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"Yoneda showed that maps in any category can be
represented as natural transformations" (Lawvere & Schanuel,
Conceptual Mathematics, p. 378). Isn't this reason enough to think of
category theory as the theory of naturality?
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That would be like saying group theory is the theory of permutations
(because of the Cayley theorem).
Perhaps my little colloquium talk entitled
``The natural transformation in mathematics''
at
would be of some interest in this connexion. I am sure lots of us have
given similar talks. The goal of the paper considered the first in category
theory was to define natural transformation. That required functor, and
that required category.
Ross