Last Updated：2025/12/30
Sentence
So
it
is
natural
to
speak
of
a
category
of
all
categories,
which
we
call
CAT,
the
objects
of
which
are
all
the
categories,
and
the
arrows
of
which
are
all
the
functors.
This
raises
genuine
problems.
Is
CAT
a
category
in
itself?
Our
answer
here
is
to
treat
CAT
as
a
regulative
idea;
that
is,
an
inevitable
way
of
thinking
about
categories
and
functors,
but
not
a
strictly
legitimate
entity.
(Compare
the
self,
the
universe,
and
God
in
Kant
1781.)
Of
course,
general
category
theory
applies
to
CAT,
and
this
category
that
we
do
not
quite
believe
in
is
the
single
one
that
we
investigate
the
most.
Quizzes for review
So it is natural to speak of a category of all categories, which we call CAT, the objects of which are all the categories, and the arrows of which are all the functors. This raises genuine problems. Is CAT a category in itself? Our answer here is to treat CAT as a regulative idea; that is, an inevitable way of thinking about categories and functors, but not a strictly legitimate entity. (Compare the self, the universe, and God in Kant 1781.) Of course, general category theory applies to CAT, and this category that we do not quite believe in is the single one that we investigate the most.
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