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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1204.0. "Category Theory" by FORTY2::BOYES ("Mr ACCVIO") Wed Mar 07 1990 12:03

This note is for discussion of category theory, entered because I
personally would like to find out about it. All that I know is that is 
is a model in which other models (vector spaces, rings, groups et al) are
manipulated as objects... this possibly leads to a theory which handles
itself as an object, but I'm a bit shaky on that point!
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1204.1HPSTEK::XIAIn my beginning is my end.Wed Mar 07 1990 20:3514
    Well, to make it as simple as possible.  This kinda thing comes up when
    you transform some difficult and inaccesable objects to something
    managable.  One example is the study of topological spaces.  
    These things are difficult because there are very few tools you can use
    in topological spaces.  So you assign (with something called a
    morphism) each topological space to a group which is easier to deal with.  
    Now you prove something about the group and then you can conclude 
    something about the original topological space.  Now you are dealing 
    with morphism that "maps" from category of topological space to the
    category of groups.  These categories are not sets (because you get
    things like Russell's paradox, if you call them sets).
    
    Eugene 
                                                      
1204.2referenceHERON::BUCHANANcombinatorial bomb squadThu Mar 08 1990 10:3420
	A book I found was a good, fun introduction to the subject is
by Arbib & Manes.   It's called "Intro to Category Theory" or something
imaginative like that.

	It starts off with the idea of sets and the definition of
morphisms between sets.   It then shows how you can obtain a more
powerful and general theory by dispensing with the idea of elements
in the sets.   So you move to the idea of categories:  but the key thing
is the idea of morphism.

	I found that it was a lot easier to come to grips with the theory
of Object-Oriented Programming having had some encounters with the
principles of Category Theory, but I wouldn't want to push the connections
too far.

	The theorems in the subject are quite pretty.   It is very much a
subject where a picture is worth a thousand words.

CHeers,
Andrew.
1204.3HPSTEK::XIAIn my beginning is my end.Thu Mar 08 1990 16:495
    It is my opinion that one should learn something about Algebraic
    Topology before starting on this stuff; otherwise, one would have a
    feeling of "why bother".
    
    Eugene
1204.4further referencesKAOH06::E_SPLETTThu Jun 28 1990 18:1740
C.A.R. Hoare,
Notes on an Approach to Category Theory for Computer Scientists;
    in 
Constructive Methods in Computing Science
NATO ASI Series, Vol. F55
Edited by M. Broy
Springer-Verlag 1989
ISBN 0-387-51369-8 (U.S.)
ISBN 3-540-51369-8


R. Goldblatt, 
Topoi: The Categorial Analysis of Logic;
North Holland, Revised Edition, 1969.


E.G. Manes and M.A. Arbib,
Algebraic Approaches to Program Semantics;
Springer-Verlag, 1986.


Saunders MacLane,
Categories for the Working Mathematician;
Springer-Verlag, 1971.


J. Lambek and P.J. Scott,
Introduction to Higher Order Categorical Logic;
Cambridge University Press, 1986.


H. Herrlich and G.E. Strecker,
Category Theory, Second Edition,
Helderman Verlag, Berlin 1979.


D.E. Rydeheard and R.M. Burstall,
Computational Category Theory.
Prentice-Hall, 1988.