Tower (mathematics)

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

[math]\displaystyle{ \cdots\rightarrow 2\rightarrow 1\rightarrow 0 }[/math]

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

In other words, a tower (of [math]\displaystyle{ \mathcal A }[/math]) is a family of objects [math]\displaystyle{ \{A_i\}_{i\geq 0} }[/math] in [math]\displaystyle{ \mathcal A }[/math] where there exists a map

[math]\displaystyle{ A_i\rightarrow A_j }[/math] if [math]\displaystyle{ i\gt j }[/math]

and the composition

[math]\displaystyle{ A_i\rightarrow A_j\rightarrow A_k }[/math]

is the map [math]\displaystyle{ A_i\rightarrow A_k }[/math]

Example

Let [math]\displaystyle{ M_i=M }[/math] for some [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M }[/math]. Let [math]\displaystyle{ M_i\rightarrow M_j }[/math] be the identity map for [math]\displaystyle{ i\gt j }[/math]. Then [math]\displaystyle{ \{M_i\} }[/math] 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