Theorem: Exact relative-lifting criterion

For Definition def:obstruction:

1. The class ${\mathrm{Ob}}_{\mathcal{A}}(F_0)$ is unchanged if $F_0$ is replaced by another element of $F_0+\mathcal{A}$.

2. A reconstruction-descending member of $F_0+\mathcal{A}$ exists if and only if

   $${\mathrm{Ob}}_{\mathcal{A}}(F_0)=0.$$

3. An individual candidate $F_0+a$ descends if and only if

   $$d_0+\partial a=0.$$

4. If $F_{\ast }$ is one descending candidate, all descending candidates are exactly

   $$F_{\ast }+{\mathcal{A}}_0,{\mathcal{A}}_0=\{a\in \mathcal{A}:a{|}_H=0\}.$$

5. Every descending candidate induces a unique map $\mathsf{M}:{\mathcal{K}}_{\mathrm{int}}\to V$ agreeing with $m^{\circ }$ after restriction to $O$. The induced ambiguity consists precisely of those maps $\bar{a}:{\mathcal{K}}_{\mathrm{int}}\to Y$ for which $\bar{a}\mathsf{Rec}\in {\mathcal{A}}_0$.

Proof. If $F_1=F_0+a_1$, then

which proves (i).

A candidate annihilates $H$ precisely when (eq:cancellation) holds. Such an $a$ exists exactly when $d_0\in \partial \mathcal{A}$, because $\partial \mathcal{A}$ is a vector subspace. This proves (ii) and (iii) by Proposition prop:descent.

If $F_{\ast }$ annihilates $H$, then $F_{\ast }+a$ annihilates $H$ exactly when $a{|}_H=0$, proving (iv).

Every correction takes values in $\ker r$, so

Proposition prop:descent gives the claimed agreement. The same proposition factors every $a\in {\mathcal{A}}_0$ uniquely through $\mathsf{Rec}$, and conversely any stated $\bar{a}$ gives such a correction. This proves (v).