Results 1  10
of
12
Natural Deduction and Coherence for Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categorie ..."
Abstract

Cited by 73 (26 self)
 Add to MetaCart
This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the twotensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansionreduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending simi...
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
"... This document was typeset using L ..."
Simple free starautonomous categories and full coherence
, 2005
"... This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence. ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence.
A Note on Actions of a Monoidal Category
, 2001
"... An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functorso that, passing to the categories of monoids (also called \algebras") in V and in [A; A], we have an adjunction MonF a MonG between the category MonV of monoids in V and the category Mon[A; A] = MndA of monads on A. We give sucient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X { and { A, and make A (at least when V is symmetric and closed) into a tensored and cotensored Vcategory A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.
Axiomatics for Data Refinement in Call By Value Programming Languages
"... We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first order fragment of the computational calculus, such as a CPS language. We give a category theoretic account of the basic setting, then show how to model contexts, then arbitrary type and term constructors, then signatures, and finally data refinement. This extends and clarifies Kinoshita and Power's work on lax logical relations for call by value languages.
Higherdimensional Mac Lane's pentagon and Zamolodchikov equations
, 1999
"... An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram whose commutativity is ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram whose commutativity is needed in order for all diagrams in somewhat weak teisi to commute. Looping twice gives a 4dimensional diagram in somewhat weak braided teisi, of which ve 3dimensional edges can be interpreted as proofs of ve dierent Zamolodchikov equations in braided monoidal 2categories. Hence higherdimensional Mac Lane's pentagon expresses the relations between these proofs concisely. 1 Introduction The coherence theorem for tricategories states that every tricategory is triequivalent to a Graycategory [6]. But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) [1], which roughly says that \all diagrams in a tricategory...
Medial Commutativity
, 2007
"... It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define assoc ..."
Abstract
 Add to MetaCart
It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor ∧. This preservation boils down to an isomorphic representation of the YangBaxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor ∧. In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups S n for 0 ≤ i ≤ n. i)
unknown title
, 2006
"... It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), which we propose to call commixing. Commixing in the presence of the unit object enables us to define associati ..."
Abstract
 Add to MetaCart
It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), which we propose to call commixing. Commixing in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of commixing by the biendofunctor ∧. This preservation boils down to an isomorphic representation of the YangBaxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor ∧. In the final section one finds coherence conditions for commixing in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups S n for 0 ≤ i ≤ n. i)