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Adjoints, wrapping, and morphisms at infinity

Tatsuki Kuwagaki, Vivek Shende

Abstract

For a localization of a smooth proper category along a subcategory preserved by the Serre functor, we show that morphisms in Efimov's algebraizable categorical formal punctured neighborhood of infinity can be computed using the natural cone between right and left adjoints of the localization functor. In particular, this recovers the following result of Ganatra--Gao--Venkatesh: morphisms in categorical formal punctured neighborhoods of wrapped Fukaya categories are computed by Rabinowitz wrapping.

Adjoints, wrapping, and morphisms at infinity

Abstract

For a localization of a smooth proper category along a subcategory preserved by the Serre functor, we show that morphisms in Efimov's algebraizable categorical formal punctured neighborhood of infinity can be computed using the natural cone between right and left adjoints of the localization functor. In particular, this recovers the following result of Ganatra--Gao--Venkatesh: morphisms in categorical formal punctured neighborhoods of wrapped Fukaya categories are computed by Rabinowitz wrapping.
Paper Structure (2 sections, 6 theorems, 50 equations)

This paper contains 2 sections, 6 theorems, 50 equations.

Table of Contents

  1. Compositions in $\widehat{S}_\infty$
  2. $\operatorname{Hom}_{\mathbb{K}[t]}(\mathbb{K}[t, t^{-1}], \mathbb{K}[t])$

Key Result

Theorem 1

Assume given a sequence as in seq, such that $\mathcal{C}$ is smooth and locally proper, and the Serre functor of $\mathcal{C}$ preserves $\mathcal{K}$. Let $i:\mathrm{Mod}\, \mathcal{S} \to \mathrm{Mod}\, \mathcal{C}$ be the pullback functor on module categories. Then for $c, d \in \mathcal{C}$, th where the map is induced by the unit maps $c \to ii^L(c)$ and $d \to ii^L(d)$.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Example 4: Coherent sheaves
  • Lemma 5
  • proof
  • Example 6: Fukaya categories
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 7 more