From inverse to coherent systems (Q5936534)

The author proves the following two results: (1) Let \(X=(X_n, p_{n, m})\) be a commutative inverse sequence, let \(f_n:X_n\to Y_n\) be a homotopy equivalence with a homotopy inverse \(g_n:Y_n\to X_n\). Let \(q_{n, m}=f_n\circ p_{n, m}\circ g_m\) for \(n<m\) and let \(Y=(Y_n, q_{n, m})\). Then \(X\) and \(Y\) are isomorphic in the level category of strong fundamental sequences (and inverse sequences commutative up to homotopy). (2) The level coherent category is isomorphic to the level category of coherent inverse sequences and strong fundamental sequences.