In category theory, a branch of abstract mathematics, a tower is defined as follows. Let I {\displaystyle {\mathcal {I}}} be the poset
⋯ → 2 → 1 → 0 {\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 A {\displaystyle {\mathcal {A}}} is a functor from I {\displaystyle {\mathcal {I}}} to A {\displaystyle {\mathcal {A}}} . In other words, a tower (of A {\displaystyle {\mathcal {A}}} ) is a family of objects { A i } i ≥ 0 {\displaystyle \{A_{i}\}_{i\geq 0}} in A {\displaystyle {\mathcal {A}}} where there exists a map
A i → A j {\displaystyle A_{i}\rightarrow A_{j}} if i > j {\displaystyle i>j}
and the composition
A i → A j → A k {\displaystyle A_{i}\rightarrow A_{j}\rightarrow A_{k}}
is the map A i → A k {\displaystyle A_{i}\rightarrow A_{k}}
Example Let M i = M {\displaystyle M_{i}=M} for some R {\displaystyle R} -module M {\displaystyle M} . Let M i → M j {\displaystyle M_{i}\rightarrow M_{j}} be the identity map for i > j {\displaystyle i>j} . Then { M i } {\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, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4
