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.

References

• Section 3.5 of Weibel, Charles A. (1994), An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4