Tower of objects

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

⋯->2->1->0

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

In other words, a tower (of 𝒜) is a family of objects {Ai}i≥0 in 𝒜 where there exists a map

Ai->Aj if i>j

and the composition

Ai->Aj->Ak

is the map Ai->Ak

## Example

Let Mi=M for some R-module M. Let Mi->Mj be the identity map for i>j. Then {Mi} 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

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Tower_of_objects&oldid=327979"