MACKEY FUNCTORS on COMPACT CLOSED CATEGORIES Groups Are
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Skew Monoidal Categories and Grothendieck's Six Operations
Skew Monoidal Categories and Grothendieck's Six Operations by Benjamin James Fuller A thesis submitted for the degree of Doctor of Philosophy. Department of Pure Mathematics School of Mathematics and Statistics The University of Sheffield March 2017 ii Abstract In this thesis, we explore several topics in the theory of monoidal and skew monoidal categories. In Chapter 3, we give definitions of dual pairs in monoidal categories, skew monoidal categories, closed skew monoidal categories and closed mon- oidal categories. In the case of monoidal and closed monoidal categories, there are multiple well-known definitions of a dual pair. We generalise these definitions to skew monoidal and closed skew monoidal categories. In Chapter 4, we introduce semidirect products of skew monoidal cat- egories. Semidirect products of groups are a well-known and well-studied algebraic construction. Semidirect products of monoids can be defined anal- ogously. We provide a categorification of this construction, for semidirect products of skew monoidal categories. We then discuss semidirect products of monoidal, closed skew monoidal and closed monoidal categories, in each case providing sufficient conditions for the semidirect product of two skew monoidal categories with the given structure to inherit the structure itself. In Chapter 5, we prove a coherence theorem for monoidal adjunctions between closed monoidal categories, a fragment of Grothendieck's `six oper- ations' formalism. iii iv Acknowledgements First and foremost, I would like to thank my supervisor, Simon Willerton, without whose help and guidance this thesis would not have been possible. I would also like to thank the various members of J14b that have come and gone over my four years at Sheffield; the friendly office environment has helped keep me sane. -
Reasoning About Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras ∗
Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras ∗ Authors: Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, Bob Coecke y Affiliation: Department of Computer Science, University of Oxford Address: Wolfson Building, Parks Road, Oxford, OX1 3QD Emails: [email protected] Abstract Compact closed categories have found applications in modeling quantum information pro- tocols by Abramsky-Coecke. They also provide semantics for Lambek's pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work Coecke-Clark-Sadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and Baez-Dolan, and the latter are used to model classical data in quantum protocols by Coecke-Pavlovic- Vicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions. 1 Introduction Compact closed categories were first introduced by Kelly [21] in early 1970's. Some thirty years later they found applications in quantum mechanics [1], whereby the vector space foundations of quantum mechanics were recasted in a higher order language and quantum protocols such as teleportation found succinct conceptual proofs. -
Categories of Quantum and Classical Channels (Extended Abstract)
Categories of Quantum and Classical Channels (extended abstract) Bob Coecke∗ Chris Heunen† Aleks Kissinger∗ University of Oxford, Department of Computer Science fcoecke,heunen,[email protected] We introduce the CP*–construction on a dagger compact closed category as a generalisation of Selinger’s CPM–construction. While the latter takes a dagger compact closed category and forms its category of “abstract matrix algebras” and completely positive maps, the CP*–construction forms its category of “abstract C*-algebras” and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*–construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*–construction fully embeds Selinger’s CPM–construction in such a way that the objects in the image of the embedding can be thought of as “purely quantum” state spaces. It also embeds the category of classical stochastic maps, whose image consists of “purely classical” state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels. 1 Introduction One of the motivations driving categorical treatments of quantum mechanics is to place classical and quantum systems on an equal footing in a single category, so that one can study their interactions. The main idea of categorical quantum mechanics [1] is to fix a category (usually dagger compact) whose ob- jects are thought of as state spaces and whose morphisms are evolutions. There are two main variations. • “Dirac style”: Objects form pure state spaces, and isometric morphisms form pure state evolutions. -
Master Report Conservative Extensions of Montague Semantics M2 MPRI: April-August 2017
ENS Cachan,Universite´ Paris-Saclay Master report Conservative extensions of Montague semantics M2 MPRI: April-August 2017 Supervised by Philippe de Groote, Semagramme´ team, Loria,Inria Nancy Grand Est Mathieu Huot August 21, 2017 The general context Computational linguistics is the study of linguistic phenomena from the point of view of computer science. Many approaches have been proposed to tackle the problem. The approach of compositional semantics is that the meaning of a sentence is obtained via the meaning of its subconstituents, the basic constituents being mostly the words, with possibly some extra features which may be understood as side effects from the point of view of the computer scientist, such as interaction with a context, time dependency, etc. Montague [43, 44] has been one of the major figures to this approach, but what is today known as Montague semantics lacks a modular approach. It means that a model is usually made to deal with a particular problem such as covert movement [47], intentionalisation [12,8], or temporal modifiers [18], and those models do not necessarily interact well one with another. This problem in general is an important limitation of the computational models of linguistics that have been proposed so far. De Groote [20] has proposed a model called Abstract Categorial Grammars (ACG), which subsumes many known frameworks [20, 21, 13] such as Tree Adjoining Grammars [24], Rational Transducers and Regular Grammars [20], taking its roots in the simply typed lambda calculus [6]. The research problem The problem we are interested in is finding a more unifying model, yet close to the intuition and the previous works, which would allow more modularity while remaining in the paradigm of compositional semantics. -
Locally Cartesian Closed Categories, Coalgebras, and Containers
U.U.D.M. Project Report 2013:5 Locally cartesian closed categories, coalgebras, and containers Tilo Wiklund Examensarbete i matematik, 15 hp Handledare: Erik Palmgren, Stockholms universitet Examinator: Vera Koponen Mars 2013 Department of Mathematics Uppsala University Contents 1 Algebras and Coalgebras 1 1.1 Morphisms .................................... 2 1.2 Initial and terminal structures ........................... 4 1.3 Functoriality .................................... 6 1.4 (Co)recursion ................................... 7 1.5 Building final coalgebras ............................. 9 2 Bundles 13 2.1 Sums and products ................................ 14 2.2 Exponentials, fibre-wise ............................. 18 2.3 Bundles, fibre-wise ................................ 19 2.4 Lifting functors .................................. 21 2.5 A choice theorem ................................. 22 3 Enriching bundles 25 3.1 Enriched categories ................................ 26 3.2 Underlying categories ............................... 29 3.3 Enriched functors ................................. 31 3.4 Convenient strengths ............................... 33 3.5 Natural transformations .............................. 41 4 Containers 45 4.1 Container functors ................................ 45 4.2 Natural transformations .............................. 47 4.3 Strengths, revisited ................................ 50 4.4 Using shapes ................................... 53 4.5 Final remarks ................................... 56 i Introduction -
The Differential -Calculus
UNIVERSITY OF CALGARY The differential λ-calculus: syntax and semantics for differential geometry by Jonathan Gallagher A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN COMPUTER SCIENCE CALGARY, ALBERTA May, 2018 c Jonathan Gallagher 2018 Abstract The differential λ-calculus was introduced to study the resource usage of programs. This thesis marks a change in that belief system; our thesis can be summarized by the analogy λ-calculus : functions :: @ λ-calculus : smooth functions To accomplish this, we will describe a precise categorical semantics for the dif- ferential λ-calculus using categories with a differential operator. We will then describe explicit models that are relevant to differential geometry, using categories like Sikorski spaces and diffeological spaces. ii Preface This thesis is the original work of the author. This thesis represents my attempt to understand the differential λ-calculus, its models and closed structures in differential geometry. I motivated to show that the differential λ-calculus should have models in all the geometrically accepted settings that combine differential calculus and function spaces. To accomplish this, I got the opportunity to learn chunks of differential ge- ometry. I became fascinated by the fact that differential geometers make use of curves and (higher order) functionals everywhere, but use these spaces of func- tions without working generally in a closed category. It is in a time like this, that one can convince oneself, that differential geometers secretly wish that there is a closed category of manifolds. There are concrete problems that become impossible to solve without closed structure. -
Dagger Symmetric Monoidal Category
Tutorial on dagger categories Peter Selinger Dalhousie University Halifax, Canada FMCS 2018 1 Part I: Quantum Computing 2 Quantum computing: States α state of one qubit: = 0. • β ! 6 α β state of two qubits: . • γ δ ac a c ad separable: = . • b ! ⊗ d ! bc bd otherwise entangled. • 3 Notation 1 0 |0 = , |1 = . • i 0 ! i 1 ! |ij = |i |j etc. • i i⊗ i 4 Quantum computing: Operations unitary transformation • measurement • 5 Some standard unitary gates 0 1 0 −i 1 0 X = , Y = , Z = , 1 0 ! i 0 ! 0 −1 ! 1 1 1 1 0 1 0 H = , S = , T = , √2 1 −1 ! 0 i ! 0 √i ! 1 0 0 0 I 0 0 1 0 0 CNOT = = 0 X ! 0 0 0 1 0 0 1 0 6 Measurement α|0 + β|1 i i 0 1 |α|2 |β|2 α|0 β|1 i i 7 Mixed states A mixed state is a (classical) probability distribution on quantum states. Ad hoc notation: 1 α 1 α + ′ 2 β ! 2 β′ ! Note: A mixed state is a description of our knowledge of a state. An actual closed quantum system is always in a (possibly unknown) “pure” (= non-mixed) state. 8 Density matrices (von Neumann) α Represent the pure state v = C2 by the matrix β ! ∈ αα¯ αβ¯ 2 2 vv† = C × . βα¯ ββ¯ ! ∈ Represent the mixed state λ1 {v1} + ... + λn {vn} by λ1v1v1† + ... + λnvnvn† . This representation is not one-to-one, e.g. 1 1 1 0 1 1 0 1 0 0 .5 0 + = + = 2 0 ! 2 1 ! 2 0 0 ! 2 0 1 ! 0 .5 ! 1 1 1 1 1 1 1 .5 .5 1 .5 −.5 .5 0 + = + = 2 √2 1 ! 2 √2 −1 ! 2 .5 .5 ! 2 −.5 .5 ! 0 .5 ! But these two mixed states are indistinguishable. -
Notions of Computation As Monoids (Extended Version)
Notions of Computation as Monoids (extended version) EXEQUIEL RIVAS MAURO JASKELIOFF Centro Internacional Franco Argentino de Ciencias de la Informaci´ony de Sistemas CONICET, Argentina FCEIA, Universidad Nacional de Rosario, Argentina September 20, 2017 Abstract There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as instances of a unifying abstract concept: monoids in monoidal categories. We demonstrate that even when working at this high level of generality one can obtain useful results. In particular, we give conditions under which one can obtain free monoids and Cayley representations at the level of monoidal categories, and we show that their concretisation results in useful constructions for monads, applicative functors, and arrows. Moreover, by taking advantage of the uniform presentation of the three notions of computation, we introduce a principled approach to the analysis of the relation between them. 1 Introduction When constructing a semantic model of a system or when structuring computer code, there are several notions of computation that one might consider. Monads [37, 38] are the most popular notion, but other notions, such as arrows [22] and, more recently, applicative functors [35] have been gaining widespread acceptance. Each of these notions of computation has particular characteristics that makes them more suitable for some tasks than for others. Nevertheless, there is much to be gained from unifying all three different notions under a single conceptual framework. In this article we show how all three of these notions of computation can be cast as monoids in monoidal categories. -
Building Closed Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 19, No 2 (1978), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MICHAEL BARR Building closed categories Cahiers de topologie et géométrie différentielle catégoriques, tome 19, no 2 (1978), p. 115-129. <http://www.numdam.org/item?id=CTGDC_1978__19_2_115_0> © Andrée C. Ehresmann et les auteurs, 1978, 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/legal.php). Toute uti- lisation 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. XIX - 2 (1978 ) ET GEOMETRIE DIFFERENTIELLE BUILDING CLOSED CATEGORIES by Michael BARR * I am interested here in examining and generalizing the construction of the cartesian closed category of compactly generated spaces of Gabriel- Zisman [1967 . Suppose we are given a category 21 and a full subcategory 5 . Suppose E is a symmetric monoidal category in the sense of Eilenberg- Kelly [1966], Chapter II, section 1 and Chapter III, section 1. Suppose there is, in addition, a «Hom» functor lil°P X 5--+ 6 which satisfies the axioms of Eilenberg-Kelly for a closed monoidal category insofar as they make sense. Then we show that, granted certain reasonable hypotheses on lbl and 21, this can be extended to a closed monoidal structure on the full subcategory of these objects of 21 which have a Z-presentation. One additional hypo- thesis suffices to show that when the original product in 5 is the cartesian product, then the resultant category is even cartesian closed. -
Hypergraph Categories
Hypergraph Categories Brendan Fong and David I. Spivak Abstract Hypergraph categories have been rediscovered at least five times, under vari- ous names, including well-supported compact closed categories, dgs-monoidal cat- egories, and dungeon categories. Perhaps the reason they keep being reinvented is two-fold: there are many applications—including to automata, databases, circuits, linear relations, graph rewriting, and belief propagation—and yet the standard def- inition is so involved and ornate as to be difficult to find in the literature. Indeed, a hypergraph category is, roughly speaking, a “symmetric monoidal category in which each object is equipped with the structure of a special commutative Frobenius monoid, satisfying certain coherence conditions”. Fortunately, this description can be simplified a great deal: a hypergraph cate- gory is simply a “cospan-algebra,” roughly a lax monoidal functor from cospans to sets. The goal of this paper is to remove the scare-quotes and make the previous statement precise. We prove two main theorems. First is a coherence theorem for hypergraph categories, which says that every hypergraph category is equivalent to an objectwise-free hypergraph category. Second, we prove that the category of objectwise- free hypergraph categories is equivalent to the category of cospan-algebras. Keywords: Hypergraph categories, compact closed categories, Frobenius algebras, cospan, wiring diagram. 1 Introduction arXiv:1806.08304v3 [math.CT] 18 Jan 2019 Suppose you wish to specify the following picture: g (1) f h This picture might represent, for example, an electrical circuit, a tensor network, or a pattern of shared variables between logical formulas. 1 One way to specify the picture in Eq. -
Categories of Mackey Functors
Categories of Mackey Functors Elango Panchadcharam M ∗(p1s1) M (t2p2) M(U) / M(P) ∗ / M(W ) 8 C 8 C 88 ÖÖ 88 ÖÖ 88 ÖÖ 88 ÖÖ 8 M ∗(p1) Ö 8 M (p2) Ö 88 ÖÖ 88 ∗ ÖÖ M ∗(s1) 8 Ö 8 Ö M (t2) 8 Ö 8 Ö ∗ 88 ÖÖ 88 ÖÖ 8 ÖÖ 8 ÖÖ M(S) M(T ) 88 Mackey ÖC 88 ÖÖ 88 ÖÖ 88 ÖÖ M (s2) 8 Ö M ∗(t1) ∗ 8 Ö 88 ÖÖ 8 ÖÖ M(V ) This thesis is presented for the degree of Doctor of Philosophy. Department of Mathematics Division of Information and Communication Sciences Macquarie University New South Wales, Australia December 2006 (Revised March 2007) ii This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This work has not been submitted for a higher degree to any other university or institution. Elango Panchadcharam In memory of my Father, T. Panchadcharam 1939 - 1991. iii iv Summary The thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally for promonoidal categories. The notion of Mackey functor was first defined by Dress [Dr1] and Green [Gr] in the early 1970’s as a tool for studying representations of finite groups. The first contribution of this thesis is the study of Mackey functors on a com- pact closed category T . We define the Mackey functors on a compact closed category T and investigate the properties of the category Mky of Mackey func- tors on T . -
Complexity of Grammar Induction for Quantum Types Antonin Delpeuch
Complexity of Grammar Induction for Quantum Types Antonin Delpeuch To cite this version: Antonin Delpeuch. Complexity of Grammar Induction for Quantum Types. Electronic Proceedings in Theoretical Computer Science, EPTCS, 2014, 172, pp.236-248. 10.4204/eptcs.172.16. hal-01481758 HAL Id: hal-01481758 https://hal.archives-ouvertes.fr/hal-01481758 Submitted on 2 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Complexity of Grammar Induction for Quantum Types Antonin Delpeuch École Normale Supérieure 45 rue d’Ulm 75005 Paris, France [email protected] Most categorical models of meaning use a functor from the syntactic category to the semantic cate- gory. When semantic information is available, the problem of grammar induction can therefore be defined as finding preimages of the semantic types under this forgetful functor, lifting the informa- tion flow from the semantic level to a valid reduction at the syntactic level. We study the complexity of grammar induction, and show that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete. Our approach could be extended to linguistic type systems such as autonomous or bi-closed categories.