Tower of objects

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

\dots \to 2 \to 1 \to 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 ${A_{i}}_{i \geq 0}$ in $𝒜$ where there exists a map

A_{i} \to A_{j}

if

i > j

and the composition

A_{i} \to A_{j} \to A_{k}

is the map $A_{i} \to A_{k}$

Example

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