Tower of objects

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


 * $$\cdots\rightarrow 2\rightarrow 1\rightarrow 0$$

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

In other words, a tower (of $$\mathcal A$$) is a family of objects $$\{A_i\}_{i\geq 0}$$ in $$\mathcal A$$ where there exists a map
 * $$A_i\rightarrow A_j$$ if $$i>j$$

and the composition
 * $$A_i\rightarrow A_j\rightarrow A_k$$

is the map $$A_i\rightarrow A_k$$

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