truncation of a chain complex

(also nonabelian homological algebra)

For $C_\bullet$ a chain complex, the *truncation* $(\tau_{\leq} C)_\bullet$ at some $n \in \mathbb{N}$ is the chain complex defined by

$(\tau_n C)_i =
\left\{
\array{
0 & | i \gt n \\
C_n/B_n & | i = n \\
C_n & | i \lt n
}
\right.
\,,$

where $B_n = im(d_n)$.

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.

Last revised on September 13, 2016 at 08:48:02. See the history of this page for a list of all contributions to it.