When constructing the cone of a morphism $f:M\rightarrow N$ (also called the mapping cone), a new complex $M[1]$ is constucted from $M$ via $$M[1]^i=M^{i+1}$$ with differentials $$d_{M[1]}^i=-d_M^{i+1}.$$ My question is why is there a change of sign?
Thanks in advance.
The change of sign is a convention, and does not affect the isomorphism class of the chain complex. The chain complex $C$ with groups $C^i = M^{i+1}$ and differentials $d_C^i = d_M^{i+1}$ is isomorphic to $M[1]$ by the chain map which is multiplication by $(-1)^i$ in degree $i$.
Likely, the choice of sign is to simplify notation later on, or to be consistent with other constructions (e.g. the tensor product of chain complexes).
