivation Matrices = arrows Abelian category Abide laws Divide & conquer Vectorization References Teaching computers to write fast numerical code In the Spiral Group (CMU), a DSL has been defined (OL) (Franchetti et al., 2009) to specify kernels in a data-independent way. • Divide-and-conquer algorithms are described as OL breakdown rules. • By recursively applying these rules a space of algorithms for a desired kernel can be generated. Rationale behind Spiral: • Target imperative code is too late for numeric processing kernel optimization. • Such optimization can be elegantly and efficiently performed well above in the design chain once the maths themselves are expressed in an index-free style. Motivation Matrices = arrows Abelian category Abide laws Divide & conquer Vectorization References Starting point Synergy: • Parallel between the pointfree notation of OL and relational algebra (relations are Boolean matrices) • Rich calculus of algebraic rules.
    [Show full text]
  • Double Cross Biproduct and Bi-Cycle Bicrossproduct Lie Bialgebras
    Ashdin Publishing Journal of Generalized Lie Theory and Applications Vol. 4 (2010), Article ID S090602, 16 pages doi:10.4303/jglta/S090602 Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras Tao ZHANG College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan Province, China Email: [email protected] Abstract We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras. 2000 MSC: 17B62, 18D35 1 Introduction As an infinitesimal or semiclassical structures underlying the theory of quantum groups, the notion of Lie bialgebras was introduced by Drinfeld in his remarkable report [3], where he also introduced the double Lie bialgebra D(g) as an important construction. Some years later, the theory of matched pairs of Lie algebras (g; m) was introduced by Majid in [4]. Its bicrossed product (or double cross sum) m ./g is more general than Drinfeld's classical double D(g) because g and m need not have the same dimension and the actions need not be strictly coadjoint ones. Since then it was found that many other structures in Hopf algebras can be constructed in the infinitesimal setting, see [5] and the references cited therein. Also in [6], Majid in- troduced the concept of braided Lie bialgebras and proved the bosonisation theorem (see Theorem 4.4) associating braided Lie bialgebras to ordinary Lie bialgebras. Examples of braided Lie bialgebras were also given there. On the other hand, there is a close relation between extension theory and cross product Lie bialgebras, see Masuoka [7].
    [Show full text]
  • WHEN IS ∏ ISOMORPHIC to ⊕ Introduction Let C Be a Category
    WHEN IS Q ISOMORPHIC TO L MIODRAG CRISTIAN IOVANOV Abstract. For a category C we investigate the problem of when the coproduct L and the product functor Q from CI to C are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category C this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in C that is invertible with respect to addition). However we show that L and Q are always isomorphic on a suitable subcategory of CI which is isomorphic to CI but is not a full subcategory. If C is only a preadditive category then we give an example to see that the two functors can be isomorphic for infinite sets I. For the module category case we provide a different proof to display an interesting connection to the notion of Frobenius corings. Introduction Let C be a category and denote by ∆ the diagonal functor from C to CI taking any object I X to the family (X)i∈I ∈ C . Recall that C is an Ab category if for any two objects X, Y of C the set Hom(X, Y ) is endowed with an abelian group structure that is compatible with the composition. We shall say that a category is an AMon category if the set Hom(X, Y ) is (only) an abelian monoid for every objects X, Y . Following [McL], in an Ab category if the product of any two objects exists then the coproduct of any two objects exists and they are isomorphic.
    [Show full text]
  • Notes on Category Theory (In Progress)
    Notes on Category Theory (in progress) George Torres Last updated February 28, 2018 Contents 1 Introduction and Motivation 3 2 Definition of a Category 3 2.1 Examples . .4 3 Functors 4 3.1 Natural Transformations . .5 3.2 Adjoint Functors . .5 3.3 Units and Counits . .6 3.4 Initial and Terminal Objects . .7 3.4.1 Comma Categories . .7 4 Representability and Yoneda's Lemma 8 4.1 Representables . .9 4.2 The Yoneda Embedding . 10 4.3 The Yoneda Lemma . 10 4.4 Consequences of Yoneda . 11 5 Limits and Colimits 12 5.1 (Co)Products, (Co)Equalizers, Pullbacks and Pushouts . 13 5.2 Topological limits . 15 5.3 Existence of limits and colimits . 15 5.4 Limits as Representable Objects . 16 5.5 Limits as Adjoints . 16 5.6 Preserving Limits and GAFT . 18 6 Abelian Categories 19 6.1 Homology . 20 6.1.1 Biproducts . 21 6.2 Exact Functors . 23 6.3 Injective and Projective Objects . 26 6.3.1 Projective and Injective Modules . 27 6.4 The Chain Complex Category . 28 6.5 Homological dimension . 30 6.6 Derived Functors . 32 1 CONTENTS CONTENTS 7 Triangulated and Derived Categories 35 ||||||||||| Note to the reader: This is an ongoing collection of notes on introductory category theory that I have kept since my undergraduate years. They are aimed at students with an undergraduate level background in topology and algebra. These notes started as lecture notes for the Fall 2015 Category Theory tutorial led by Danny Shi at Harvard. There is no single textbook that these notes follow, but Categories for the Working Mathematician by Mac Lane and Lang's Algebra are good standard resources.
    [Show full text]
  • Almost Abelian Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 42, No 3 (2001), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES WOLFGANG RUMP Almost abelian categories Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001), p. 163-225 <http://www.numdam.org/item?id=CTGDC_2001__42_3_163_0> © Andrée C. Ehresmann et les auteurs, 2001, 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/ CAHIERSDE TtOPOLOGIE ET Volume XLII-3 (2001) GEOMEl’RIE DIFFERENTIEUE CATEGORIQUES ALMOST ABELIAN CATEGORIES By Wolfgang RUMP Dedicated to K. W. Roggenkamp on the occasion of his 6e birthday RESUME. Nous introduisons et 6tudions une classe de categories additives avec des noyaux et conoyaux, categories qui sont plus générales que les categories ab6liennes, et pour cette raison nous les appelons presque ab6liennes. L’un des objectifs de ce travail est de montrer que cette notion unifie et generalise des structures associ6es aux categories ab6liennes: des theories de torsion (§4), des foncteurs adjoints et des bimodules (§6), la dualite de Morita et la th6orie de "tilting" (§7). D’autre part, nous nous proposons de montrer qu’il y a beaucoup de categories presque ab6liennes: en algebre topologique (§2.2), en analyse fonctionnelle (§2.3-4), dans la th6orie des modules filtr6s (§2.5), et dans la th6orie des représentations des ordres sur les anneaux de Cohen-Macaulay de dimension inf6rieure ou 6gale a 2 (§2.1 et §2.9).
    [Show full text]
  • On Adjoint and Brain Functors
    On Adjoint and Brain Functors David Ellerman Philosophy Department University of California at Riverside Abstract There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts. The Mathematical Appendix is of general interest to category theorists as it is a defense of the use of heteromorphisms as a natural and necessary part of category theory. Contents 1 Category theory in the life and cognitive sciences 1 2 The ubiquity and importance of adjoints 2 3 Adjoints and universals 3 4 The Hom-set definition of an adjunction 4 5 Heteromorphisms and adjunctions 6 6 Brain functors 8 7 A mathematical example of a brain functor 12 8 Conclusion 13 9 Mathematical Appendix: Are hets really necessary in category theory? 13 9.1 Chimeras in the wilds of mathematical practice . 14 9.2 Hets as "homs" in a collage category . 15 9.3 What about the homs-only UMPs in adjunctions? . 16 9.4 Are all UMPs part of adjunctions? . 16 1 Category theory in the life and cognitive sciences There is already a considerable but widely varying literature on the application of category theory to the life and cognitive sciences—such as the work of Robert Rosen ([31], [32]) and his followers1 as 1 See [38], [20], and [21] and their references.
    [Show full text]
  • Completely Positive Projections and Biproducts
    Completely positive projections and biproducts Chris Heunen Aleks Kissinger Peter Selinger Department of Computer Science Department of Computer Science Department of Mathematics and Statistics University of Oxford University of Oxford Dalhousie University [email protected] [email protected] [email protected] The recently introduced CP*–construction unites quantum channels and classical systems, subsum- ing the earlier CPM–construction in categorical quantum mechanics. We compare this construction to two earlier attempts at solving this problem: freely adding biproducts to CPM, and freely split- ting idempotents in CPM. The CP*–construction embeds the former, and embeds into the latter, but neither embedding is an equivalence in general. 1 Introduction Two of the authors recently introduced the so-called CP*–construction, turning a category V of ab- stract state spaces into a category CP∗[V] of abstract C*-algebras and completely positive maps [4]. It accommodates both quantum channels and classical systems in a single category. Moreover, it allows nonstandard models connecting to the well-studied theory of groupoids. In particular, it subsumes the earlier CPM–construction, which gives the subcategory CPM[V] of the CP*–construction of abstract matrix algebras [8], and adds classical information to it. There have been earlier attempts at uniting quantum channels and classical systems [8, 9]. This paper compares the CP*–construction to two of them: freely adding biproducts to CPM[V], and freely splitting the dagger idempotents of CPM[V]. These new categories are referred to as CPM[V]⊕ and Split[CPM[V]], respectively. We will prove that the CP*–construction lies in between these two: there are full and faithful functors CPM[V]⊕ CP∗[V] Split[CPM[V]].
    [Show full text]
  • How Can We Understand an Abelian Category?
    HOW CAN WE UNDERSTAND AN ABELIAN CATEGORY? Jackson Ryder Supervisor: Associate Professor Daniel Chan School of Mathematics and Statistics UNSW Sydney November 2020 Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science with Honours Plagiarism statement I declare that this thesis is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere. I acknowledge that the assessor of this thesis may, for the purpose of assessing it: • Reproduce it and provide a copy to another member of the University; and/or, • Communicate a copy of it to a plagiarism checking service (which may then retain a copy of it on its database for the purpose of future plagiarism check- ing). I certify that I have read and understood the University Rules in respect of Student Academic Misconduct, and am aware of any potential plagiarism penalties which may apply. By signing this declaration I am agreeing to the statements and conditions above. Signed: Date: i Acknowledgements Firstly, I would like to thank my supervisor Daniel Chan for exposing me to such a wonderful area of mathematics, and for his constant support and guidance throught this challenging year. I would also like to thank my highschool maths teacher, Phil Baillie, for first introducing me to higher maths and taking the time to foster my interest in the sub- ject. Without him I would never have thought of pursuing a degree in mathematics to begin with. Last, but certainly not least, I thank my family, particularly my mother and brother, for their unconditional love and support.
    [Show full text]