I am looking for the definition of $End(C_*)$, where $(C_*,d)$ is a chain complex of modules. In degree $n$, $End(C_*)$ is the set of endomorphisms (not chain complexes) $C_* \to C_{*+n}$. It seems like the differential must be of the form $d_{End} : f \longmapsto f \circ d - d \circ f$, with some signs issues depending on the degree $n$. Maybe something like $d_{End} : f \longmapsto (-1)^n f \circ d - d \circ f$. Does somebody can confirm that please ? I canot find any reference.
thanks !
Note that $d$ is a morphism of degree $-1$ (or $1$), depending on your conventions. If $f :C\to C$ is of degree $f$, then to compute $\partial(f)$ you compute a graded commutator $[d,f]$. This is $df-(-1)^{|f|}fd$, since to compute $fd$ from $d\otimes f$ we have to first switch $d\otimes f$ with the braiding given by $\beta(a\otimes b) = (-1)^{|a||b|}b\otimes a$ and then compose. You can also do $[f,d]$, and this will create the same sign, of course, but on $df$.