A quasi-isomorphism is a map 
of cochain complexes
inducing an isomorphism 
on cohomology
in every degree 
(Stacks Project, Definition 12.13.10). The analogous definition for chain
complexes uses homology. A quasi-isomorphism need
not be an isomorphism in each degree. For example,
the complex 
in degrees 
and 0 maps by reduction modulo 2 to 
in degree 0, with zero terms elsewhere. The map induces
an isomorphism on cohomology,
but its degree-zero map is not injective. The derived
category is obtained by formally inverting quasi-isomorphisms in the homotopy
category.
Quasi-Isomorphism
See also
Chain Complex, Cochain Complex, Cohomology, Derived Category, Homotopy Category, IsomorphismExplore with Wolfram|Alpha
References
Stacks Project. "Complexes." Definition 12.13.10. https://stacks.math.columbia.edu/tag/0115.Cite this as:
Weisstein, Eric W. "Quasi-Isomorphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quasi-Isomorphism.html