Tower of objects

In category theory, a branch of abstract mathematics, a tower is defined as follows. Let ${\displaystyle \mathcal{I}}$ 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 ${\displaystyle \mathcal{A}}$ is a functor from ${\displaystyle \mathcal{I}}$ to ${\displaystyle \mathcal{A}}$.

In other words, a tower (of ${\displaystyle \mathcal{A}}$) is a family of objects ${\displaystyle \{A_i{\}}_{i\ge 0}}$ in ${\displaystyle \mathcal{A}}$ 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 ${\displaystyle A_i\to A_k}$

Example

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.

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 

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Tower of objects. Read more