Tower (mathematics)

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}

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, 38, Cambridge University Press, ISBN 978-0-521-55987-4

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