Benjamin C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 1 : 1 m 1 maps to 0, and m : m m 1 maps i 1 to (i) 1 for any 0 i m 1. We leave the definition of , and all other calculations, as a long(ish) Exercise. 4.5 Where to now There are a number of books which cover basic category theory. For a short and gentle introduction, see [29]. For a longer first text see [19] Both of these books are intended for computer scientists. The original and recommended general reference for category theory is [25] which was written for mathematicians. A very concise and fast paced introduction can be found in [21] which also covers the ....

B.C. Pierce. Basic Category Theory for Computer Scientists. Foundations of Computing Series. The MIT Press, 1991.

....introduction and the proofs of the theorems. 3.1 Category theory Category theory was developed in an attempt to unify simple abstract concepts that were applicable in many branches of mathematics. Excellent introductions to category theory and its application in Computer Science can be found in [Pier90, Gogu91, Pier91, Aspe91, Barr96]. 3.1.1 Basic definitions Definition 3.1. A category C is a collection of objects and a collection of arrows satisfying the CATEGORIES following properties: ffl For each arrow f there is a domain object dom(f) and a codomain object codom(f) and by writing f : x y it is indicated that x = ....

B. Pierce, Basic Category Theory for Computer Scientists, Foundations of Computing Series, MIT Press, Cambridge, MA, 1991.

....some of the mathematical structures which are used in this thesis. We presuppose some fundamental definitions of category theory such as category, functor, natural transformation, co )limit, adjunction, and cartesian closed category, which can be found in any textbook on category theory, such as [ML71, Pie91, BW95, Bor94a]. 2.1 Monoidal Categories Definition 2.1. A monoidal category is a category C with a functor : C C C , called the tensor product, an object I in C , called the tensor unit, and natural isomorphisms a, l and r with components B) B I A satisfying the coherence conditions given ....

Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1991.

....for the rest of the paper. We introduce monads and monad transformers and discuss their connection with computations and the semantics of programming languages. The reader is referred to [Papa01] for a more complete and informative introduction, and to the literature related to category theory [Pier91, Aspe91, Barr96] and domain theory [Scot82, Gunt90] 2.1 Monads and monad transformers The notion of monad, also called triple, is not new in the context of category theory. In Computer Science, monads became very popular in the 1990s. The categorical properties of monads are discussed in most books on category ....

B. Pierce, Basic Category Theory for Computer Scientists, Foundations of Computing Series, MIT Press, Cambridge, MA, 1991.

....than just two; see Winskel s textbook for details [92] In Chapter 6, we consider fixed point operators in more abstract settings, i.e. without assuming that the underlying structure is domains and continuous maps. We assume a minimal acquaintance with category theory in the following discussion [2, 70]. The basic structure we work with is a category with finite products. We write for the terminal object. The set of arrows between two objects A and B is denoted B) We will need the following basic definitions [33, 79] Definition A.3 A fixed point operator is a family of functions ( # ....

Pierce, B. C. Basic Category Theory for Computer Scientists. MIT Press, Cambridge, MA, 1991. (141)

....of type assignments G, into a product of the sets denoting the finite number of types in the assignment. Note that the semantics of programs is defined on typing judgments, and maps to elements of the meanings of their types. This is the standard way of defining the semantics of typed languages [56, 18, 39], and the implementation in the next section will be a direct codification of this definition. 2.2 Interpreters in Meta D An interpreter for SL can be simply an implementation of the definition in Figure 1. We begin by defining the datatypes that will be used to interpret the basic types (and ....

Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, Cambridge, Mass., 1991.

.... to form a category) 4 Using category theoretical notions for model constructions In this section we assume given any class of objects MOD (we will call them models) and a notion of morphisms (we will call them homomorphisms) between them, satisfying the basic rules of category theory (e.g. see [Pi91]) a) A homomorphism p is called monic or a monomorphism if for any two homomorphisms f and g pf = pg f = g A homomorphism p is called epic or an epimorphism if for any two homomorphisms f and g fp = gp f = g 14 b) The homomorphism b : B C is an isomorphism if there is a c : C B such that bc = ....

B.C. Pierce, Basic Category Theory for Computer Scientists, MIT Press, 1991. 40

....whose nodes are called worlds and whose arcs are called transitions, with a distinguished node called initial world. Notation A.12. We use the symbols W, T, and w 0 to represent the collections of worlds and transitions, and the initial world, respectively. A. 2 Category Theory Category Theory [Pei91, Mar96, FM94] is the mathematical discipline that studies, in a general and abstract way, relationships between arbitrary entities. Basically, a category is a graph, with nodes called objects and arcs called morphisms, such that paths are closed under transitivity and reflexivity. Definition ....

Benjamin C. Peirce. Basic Category Theory for Computer Scientists. The MIT Press, 1991.

....[12] a system for the specification and formal development of software. It has a rigorous mathematical foundation, based on logic and category theory. It provides an ordersorted higher order logic representation language called Slang, whose semantics are founded on categorical type theory [3, 10]. It includes a rich set of primitives for composing specifications by reusing and parameterizing one or more copies of other specifications. The user specifies what and how various component specifications are to be included in the whole, and the colimit operation from category theory is used to ....

B. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1991.

....section) 5 Ongoing Theoretical Developments The formal concepts surveyed in the previous sections are currently being extended and re ned to better address the fundamental issues that inspired them. Such extensions and re nements are being carried out in a categorical framework. Category theory [33,3,27] is getting increasingly popular in many areas of Computer Science, because ideas in those areas can be conveniently expressed and reasoned about by means of the plethora of concepts and results of this branch of mathematics. In particular, category theory has proven very successful in the eld of ....

Benjamin C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press, Cambridge, Massachussets, 1991.

....Special Research Programme (Sonderforschungsbereich) 441. A Category Theory Basics In this appendix, we will just provide the basic category theoretic notions necessary for the current paper. A reader interested in a more elaborate introduction to category theory may consult the book by Pierrce [15], for example. 21 Definition 43 A category C consists of a collection of objects and a collection of arrows (also called morphisms) with the following properties. For each two arrows f : A B and g : B C there exists a composite arrow g f : A C . Composition is associative. And each object ....

Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, Cambridge, USA, 1991.

....would be easy to reproduce in other KR languages. There are parallels with the work on capturing design patterns in object oriented programming [18] The mathematical field of Category Theory provides a full, formal foundation for representing, morphing, and composing theory structures together [19], and KM s implementation can be seen as a simple example of this. 59 17 Prototypes 17.1 Introduction Prototypes are a new and significantly di#erent style of knowledge representation in KM, o#ering some important advantages, which we now describe. They do not o#er any new features in terms of ....

B. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press, 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, Cambridge, Mass., 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press, Cambridge, Massachusetts, 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT, 1991. 15

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B. Pierce. Basic Category Theory for Computer Scientists. Foundations of Computing. The MIT Press, 1991.

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Benjamin C. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1993.

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B. C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press.

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Benjamin Pierce. Basic category Theory for Computer Scientists. Foundations of Computing Series. MIT Press, 1991.

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B. C. Pierce. Basic Category Theory for Computer Scientists. The MIT Press.

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B.C. Pierce. Basic Category Theory for Computer Scientists. Foundations of Computing Series, 1991.

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B. Pierce. Basic Category Theory for Computer Scientists. MIT Press, 1991.

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Pierce, Benjamin C., Basic Category Theory for Computer Scientists, MIT Press, 1994.

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