In section 14, an exact couple, \begin{equation} \begin{array}{c} A\overset{i}\longrightarrow A'\\ k\nwarrow\quad\swarrow j\\ B \end{array} \end{equation} is introduced.
I have a question about the map $i$. Is it an inclusion? $i:A\hookrightarrow A'$.I
I also would like to ask follows: Here the author defined the map $d := j\circ k$ and says $d^2 = j(kj)k = 0$. Does the last equality follow from the exactness of the sequence?: ($\text{im}j = \ker k$). \begin{equation} \cdots\rightarrow A\overset{i}\rightarrow A'\overset{j}\rightarrow B \overset{k}\rightarrow A\overset{i}\rightarrow\cdots \end{equation} Is the first triangle diagram a rewritten of this exact sequence? If so, I would like to ask why this definition is useful and its motivation.