Charles Wells

A generalization of the concept of sketch

This paper introduces an extension of the concept of sketch, called a form, which allows the specification of entities other than limits and colimits in a model. A form can require that a diagram become (in a model) an instance of any categorial construction specifiable in an essentially algebraic way. Constructions which can be specified in this way include function space objects in and reflexive objects in a cartesian closed category, power objects in a topos, and list objects in a locos. This generalization is motivated by the desire to specify functional programming languages by sketches.

The Formal Description of Data Types Using Sketches

This paper is an exposition of the basic ideas of the mathematical theory of sketches and a detailed description of some of the ways in which this theory can be used in theoretical computer science to specify datatypes. In particular, this theory provides a convenient way of introducing datatypes which have variants, for example in case of errors or nil pointers. The semantics is a generalization of initial algebra semantics which in some cases allows initial algebras depending on a parameter such as a bound for overflow.

Communicating Mathematics: Useful Ideas from Computer Science

1.1. Purpose. This article describes certain ideas originating in the theory and practice of computer science, and shows how the teaching and exposition of mathematics could benefit if these ideas were widely understood by mathematicians and used in their teaching and writing. These ideas are discussed here because I believe they are important for mathematicians to understand. Some of them are based on theoretical work by computer scientists and others are based on the practice of computer professionals inside and outside academia. Computer scientists would not regard the various concepts as of equal importance, and the whole collection of ideas is nothing like a fair presentation of the current state of computing.

The degrees of permutation polynomials over finite fields

Abstract A number of theorems are proved concerning the connection between the cycle structure of a permutation of a finite field GF( q ) and the degree of the polynomial representing it. In particular (Section 4) if K r is the set of permutations of GF( q ) moving≤ r elements, then, if r grows slowly enough with respect to q as q →∞, almost all polynomials of degree≤ q −1 representing permutations in K r have degree q −2.

Varieties of mathematical prose

This article begins the development of a taxonomy of mathematical prose, describing the precise function and meaning of specic types of mathematical exposition. It further discusses the merits and demerits of a style of mathematical writing that labels each passage according to its function as described in the taxonomy.

A formalism for the specification of essentially-algebraic structures in 2-categories

A type of higher-order two-dimensional sketch is defined which has models in suitable 2-categories. It has as special cases the ordinary sketches of Ehresmann and certain previously defined generalizations of one-dimensional sketches. These sketches allow the specification of constructions in 2-categories such as weighted limits, as well as higher-order constructions such as exponential objects and subobject classifiers, that cannot be sketched by limits and colimits. These sketches are designed to be the basis of a category-based methodology for the description of functional programming languages, complete with rewrite rules giving the operational semantics, that is independent of the usual specification methods based on formal languages and symbolic logic. A definition of ‘path grammar’, generalizing the usual notion of grammar, is given as a step towards this goal.

On the Communication of Mathematical Reasoning.

ABSTRACT This article discusses some methods of describing and referring to mathematical objects and of consistently and unambiguously signaling the logical structure of mathematical arguments.

A triple in CAT

A triple (or monad ) in a category K is a triple = ( T , μ, η) where, T : K → K is a functor and μ: TT →, T , η: 1 k → T are natural transformations for which (1.1) and (1.2) commute: In these diagrams the component of a natural transformation α at an object x is denoted xα. Thus for example ( k η) T is the value of the functor T applied to the component of η at k , whereas ( kT )η is the component of η at the object kT . I write functions and functors on the right and composition from left to right.