inverse limit

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inverse limit

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inverse limit (plural inverse limits)

1. (algebra) A 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 ${\displaystyle i\leq j}$ in the indexing poset then ${\displaystyle f_{ij}:A_{j}\rightarrow A_{i}}$ in the inverse system and if ${\displaystyle a_{i}\in A_{i}}$, ${\displaystyle a_{j}\in A_{j}}$ are components of M then ${\displaystyle f_{ij}(a_{j})=a_{i}}$).

   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.

2. (category theory) a limit

• projective limit

• direct limit

• inverse system