Naturality of singular mayer vietoris connectors

Postcomposition by $f$ takes an $X$-simplex in $U$ to a $Y$-simplex in $U^{\mathrm{\prime }}$, and likewise for $V$ and the intersections. It therefore induces chain maps on all three terms of the small-chain sequence. Composition commutes with face restriction, the signed overlap inclusion and sum. Dualizing gives cochain maps $D_Y\to D_X$, $E_Y\to E_X$, $F_Y\to F_X$ commuting with the two arrows in [F1].

By [F2] this cochain diagram commutes with the connectors into $H(D)$. Explicitly, if $be=c$ and $\delta e=a(d)$ in the $Y$ row, the images satisfy the same equations in the $X$ row, so the image of $d$ is the connecting representative of the image of $c$. This uses an image of a supplied lift; it does not claim that zero extension is natural.

Write $I_X,I_Y$ for the small-chain inclusions and $f_{\mathrm{s}\mathrm{m},\mathrm{\#}}$ for the small-chain map. The equation $f_{\mathrm{\#}}{I}_X=I_Y{f}_{\mathrm{s}\mathrm{m},\mathrm{\#}}$ holds on every generator. Hence the canonical isomorphisms $\theta =H(I^{\ast })$ of [F1] commute with pullback. Transporting step 2.1 by their inverses proves the displayed connector identity. The restriction and difference squares already commute by step 1.1, giving the full sequence claim.

Empty intersections yield zero connector domains; empty spaces yield zero groups. In degree zero the same lift calculation applies, with negative primitives zero. On one-point spaces or identical covers the connector is zero by [F1]. Constant or degenerate simplices still satisfy the generator equation in step 3.1. All maps are supplied by postcomposition; neither natural choices of chain inverses nor AC are required.