Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
Isomorphisms of Categories
-
Andrzej Trybulec
-
Warsaw University, Bialystok
Summary.
-
We continue the development of the category theory basically
following [8]
(compare also [7]). We define the concept of isomorphic
categories and prove basic facts related, e.g. that the Cartesian product
of categories is associative up to the isomorphism. We introduce the
composition of a functor and a transformation, and of transformation
and a functor,
and afterwards we define again those concepts for natural transformations.
Let us observe, that we have to duplicate those concepts because of the
permissiveness: if a functor $F$ is not naturally transformable to $G$, then
natural transformation from $F$ to $G$ has no fixed meaning, hence we cannot
claim that the composition of it with a functor as a transformation
results in a natural transformation.
We define also the so called horizontal composition of transformations
([8], p.~140, exercise {\bf 4}.{\bf 2},{\bf 5}(C)) and prove
{\em interchange law} ([7], p.44).
We conclude with the definition of equivalent categories.
The terminology and notation used in this paper have been
introduced in the following articles
[9]
[5]
[11]
[2]
[3]
[1]
[4]
[6]
[10]
Contents (PDF format)
Bibliography
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Czeslaw Bylinski.
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1, 1989.
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Springer Verlag, New York, Heidelberg, Berlin, 1971.
- [8]
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\em Wst\c ep do teorii kategorii i funktorow, volume 45 of
\em Biblioteka Matematyczna.
PWN, Warszawa, 1978.
- [9]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [10]
Andrzej Trybulec.
Natural transformations. Discrete categories.
Journal of Formalized Mathematics,
3, 1991.
- [11]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received November 22, 1991
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