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Induction in perverse schobers and cluster tilting theory

Merlin Christ

Abstract

We exhibit gluing properties of cluster tilting subcategories in exact $\infty$-categories within the framework of perverse schobers on surfaces with boundary. These results are based on a study of the restriction functors from global sections of perverse schobers to local sections and their adjoint induction functors. New examples include cluster tilting subcategories in higher rank topological Fukaya categories related to higher Teichmüller theory, and cluster tilting subcategories arising from marked surfaces with punctures.

Induction in perverse schobers and cluster tilting theory

Abstract

We exhibit gluing properties of cluster tilting subcategories in exact -categories within the framework of perverse schobers on surfaces with boundary. These results are based on a study of the restriction functors from global sections of perverse schobers to local sections and their adjoint induction functors. New examples include cluster tilting subcategories in higher rank topological Fukaya categories related to higher Teichmüller theory, and cluster tilting subcategories arising from marked surfaces with punctures.

Paper Structure

This paper contains 22 sections, 30 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Induction and restriction functors for perverse schobers
  3. Gluing cluster tilting subcategories
  4. Examples
  5. Outlook
  6. Higher categorical preliminaries
  7. Limits of infinity-categories via sections of the Grothendieck construction
  8. Exact infinity-categories and cluster tilting subcategories
  9. Perverse schobers on marked surfaces
  10. Surfaces, ribbon graphs and induced trajectories
  11. Parametrized perverse schobers
  12. Induction functors for parametrized perverse schobers
  13. Boundary cap and cup functors
  14. Induction from vertices and internal edges
  15. Induction for subgraphs/subsurfaces
  16. ...and 7 more sections

Key Result

Theorem 1

Let ${\bf G}$ be a spanning ribbon graph of a marked surface without $1$-valent vertices. Let $\mathcal{F}$ be a ${\bf G}$-parametrized perverse schober. Given for every vertex $v\in {\bf G}_0$ a cluster tilting subcategory $\mathcal{T}_v\subset \mathcal{F}(v)$, the additive closure of the union of defines a cluster tilting subcategory of the Frobenius exact $\infty$-category $\mathcal{H}({\bf G}

Figures (5)

  • Figure 1: The line field $\nu_{\bf G}$ for a spanning graph ${\bf G}$ of the $5$-gon.
  • Figure 2: The clockwise web trajectory $\gamma^\circlearrowright_v$ starting at a trivalent vertex $v$ and the clockwise curve trajectory $\gamma^\circlearrowright_e$ starting at an incident edge $e$.
  • Figure 3: An example of a ribbon graph where the external successor edge $e_m$ of the external edge $e$ coincides with $e$. The corresponding clockwise trajectory $\gamma^\circlearrowright_e$ is in blue.
  • Figure 4: The vanishing paths associated with thimbles arising from left induction are described by clockwise trajectories.
  • Figure 5: A collection of curve and web trajectories in the $4$-gon corresponding to a cluster tilting object in $\operatorname{Fun}(\Delta^2,\operatorname{CoSing}(\Pi_2(A_2)))$.

Theorems & Definitions (95)

  • Theorem 1
  • Proposition 1: See \ref{['prop:evv']}
  • Definition 2.1: $\!\!$Bar15
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6: $\!\!$Chr22b
  • Definition 2.7
  • Definition 2.8
  • ...and 85 more