Tower (mathematics)

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

${\displaystyle \cdots \rightarrow 2\rightarrow 1\rightarrow 0}$

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

In other words, a tower (of ${\displaystyle {\mathcal {A}}}$) is a family of objects ${\displaystyle \{A_{i}\}_{i\geq 0}}$ in ${\displaystyle {\mathcal {A}}}$ where there exists a map

${\displaystyle A_{i}\rightarrow A_{j}}$ if ${\displaystyle i>j}$

and the composition

${\displaystyle A_{i}\rightarrow A_{j}\rightarrow A_{k}}$

is the map ${\displaystyle A_{i}\rightarrow A_{k}}$

Example

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

References

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