Is this a typo in Weibel, page 1?

Question

It says a morphism $u : C_{\cdot } \to D_{\cdot}$ of chain complexes is a family of homomorphisms $u_n : C_n \to D_n$ such that $u_{n-1} d_n = d_{n-1} u_{n}$, but shouldn't it just be that $u_{n-1} d_n = d_n u_{n}$? I want to make sure my understanding of the differentials $d$ is correct before moving on.

Answer 1

What is written in the text is an error. The composition $d_{n-1}u_n$ doesn't make sense since $u_n:C_n\to D_n$ and $d_{n-1}:D_{n-1}\to D_{n-2}$. It should be $d_nu_n$. The convention is that the index of the map should match the index on the domain. It's often easier, as sometimes done in that book, just to write $ud = du$ and omit the indices altogether.

The error is also already written in his 1995 version errata sheet. If you have the paper copy I just recommend fixing the errors in each chapter before you read it.