Suppose that we have a directed set $(I, \leq)$, and a set of maps \begin{equation} f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j, \end{equation} of singular cochain complexes of topological spaces $A_k$, such that $f_{i,i} = Id_{C^*(A_i)}$, and satisfying the following "homotopy cocycle" property \begin{equation} \text{for all } i \leq j \leq k, \quad f_{j,k} \circ f_{i,j} \text{ is homotopic to } f_{i,k}. \end{equation} Can we define the limit \begin{equation} \underset{i \in I}{\lim} H^*(A_i) \end{equation} of of singular cohomology groups ?
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1$\begingroup$ Yes, in the sense that these maps induce a direct system on homology that you can take the colimit of. $\endgroup$– Phil TostesonCommented Aug 13, 2018 at 13:52
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$\begingroup$ Thanks for your comment. Could you extend your argument please ? I'm not very familiar with these things. $\endgroup$– BrianTCommented Aug 13, 2018 at 14:12
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1$\begingroup$ Homotopic maps induce the same map on homology, therefore after passing to homology you just have an ordinary diagram indexed by $I$. Take the limit of this diagram. $\endgroup$– Phil TostesonCommented Aug 13, 2018 at 16:27
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