Proposition: Descent onto the reconstructed image

Let ${\mathcal{P}}_{\mathrm{int}}$ be a vector space of presentations, let

be linear, and let

be linear. There is a unique linear map

satisfying $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$ if and only if

If ${\mathcal{K}}_{\mathrm{int}}=\mathrm{im}\mathsf{Rec}$, this is descent onto ${\mathcal{K}}_{\mathrm{int}}$. If not, uniqueness is asserted only on the reconstructed image.

Proof. If the factorization exists and $\mathsf{Rec}A=0$, then $\tilde{\mathsf{M}}(A)=\mathsf{M}(0)=0$.

Conversely, assume (eq:descent). Define

If $\mathsf{Rec}A=\mathsf{Rec}B$, then $A-B\in \ker \mathsf{Rec}$, so their proposed images agree. The map is therefore well defined, linear, and uniquely determined on $\mathrm{im}\mathsf{Rec}$.