Confusion about the differential of the mapping cone

Let $$f:B\to C$$ be a map of chain complexes and let $$C(f)$$ denote the mapping cone of $$f.$$ In Weibel's book, he defines the differential of $$C(f)$$ by the formula \begin{align*} d_{C(f)}:C(f)_n&=B_{n-1}\oplus C_n\to C(f)_{n-1}\\ d_{C(f)}(b,c)&=(-d_B(b),d_C(c)-f(b)). \end{align*} Notice the minus sign attached to the $$d_B.$$ This makes bad things happen, like the fact that the canonical surjection $$C(f)\to B[-1]$$ is not a chain map (it anticommutes with $$d$$). Furthermore, wikipedia gives different signs: $$d_C(f)(b,c)=(-d_B(b),d_C(c)+f(b)).$$ Not only is this fundamentally different from the differential Weibel gives, it doesn't even alleviate the issue that I have!

So here's my question: what are the correct signs for the mapping cone? Furthermore, does it even matter? It's my understanding that one will run in to signs a lot while studying homological algebra, so this may just be a pedantic question altogether. However I'm worried that these signs will neverendingly trip me up if I don't figure them out early on.

It is usual for the differential in $$B[s]$$ to have a minus sign $$(-1)^s$$. I.e., if $$d$$ is one of the differentials of $$B$$, then $$(-1)^sd$$ will be one of the differentials of $$B[s]$$. With this convention, which both Weibel and Wikipedia use, the natural map $$C(f)\to B[-1]$$ is a chain map.
By the way, a warning. You and Weibel use the convention that $$B[1]$$ means "shift one place to the right" (i.e., in the direction of the arrows). I think pretty much everybody else uses the convention that $$B[1]$$ means "shift one place to the left".
The difference between Weibel's $$d_C(c)-f(b)$$ and Wikipedia's $$d_C(c)+f(b)$$. Doesn't really matter. The two versions of the mapping cone are naturally isomorphic via the isomorphism $$B_{n-1}\oplus C_n\to B_{n-1}\oplus C_n$$ given by $$(b,c)\mapsto(b,-c)$$
• I see so there was a different sign that I was forgetting. That clears it up. Thanks! (Also thank you for the note on $B[1]$. That is probably something that would have taken me a while to figure out.) Apr 22 '20 at 15:44