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truncation of a chain complex

Definition

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

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

where $B_n=\mathrm{im}(d_n)$.

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.

References

• Charles Weibel, An Introduction to Homological Algebra