f703c96ee5ea — Benedikt Fluhr <http://bfluhr.com> 7 years ago
Proof of Compatibility Lemma
1 files changed, 34 insertions(+), 1 deletions(-)

M 00_12_EqualityOfInterlDist.md

M 00_12_EqualityOfInterlDist.md +34 -1
@@ 29,6 29,7 @@ yields the following

(@CpiEIso) **Lemma.**
The homomorphism $(\mathcal{C} \circ \pi \circ \mathcal{E})_f$
+from $\mathcal{C} \mathcal{E} f$ to $\mathcal{C} \mathcal{R} \mathcal{E} f$
is an isomorphism of precosheaves.

Now $\mathbf{C}$ is closed under isomorphisms and thus also

@@ 37,7 38,7 @@ Now we consider the case of $r$ not nece
We note that for any continuous function
$g \colon Y \rightarrow (-\infty, \infty]$ and
$r \in (-\infty, \infty]$ we have
-$r + g = \mathcal{S}((-r, -r)) (g)$
+$r + g = \mathcal{S}((\infty, -r)) (g)$
by definition of $\mathcal{S}$.
We now recall that we also defined the endofunctors
associated to the smoothing functor

@@ 71,6 72,38 @@ We have
(\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b})
\circ \mathcal{E})_f$. +* *Proof.* +First we set$(a', a) := \mathbf{a}$and$(b', b) := \mathbf{b}$. +Now let$r \in \R$. +If we unravel the definitions we obtain +$(\overline{\mathcal{S}}(\mathbf{a}) \pi^2_* \mathcal{C} \mathcal{E} f)(
+  [-\infty, r)
+ ) =
+ \Lambda(\epi f \cap X \times [-\infty, r + a))$+and +$(\overline{\mathcal{S}}(\mathbf{b}) \pi^2_* \mathcal{C} \mathcal{E} f)(
+  [-\infty, r)
+ ) =
+ \Lambda(\epi f \cap X \times [-\infty, r + b))$. +Let$i$be the inclusion of$\epi f \cap X \times [-\infty, r + a)$+into$\epi f \cap X \times [-\infty, r + b)$+and let +$\tau \colon \epi f \cap X \times [-\infty, r + a) \rightarrow
+ \epi f \cap X \times [-\infty, r + b),
+ (p, t) \mapsto (p, t - a + b)$, +then +$(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ
+  \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_{f [-\infty, r)} = \Lambda(i)$+and +$(\pi^2_* \circ \mathcal{C} \circ \mathcal{S}(\mathbf{a} \preceq \mathbf{b})
+  \circ \mathcal{E})_{f [-\infty, r)} = \Lambda(\tau)$. +Now let$(p, t) \in \epi f \cap X \times [-\infty, r + a)$, +then$\{p\} \times [t, t - a + b]$is contained in +$\epi f \cap X \times [-\infty, r + b)$. +Moreover we have +$(p, t), (p, t - a + b) \in \{p\} \times [t, t - a + b]$, +hence$\Lambda(i) = \Lambda(\tau)$. + * **Corollary.** We have$(\mathcal{S}'(\mathbf{a} \preceq \mathbf{b}) \circ