Theorem: Reconstruction descent

Let $\mathsf{Rec}:\mathcal{P}\to \mathcal{K}$ be a surjective linear presentation map. A linear map $\tilde{\mathsf{M}}:\mathcal{P}\to Y$ factors uniquely as

if and only if

If $\mathcal{K}$ has the quotient topology and $\tilde{\mathsf{M}}$ is continuous, the induced map is continuous.

Proof. Factorization annihilates every null presentation. Conversely, two presentations of the same kernel differ by an element of $\ker \mathsf{Rec}$, so (eq:reconstruction-condition) makes their images equal. Set $\mathsf{M}(\mathsf{Rec}p)=\tilde{\mathsf{M}}(p)$. Surjectivity gives uniqueness, and the quotient topology gives continuity.