Definition of an exact sequence

Question

I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf

I see the following definitions for exact sequences and short exact sequences :

But then later on I see this:

This is where I get confused :

• How can that sequence be exact ? from the earlier definition I thought that this would require 2 mappings ?

• why is g surjective ? I see surjectivity appear in the context of short exact sequences, not in the context of (merely) exact sequences (and that sequence is not a short exact sequence either since it does not start with 0 as per the earlier definition ?)

Answer 1

That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.