title: Tower of objects
text: # 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.
  
  