Proposition: From values to maps: what is and is not fixed

1. If each ${\mathcal{W}}_{\Gamma }$ is a vector space, then ${\mathcal{C}}_{\mathrm{part}}$ is a linear subspace of ${\mathcal{D}}^{\prime }$ (finite sums of linear images), and for any presentation space ${\mathcal{P}}_{\mathrm{int}}$ and $Y=\ker r_O$ the set

   $${\mathcal{A}}_{\max }:=\{a\in \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y):a({\mathcal{P}}_{\mathrm{int}})\subseteq {\mathcal{C}}_{\mathrm{part}}\}$$

   is a vector space; it is the largest class satisfying hypothesis (H2) of the patch's conditional descent test.

2. XIII-C defines no map on presentations: it states neither linearity, nor continuity, nor tensor type, nor covariance for any $a\in \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$, and it does not state that the frozen class is ${\mathcal{A}}_{\max }$ or any subspace of it. No exhaustiveness theorem is proved there, and none is claimed here.

3. Since ${\mathcal{P}}_{\mathrm{int}}$ is open (Proposition prop:Q2), ${\mathcal{A}}_{\max }$ is not yet an object.

Proof. (i) Linear images of vector spaces under $w\mapsto D_{\Gamma }[w{\partial }^{\alpha }\delta (r_I)S_{\Gamma /I}]$ and finite sums are vector spaces; preimage conditions $a({\mathcal{P}}_{\mathrm{int}})\subseteq {\mathcal{C}}_{\mathrm{part}}$ for a subspace ${\mathcal{C}}_{\mathrm{part}}$ define a subspace of $\mathrm{Hom}$. (ii), (iii) are the record and Proposition prop:Q2.