Tower (Mathematics)

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

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

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

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

${\displaystyle A_i\to A_j}$ if ${\displaystyle i>j}$

and the composition

${\displaystyle A_i\to A_j\to A_k}$

is the map ${\displaystyle A_i\to A_k}$

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

• 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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