# nLab truncation of a chain complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

For ${C}_{•}$ a chain complex, it truncation $\left({\tau }_{\le }C{\right)}_{•}$ at some $n\in ℕ$ is the chain complex defined by

$\left({\tau }_{n}C{\right)}_{i}=\left\{\begin{array}{cc}0& \mid i>n\\ {C}_{n}/{B}_{n}& \mid i=n\\ {C}_{n}& \mid i(\tau_n C)_i = \left\{ \array{ 0 & | i \gt n \\ C_n/B_n & | i = n \\ C_n & | i \lt n } \right. \,,

where ${B}_{n}=\mathrm{im}\left({d}_{n}\right)$.

For connective chain complexes this is the notion of truncated object in an (infinity,1)-category realized in the (infinity,1)-category of chain complexes.

## References

Created on August 24, 2012 14:13:43 by Urs Schreiber (82.113.106.22)