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A note on gluing: a pillar of algebraic geometry

Sophie Marques, Damas Mgani

TL;DR

This work develops a comprehensive categorical framework for gluing in algebraic geometry, introducing gluing functors on indexing categories $\mathbb{P}_2(\mathrm{I})$ and $\mathbb{S}_2(\mathrm{I})$ and showing that glued objects arise as limits or equalizers. It connects gluing to Grothendieck topologies, effective descent data, and presheaf theory, and it develops universal and strong notions of gluing, refinements, and composition. The results include criteria for when gluing is representable as an equalizer, examples of gluable categories, and a systematic treatment of gluing for (pre)sheaves and enriched structures, providing a robust foundation for descent and descent-like constructions in algebraic geometry. Overall, the paper clarifies the logical structure of gluing, links it to standard descent data, and extends the framework to enriched and presheaf contexts with explicit canonical representations.

Abstract

This paper examines the concept of gluing, placing it within its most general categorical context and tracing its foundational role in the broader architecture of algebraic geometry.

A note on gluing: a pillar of algebraic geometry

TL;DR

This work develops a comprehensive categorical framework for gluing in algebraic geometry, introducing gluing functors on indexing categories and and showing that glued objects arise as limits or equalizers. It connects gluing to Grothendieck topologies, effective descent data, and presheaf theory, and it develops universal and strong notions of gluing, refinements, and composition. The results include criteria for when gluing is representable as an equalizer, examples of gluable categories, and a systematic treatment of gluing for (pre)sheaves and enriched structures, providing a robust foundation for descent and descent-like constructions in algebraic geometry. Overall, the paper clarifies the logical structure of gluing, links it to standard descent data, and extends the framework to enriched and presheaf contexts with explicit canonical representations.

Abstract

This paper examines the concept of gluing, placing it within its most general categorical context and tracing its foundational role in the broader architecture of algebraic geometry.
Paper Structure (24 sections, 11 theorems, 38 equations, 4 figures)

This paper contains 24 sections, 11 theorems, 38 equations, 4 figures.

Table of Contents

  1. Notation
  2. Defining gluing categorically
  3. Categories of interest
  4. Truncated power set category of $\mathrm{I}$
  5. Split truncated power set category of $\rm{I}$ of order $2$
  6. Gluing functors
  7. Universal Gluing
  8. Glued up objects as equalizer
  9. Examples of gluable categories
  10. $\textsf{Hom}$ functor induced by a gluing functor
  11. Operations between gluing functors
  12. Refinements
  13. Formally composing gluing
  14. Grothendieck topologies
  15. Canonical functor associated with a sink
  16. ...and 9 more sections

Key Result

Proposition 2.13

Let $\mathbb{C}$ be a category that admits products. Suppose $\mathbf{G}$ is a $\mathbb{C}$-gluing functor of type $\mathrm{I}$, and let $L \in \mathbb{C}$ with a family of morphisms ${(\pi_i)}_{\substack{\text{$\mkern-0.8mu i\! \in\! \mathrm{I}$}}}$, where $\pi_i: L \to \mathbf{G}(i)$ is a morphism

Figures (4)

  • Figure 1: Gluing a square into a torus
  • Figure 2: Diagram representation of $\mathcal{G}$ and its limits, where the top pushout squares each describe $\mathbf{G}$, the bottom-left pushout square represents ${\mathbf{G}}_{\substack{\text{$\mkern-0.8mu V_1$}}}$, the bottom-right pushout square represents ${\mathbf{G}}_{\substack{\text{$\mkern-0.8mu V_2$}}}$, the pointed arrows indicate the refinement maps, the bold front diagram describes $\lim \mathcal{G}$ and the other arrows are as described above.
  • Figure 3: Representation of the process of an effective gluing of three topological spaces in $\mathbb{Top}^{\operatorname{op}}$
  • Figure 7: Gluing diagram for natural transformations

Theorems & Definitions (64)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Example 2.10
  • ...and 54 more