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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