# A Sheaf-Theoretical Happel Functor

• Ulrich Bauer
• Benedikt Fluhr

January 27th, 2023

https://bfluhr.com/ARTNeM-23/#title

• $\htmlStyle{color: var(--color-quiver, #007aff);}{Q}$: $A_n$-quiver
• $\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$: path algebra
• $D^b(\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$: (bounded) derived category of finite-dimensional right $\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$-modules
• $\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$: repetition quiver
• $\k (\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$: path category of $\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$
• $\mathrm{mesh}(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$: mesh ideal of $\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}$
• $\k \langle \Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q} \rangle \coloneqq \k (\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) / \mathrm{mesh}(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$: mesh category of $\htmlStyle{color: var(--color-quiver, #007aff);}{Q}$
• $H \colon \k \langle \Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q} \rangle \xrightarrow{\sim~} \mathrm{ind}(D^b(\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q})) \hookrightarrow D^b(\k \htmlStyle{color: var(--color-quiver, #007aff);}{Q})$: Happel functor

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.

Let $\mathrm{mesh}'(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) \coloneqq \alpha(\mathrm{mesh}(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}))$. Then $\alpha$ induces a $\k$-linear isofunctor $\bar{\alpha} \colon \k \langle \Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q} \rangle = \k(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) / \mathrm{mesh}(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) \to \k(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) / \mathrm{mesh}'(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) .$

We have the lattice $\L$ and the $k$-linear functor $\beta \colon \k(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) \to \k \L$.

$\beta$ induces a fully faithful functor $\bar{\beta} \colon \k(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) / \mathrm{mesh}'(\Z \htmlStyle{color: var(--color-quiver, #007aff);}{Q}) \to \k \L / \partial \L$ merely adjoining zero objects.

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.

$\htmlStyle{color: var(--color-quiver, #007aff);}{C} \subset \M$ closed proper downset

$\htmlStyle{color: var(--color-quiver, #007aff);}{q \coloneqq \partial C}$

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

$\iota_0 \colon D \rightarrow \mathrm{Sh}_t(\htmlStyle{color: var(--color-quiver, #007aff);}{q_{\gamma}}, \htmlStyle{color: var(--color-quiver, #007aff);}{\partial q}),\, u \mapsto \k_{\htmlStyle{color: var(--color-quiver, #007aff);}{Z(u)}}$

The short exact sequence $0 \rightarrow \iota_0 (u) \xrightarrow{\begin{pmatrix} 1 \\ 1 \end{pmatrix}} \iota_0 (v_1) \oplus \iota_0 (v_2) \xrightarrow{\begin{pmatrix} 1 & -1 \end{pmatrix}} \iota_0 (w) \rightarrow 0$

yields the distinguished triangle $\iota_0 (u) \xrightarrow{\begin{pmatrix} 1 \\ 1 \end{pmatrix}} \iota_0 (v_1) \oplus \iota_0 (v_2) \xrightarrow{\begin{pmatrix} 1 & -1 \end{pmatrix}} \iota_0 (w) \xrightarrow{\partial'} \iota_0 (u)[1] .$

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.

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