inverse limit
noun1
Algebra, Mathematics, Sciences 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 i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then 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, Computing, Engineering, Mathematics, Natural sciences, Physical sciences, Sciences a limit
- Synonyms
- projective limit
- Antonyms
- direct limit
- Related terms
- inverse system