1.NounA subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If
i
≤
j
{\displaystyle i\leq j}
in the indexing poset then
f
i
j
:
A
j
→
A
i
{\displaystyle f_{ij}:A_{j}\rightarrow A_{i}}
in the inverse system and if
a
i
∈
A
i
{\displaystyle a_{i}\in A_{i}}
,
a
j
∈
A
j
{\displaystyle a_{j}\in A_{j}}
are components of M then
f
i
j
(
a
j
)
=
a
i
{\displaystyle f_{ij}(a_{j})=a_{i}}
).
"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”.; Inverse limits are concrete-categorical versions of limits."