Tower (mathematics) - Alchetron, The Free Social Encyclopedia

In category theory, a branch of abstract mathematics, a tower is defined as follows. Let I be the poset

⋯ → 2 → 1 → 0

of whole numbers in reverse order, regarded as a category. A (countable) tower of objects in a category A is a functor from I to A .

In other words, a tower (of A ) is a family of objects { A i } i ≥ 0 in A where there exists a map

A i → A j if i > j

and the composition

A i → A j → A k

is the map A i → A k

Example

Let M i = M for some R -module M . Let M i → M j be the identity map for i > j . Then { M i } forms a tower of modules.

(Text) CC BY-SA