I would like to ask you about a special kind of direct systems $ (A_i, f_{i}^{j} )_{ i,j \in ( I , \leq ) } $ looking like a cochaîn complex $ (A_i , f_{i}^{j} )_{ i,j \in ( \mathbb{N}^* , \leq ) } $ such that : $ f_{i+1}^{i+2} \circ f_{i}^{i+1} = 0_{A_{i}} $ for all $ i \in \mathbb{N}^* $.
How do we obtain generally and explicitly his direct limit : $ A = \displaystyle \lim_{ \to } A_i $ ? Can you give me an example showing me that ?
I would like to ask you the same question but, replacing a direct limit by an inverse limite, and a cochain complex by a chain of complex.
Thanks in advance for your help.
The direct limit is always $0$. More generally, suppose you have a direct system $(A_i, f^j_i)$ such that for each $i$ there is some $j\geq i$ such that $f^j_i=0$ (in your case, you can take $j=i+2$). Then $\varinjlim A_i=0$. Indeed, given any object $B$, a map from the direct system to $B$ consists of maps $g_i:A_i\to B$ for each $i$ such that $g_i=g_jf^j_i$ whenever $i\leq j$. Choosing $j$ such that $f^j_i=0$, you get that $g_i=0$ for all $i$. It follows that every map from the direct system to any object factors uniquely through the $0$ object, so $0$ is the direct limit.