Interleavings and cluster categories
arising from the circle

Université de Sherbrooke

Christian Chávez
Université de Sherbrooke RTA2026

Plan

  • $\operatorname{Fun}(\R, \operatorname{vect}_{\kk})$ and $\mathcal{C}_{A_n}$
  • Rolling the line onto a circle
  • The category of chords
  • Connection to $\mathcal{C}_{A_n}$
  • The interleaving distance on $S^1$

Persistence modules

Cluster categories

$\operatorname{Fun}(\R, \operatorname{vect}_{\kk})$

$\mathcal{C}_{A_n}$

$A_n\colon\ \circ \to \circ \to \cdots \to \circ$

ind: interval modules $\kk^{[a,b]}$
ind: diagonals in the $(n+3)$-gon

$A_2\colon\ \circ \to \circ$

Interleavings

Definition

A $t$-interleaving of $\kk^I$ and $\kk^J$ is a pair of morphisms ${\phi\colon \kk^I \to \kk^J[t],\ \psi\colon \kk^J \to \kk^I[t]}$ that make the following diagrams commute.

$\kk^I$
$\kk^I[2t]$
$\kk^J[t]$
$\kk^I[0\to 2t]$
$\phi$
$\psi[t]$
$\kk^J$
$\kk^J[2t]$
$\kk^I[t]$
$\kk^J[0\to 2t]$
$\psi$
$\phi[t]$

The interleaving distance is $$d(\kk^I, \kk^J) = \inf\{\, t \geqslant 0 \mid \kk^I \text{ and } \kk^J \text{ are } t\text{-interleaved} \,\}.$$

Here $\kk^I[t] = \kk^{I-t}$.

Definition

The interleaving distance is $d(\kk^I, \kk^J) =$ “the least energy needed to align $I$ and $J$ to each other”.

$[c,d]$ is aligned to $[a,b]$ $\iff c \leqslant a \leqslant d \leqslant b \iff \Hom\bigl(\kk^{[a,b]}, \kk^{[c,d]}\bigr) \neq 0$

Rolling the line onto a circle

Rolling the line onto a circle

Admissible bijections

Definition
The category $\X$ consists of the following data:
  1. Objects: two-element subsets of $S^1$, called chords.
  2. Morphisms: for two chords $A$ and $B$, we define $\Hom(A,B)$ as the free $\kk$-vector space with basis $\operatorname{Adm}(A,B)$. The identity morphism of $A$ is the admissible bijection $\operatorname{Id}_A$.
  3. Composition: for two basis morphisms $f \in \operatorname{Adm}(A,B)$ and $g \in \operatorname{Adm}(B,C)$ we define $$g \circ f = \begin{cases} gf & \text{if } gf \in \operatorname{Adm}(A,C), \\ 0 & \text{else.} \end{cases}$$ We extend $\circ$ bilinearly.

The Auslander–Reiten quiver of $\X_5$

Irreducible morphisms move one endpoint of a chord to the next vertex counterclockwise, keeping the other fixed. Identical chords are identified.

McFaddin, Needham: Interleaving distances, monoidal actions and 2-categories arXiv:2311.11936

The interleaving distance in general

  • Let $(G, \otimes, e)$ be a monoidal category. A $G$-actegory is a pair $(\mathbf{A}, T)$ where $\mathbf{A}$ is a category and $T$ is a monoidal functor $G \to \operatorname{End}(\mathbf{A})$.
  • A monoidal weight on $G$ is a function $w\colon G_0 \to [0, \infty]$ such that $w(e) = 0$ and ${w(g \otimes h) \leqslant w(g) + w(h)}$ for all $g, h \in G_0$.
Definition
Let $T\colon G \to \operatorname{End}(\mathbf{A})$ be a monoidal functor and $w$ a monoidal weight on $G$.
  1. Two objects $A, B \in \mathbf{A}$ are $(g, h)$-interleaved for $g, h \in G_0$, if there exist ${\phi\colon A \to T_g(B),\ \psi\colon B \to T_h(A)}$ in $\mathbf{A}$ and ${\alpha\colon e \to g \otimes h,\ \beta\colon e \to h \otimes g}$ in $G$ such that the diagrams
    $A$
    $T_{g\otimes h}(A)$
    $T_g(B)$
    $T(\alpha)_A$
    $\phi$
    $T_g(\psi)$
    $B$
    $T_{h\otimes g}(B)$
    $T_h(A)$
    $T(\beta)_B$
    $\psi$
    $T_h(\phi)$
         commute.
  2. The associated $G$-interleaving distance is the function $d\colon \mathbf{A}_0 \times \mathbf{A}_0 \to [0, \infty]$ given by $$d(A, B) = \inf\{\, \max\{w(g), w(h)\} \mid A \text{ and } B \text{ are } (g, h)\text{-interleaved} \,\}.$$

Rotation of a chord

Caldero, Chapoton, Schiffler: Quivers with relations arising from clusters (A_n case)arXiv:math/0401316
Igusa, Todorov: Continuous Frobenius categoriesarXiv:1209.0038
Garcia, Igusa: Continuously triangulating the continuous cluster categoryarXiv:1907.11365

Persistence modules

The category of chords

Cluster categories

$\operatorname{Fun}(\R, \operatorname{vect}_{\kk})$

$\X$

$\mathcal{C}_{A_n}$

$A_n\colon\ \circ \to \circ \to \cdots \to \circ$

ind: interval modules $\kk^{[a,b]}$
ind: chords
ind: diagonals in the $(n+3)$-gon

$A_2\colon\ \circ \to \circ$

If you want to learn more...
QR code to the blog post on the interleaving distance between interval modules