By D. E. Rydeheard


1 Introduction
1.1 The contents
1.2 Accompanying texts
1.2.1 Textbooks on classification theory
1.2.2 ML references and availability
1.2.3 a variety of textbooks on practical programming
1.3 Acknowledgements
2 practical Programming in ML
2.1 Expressions, values and environments
2.2 Functions
2.2.1 Recursive definitions
2.2.2 better order functions
2.3 Types
2.3.1 Primitive types
2.3.2 Compound types
2.3.3 kind abbreviation
2.4 style polymorphism
2.5 Patterns
2.6 Defining types
2.7 summary types
2.8 Exceptions
2.9 different facilities
2.10 Exercises
3 different types and Functors
3.1 Categories
3.1.1 Diagram chasing
3.1.2 Subcategories, isomorphisms, monics and epis
3.2 Examples
3.2.1 units and finite sets
3.2.2 Graphs
3.2.3 Finite categories
3.2.4 kinfolk and partial orders
3.2.5 Partial orders as categories
3.2.6 Deductive systems
3.2.7 common algebra: phrases, algebras and equations
3.2.8 units with constitution and structure-preserving arrows
3.3 different types computationally
3.4 different types as values
3.4.1 the class of finite sets
3.4.2 phrases and time period substitutions: the class T_tau^Fin
3.4.3 A finite category
3.5 Functors
3.5.1 Functors computationally
3.5.2 Examples
3.6 Duality
3.7 An assessment*
3.8 Conclusion
3.9 Exercises
4 Limits and Colimits
4.1 Definition via universality
4.2 Finite colimits
4.2.1 preliminary objects
4.2.2 Binary coproducts
4.2.3 Coequalizers and pushouts
4.3 Computing colimits
4.4 Graphs, diagrams and colimits
4.5 A basic development of colimits
4.6 Colimits within the class of finite sets
4.7 A calculation of pushouts
4.8 Duality and limits
4.9 Limits within the classification of finite sets
4.10 An program: operations on relations
4.11 Exercises
5 developing Categories
5.1 Comma categories
5.1.1 Representing comma categories
5.2 Colimits in comma categories
5.3 Calculating colimits of graphs
5.4 Functor categories
5.4.1 average transformations
5.4.2 Functor categories
5.5 Colimits in functor categories
5.6 Duality and limits
5.7 summary colimits and limits*
5.7.1 summary diagrams and colimits
5.7.2 classification constructions
5.7.3 listed colimit structures
5.7.4 Discussion
5.8 Exercises
6 Adjunctions
6.1 Definitions of adjunctions
6.2 Representing adjunctions
6.3 Examples
6.3.1 flooring and ceiling capabilities: changing genuine numbers to integers
6.3.2 elements of a graph
6.3.3 loose algebras
6.3.4 Graph theory
6.3.5 Limits and colimits
6.3.6 Adjunctions and comma categories
6.3.7 Examples from algebra and topology
6.4 Computing with adjunctions
6.5 unfastened algebras
6.5.1 developing loose algebras
6.5.2 A program
6.5.3 An instance: transitive closure
6.5.4 different buildings of loose algebras
6.6 Exercises
7 Toposes
7.1 Cartesian closed categories
7.1.1 An instance: the class of finite sets
7.2 Toposes
7.2.1 An instance: the topos of finite sets
7.2.2 Computing in a topos
7.2.3 common sense in a topos
7.2.4 An instance: a three-valued logic
7.3 Conclusion
7.4 Exercises
8 A specific Unification Algorithm
8.1 The unification of terms
8.2 Unification as a coequalizer
8.3 On developing coequalizers
8.4 A specific program
9 developing Theories
9.1 Preliminaries
9.2 developing theories
9.3 Theories and institutions
9.4 Colimits of theories
9.5 Environments
9.6 Semantic operations
9.7 imposing a express semantics
10 Formal structures for class Theory
10.1 Formal facets of class theory
10.2 classification conception in OBJ
10.3 class concept in a kind theory
10.4 express info types
A ML Keywords
B Index of ML Functions
C different ML Functions
D solutions to Programming routines 231

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In which each number is the sum of its two immediate predecessors. Write a recursive definition of the n-th entry in the sequence. Exercise 3. Natural numbers Define the type of natural numbers as follows: datatype Num = zero | succ of Num Define a function numprint : numbers as integers. Num -> int which displays natural In the text of the chapter we show how to define addition of natural numbers. Use this addition operation and the same cases as in its definition to define the multiplication of natural numbers.

The function which sums a list of integers. 3. The function which takes a list of coefficients a 0 , a1 , . . , an and a value x and evaluates the polynomial a 0 +a1 ×x+. . an ×xn . 10 EXERCISES 4. Use the append function, concatenating lists end to end, to define the function which reverses a list. 5. The function maplist which applies a function to all items in a list returning the list of results. What is its most general type? 6. The function calculating the sum of a list of integers can be generalized.

More precisely, since arrows determine their source and target objects, arrows in Set are typed total functions, which we may consider to be triples (a, f, b), a and b being sets and f a function defined on all elements of a and whose results lie in b. g(f (x)), c) A subcategory of Set is that of finite sets, FinSet, whose arrows are again typed total functions. There are other categories whose objects are sets. For instance, we may consider arrows to be not total functions but partial functions to get a category SetP f .

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