Tower of objects explained

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

l 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

l A is a functor from l I to l A .

In other words, a tower (of

l A ) is a family of objects \{A _{i\}} _{i\geq} in l A where there exists a mapA _{i →} A _{j} if i>j and the compositionA _{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.

References

• Section 3.5 of