Let
$\alpha \colon \k (\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) \to \k (\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$
be the $\k$-linear autofunctor
flipping the signs of arrows in every other column of $\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$.
Then $\alpha$ turns most of the mesh relations
into commutativity relations.
We transform the $k$-linear Happel functor
$H \colon \k \langle \Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q} \rangle \to
D^b(\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$
to an ordinary functor
$H'' \colon \L \to D^b(\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$
vanishing on $\partial \L$.
The lattice
$\M \subset \R^{\circ} \times \R$
is our continuously indexed counterpart to $\L$.
The glide reflection
$\Sigma \colon \M \to \M,\, u \mapsto \Sigma(u)$
corresponds to the automorphism
of $\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$ that is induced by suspension.
The open sets of
$\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}$
are the open downsets of $\htmlStyle{color: var(--color-quiver, #007aff);}{q}$.
$D^+_t(\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}, \htmlStyle{color: var(--color-quiver, #007aff);}{\partial q})$
is the full subcategory of "derived sheaves"
in $D^+(\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}})$
vanishing on
$\htmlStyle{color: var(--color-quiver, #007aff);}{\partial q}$
with degree-wise finite-dimensional sheaf cohomology
on connected opens.
$\textcolor{DimGrey}{D}
:= \htmlStyle{color: var(--color-quiver, #007aff);}{C} \setminus \htmlStyle{color: var(--color-shifted-downset, #007355);}{\Sigma^{-1}(C)}$:
fundamental domain with respect to
$\langle \Sigma \rangle \curvearrowright \M$
corresponds to the Auslander-Reiten quiver
\[
\iota_0 \colon D \rightarrow \mathrm{Sh}_t(\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}, \htmlStyle{color: var(--color-quiver, #007aff);}{\partial q}),\,
u \mapsto \k_{\htmlStyle{color: var(--color-quiver, #007aff);}{Z(u)}}
\]
Using $\partial'$ to connect the tiles we obtain a functor
\[
\iota \colon \M \rightarrow \Der_t (\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}, \htmlStyle{color: var(--color-quiver, #007aff);}{\partial q})
\]
with
$
\iota \circ \Sigma = \Sigma \circ \iota = \iota(-)[1]
\quad \text{and} \quad
\iota |_D = \iota_0
.
$
As $\iota$ vanishes on $\partial \M$
it can be transformed to a $\k$-linear functor:
Proposition
The $\k$-linear functor $\iota^{\flat}$
yields an equivalence of categories:
\[
\iota^{\flat}
\colon
\k \mathbb{M} / \partial \mathbb{M}
\xrightarrow{\sim~}
\mathrm{ind}\left(D^+_t (q_{\gamma}, \partial q)\right)
.
\]
Theorem
The category
$D^+_t (q_{\gamma}, \partial q)$
is equivalent to the corresponding category
for any other choice of $C$.
If $\htmlStyle{color: var(--color-quiver, #007aff);}{\partial q}$ is closed in $\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}$,
then there is an adjoint equivalence
Corollary
There is a space $q_{\gamma}$
with totally ordered topology
such that
$D^+_t (q_{\gamma}, \partial q)$
is equivalent to
$D^+_t (\R)$.
Roughly speaking,
Leray sheaves of Morse functions,
aka
level set persistent cohomology,
can be seen as objects of
$D^+_t (\R)$
and extended persistent cohomology as objects of
$D^+_t (q_{\gamma}, \partial q)$
for a $q_{\gamma}$ with totally ordered topology.
