Last Updated：2025/12/31
Sentence
An
inverse
limit
has
“natural
projections”
which
are
restrictions
of
the
projections
of
the
Cartesian
product
(to
a
domain
which
is
the
inverse
limit).
The
reason
why
the
projections
are
described
as
“natural”
would
be
the
following:
besides
the
functor
from
an
index
poset
to
the
inverse
system,
there
is
another
functor
from
the
same
index
poset
to
the
inverse
limit
of
that
system,
this
functor
being
a
constant
functor.
Then
there
is
a
natural
transformation
from
the
constant
functor
to
the
inverse
limit’s
functor:
the
components
of
such
natural
transformation
are
the
said
“natural
projections”.
Quizzes for review
An inverse limit has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the inverse limit). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the inverse limit of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the inverse limit’s functor: the components of such natural transformation are the said “natural projections”.
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