Simplicial vs. singular compactly supported cohomology

Let \(X\) be a locally compact \(\Delta\)-complex. A compactly supported simplicial cochain is nonzero on only finitely many simplices, since local compactness implies that each compact set meets only finitely many simplices. Hence \[ \Delta_c^i(X;G)=\bigcup_A \Delta^i(X,A;G), \] where \(A\) ranges over subcomplexes containing all but finitely many simplices. The same family is cofinal for compactly supported singular cochains, and \[ C_c^i(X;G)=\bigcup_A C^i(X,A;G). \] These unions are filtered direct limits over \(A\) ordered by reverse inclusion.

For each such subcomplex \(A\), the usual comparison theorem between simplicial and singular cohomology of a \(\Delta\)-complex pair gives an isomorphism \[ H^i_{\Delta}(X,A;G)\xrightarrow{\cong}H^i_{\mathrm{sing}}(X,A;G), \] natural with respect to inclusions of pairs. Since filtered direct limits are exact, cohomology commutes with these direct limits. Therefore \[ H^i(\Delta_c^*(X;G)) \cong \varinjlim_A H^i_{\Delta}(X,A;G) \cong \varinjlim_A H^i_{\mathrm{sing}}(X,A;G) \cong H^i(C_c^*(X;G)). \] Thus the simplicial and singular compactly supported cohomology groups are naturally isomorphic.