## Elements of Basic Category Theory (1996)

Citations: | 2 - 2 self |

### BibTeX

@TECHREPORT{Martini96elementsof,

author = {Alfio Martini},

title = {Elements of Basic Category Theory},

institution = {},

year = {1996}

}

### OpenURL

### Abstract

Category theory provides an elegant and powerful means of expressing relationships across a wide area of mathematics. But further than this it has had a considerable impact on the conceptual basis both of mathematics and many parts of theoretical computer science. Important connections in computer science include the design of both functional and imperative programming languages, semantic models of programming languages, semantics of concurrency, specification and development of algorithms, type theory and polymorphism, specification languages, algebraic semantics, constructive logic and automata theory. The purpose of this text is to provide a soft stairway to this infectious and attractive field of mathematics. We provide here a careful and detailed explanation of "basic elements", or more precisely, from the elementary definitions to adjoint situations. The general approach used here is to provide a careful motivation for the majority of constructions as well as a detailed presentat...

### Citations

966 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...tration of the constructions. For a short and highly readable discussion on foundational aspects of category theory (which we don't treat in this text), we especially recommend [AHS90], chapter 2 and =-=[MaC71]-=- pp. 21-26. 28 Chapter 3 Basic concepts Commutative diagrams are the categorist's way of expressing equations. The first section introduces the concepts of diagrams and commutative diagrams. In this w... |

448 | Introduction to higher order categorical logic, volume 7 of Cambridge studies in advanced mathematics - Lambek, Scott - 1986 |

273 | Semantics of Programming Languages: Structures and Techniques - Gunter - 1992 |

187 |
Introduction to the algebraic theory of graph grammars
- Ehrig
- 1979
(Show Context)
Citation Context ...rtunately, one can not claim, in general, that (1) is a pushout if (1)+(2) and (2) are pushouts. However, at least in Set and Graph this is true, provided that b : B ! C is injective (see [EK76a] and =-=[EK76b]-=-, for instance). Example 4.8.6 Pushouts as amalgamations. If the effect of coequalizers is that of forcing identifications, that of pushouts is to form the so called amalgamated sums. First, consider ... |

184 |
Basic Category Theory for Computer Scientists
- Pierce
- 1991
(Show Context)
Citation Context ...ory. For a very nice presentation of limits the reader may look at [AL91], [Gol86], [MaC71] and [AHS90]. The concept of completeness is treated in deep in [AHS90]. Theorem 5.2.7 is an adaptation from =-=[Pie90]-=-. A more interesting construction of limits/colimits, especially for practical applications, is when one first constructs the product/coproduct of all objects in the diagram and then iteratively compu... |

129 |
Fundamentals of Algebraic Specification
- Ehrig, Mahr
- 1985
(Show Context)
Citation Context ...1] present in some detail a extensive number of examples both from discrete mathematical structures and computer science. The presentation of basic concepts of algebraic specification is adapted from =-=[EM85]-=-. [EP72] is especially recommended for computer scientists interested in a categorical treatment of the main concepts related to automata theory. [Poi92] is a very modern and detailed introduction to ... |

90 | Algebraic approaches to program semantics - Arbib, Manes - 1986 |

89 | General theory of natural equivalences - Eilenberg, Lane - 1945 |

59 | Abstract and Concrete Categories: The Joy of Cats
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ...as well as in the illustration of the constructions. For a short and highly readable discussion on foundational aspects of category theory (which we don't treat in this text), we especially recommend =-=[AHS90]-=-, chapter 2 and [MaC71] pp. 21-26. 28 Chapter 3 Basic concepts Commutative diagrams are the categorist's way of expressing equations. The first section introduces the concepts of diagrams and commutat... |

59 | Category theory - Herrlich, Strecker - 1973 |

51 |
A Model Theoretic Oriented Approach to Partial Algebras
- Burmeister
- 1986
(Show Context)
Citation Context ...hisms between them. A partial algebra for a given signature is just like a total algebra with the crucial difference that the operation symbols are interpreted as partial and not total functions (see =-=[Bur86]-=- for the details). Proposition 3.2.12 Every identity morphism is iso. Proof: Only define id \Gamma1 A = id A and then we have that id A ffi id \Gamma1 A = id A and id \Gamma1 A ffi id A = id A . 2 Pro... |

17 |
Basic category theory
- Poigné
- 1992
(Show Context)
Citation Context ...epts of algebraic specification is adapted from [EM85]. [EP72] is especially recommended for computer scientists interested in a categorical treatment of the main concepts related to automata theory. =-=[Poi92]-=- is a very modern and detailed introduction to an quite extensive part of category theory. It is especially useful for computer scientists since it uses many concepts from functional programming langu... |

11 |
Categories and Computer Science
- Walters
- 1991
(Show Context)
Citation Context ...unctors, that play a central role in the development of categoy theory, as we shall se later on, when we introduce the concept of functors. 2.4 Bibliographic notes [Gol86], [EP72], [AM85], [BW90] and =-=[Wal91]-=- are very nice introductions to category theory. Especially, [Gol86] is an excellent beginner's book, since it makes liberal use of simple, settheoretic examples and motivating intuitions. [BW90] and ... |

7 |
Functors: The Categorical Imperative
- Arbib, Manes, et al.
- 1975
(Show Context)
Citation Context ... However, the reader may wish to look at [AHS90], chapter 7, where several other types of morphisms are presented. The concept of duality is given a detailed presentation with a careful motivation in =-=[AM75]-=-. 49 Chapter 4 Basic constructions In this chapter we consider some fundamental categorical constructions, i.e., particular objects (and morphisms) that satisfy a given set of axioms described in the ... |

7 |
H.J.: Parallelism of manipulations in multidimensional information structures
- Ehrig, Kreowski
- 1976
(Show Context)
Citation Context ...k 4.8.5 Unfortunately, one can not claim, in general, that (1) is a pushout if (1)+(2) and (2) are pushouts. However, at least in Set and Graph this is true, provided that b : B ! C is injective (see =-=[EK76a]-=- and [EK76b], for instance). Example 4.8.6 Pushouts as amalgamations. If the effect of coequalizers is that of forcing identifications, that of pushouts is to form the so called amalgamated sums. Firs... |

5 |
The algebra of functional programs: Function level reasoning, linear equations, and extended definitions. In: Formalization of Programming Concepts
- Backus
- 1981
(Show Context)
Citation Context ...f(s)] List(S) is (often) called the free monoid generated by S (see 2.3.8). Thus any set function between sets induces a monoid homomorphism between the corresponding free monoids. f is called fff in =-=[Bac81]-=- and is often called applytoall or maplist in the computing science literature. 99 Proposition 6.1.3 The map List : Set ! Mon defined in 6.1.1.7 is a functor. Proof: Let S be a set, [s 1 ; : : : ; s n... |

4 | Structures: An Introduction to Category Theory for the Working Computer Scientist - Categories - 1991 |

3 |
Kolimeskonstruktionen in Algebraischen Spezifikationssprachen
- Padberg
- 1991
(Show Context)
Citation Context ... of limits/colimits, especially for practical applications, is when one first constructs the product/coproduct of all objects in the diagram and then iteratively computes equalizers/coequalizers (see =-=[Pad91]-=- for details). 96 Chapter 6 Functors and natural Transformations The starting point of Category Theory is the premise that every kind of mathematically structured object comes equipped with a notion o... |

2 |
M.: “Kategorien und Automaten
- Ehrig, Pfender
- 1972
(Show Context)
Citation Context ...from C to Set, called hom-functors, that play a central role in the development of categoy theory, as we shall se later on, when we introduce the concept of functors. 2.4 Bibliographic notes [Gol86], =-=[EP72]-=-, [AM85], [BW90] and [Wal91] are very nice introductions to category theory. Especially, [Gol86] is an excellent beginner's book, since it makes liberal use of simple, settheoretic examples and motiva... |

2 |
TOPOI: The Categorial Analysis of Logic: Studies in logic and the foundations of mathematics
- Goldblatt
- 1983
(Show Context)
Citation Context ...functors from C to Set, called hom-functors, that play a central role in the development of categoy theory, as we shall se later on, when we introduce the concept of functors. 2.4 Bibliographic notes =-=[Gol86]-=-, [EP72], [AM85], [BW90] and [Wal91] are very nice introductions to category theory. Especially, [Gol86] is an excellent beginner's book, since it makes liberal use of simple, settheoretic examples an... |