Gluing Data categories and gluing data functors

Abstract

We present a novel approach to the concept of gluing in mathematics by introducing the notions of a gluing data category and a gluing data functor. Our work provides a formal categorical characterization of the notion of gluing in algebraic geometry. By using this characterization, we are able to describe gluing in a unified way that applies to a wide range of mathematical structures, including topological spaces, presheaves, sheaves, ringed topological spaces, locally ringed topological spaces, and schemes. Our results provide a fresh perspective on gluing that is both abstract and formal, offering a deeper understanding of this fundamental concept in mathematics.

Figures (12)

• Figure 1: Diagram representation of a gluing Data category of type $\mathrm{I}=\{ i, j, k\}$. The identity maps on every element are not presented in this diagram for the sake of clarity. The arrows in both directions each compose into the identity map.
• Figure 2: Representation for gluing and the glued-up object in $\mathbf{Top}^{\substack{\text{$\operatorname{op}$}}}$ is as follows: The glued-up object $Q$ is situated in the center and is obtained from the three topological spaces, namely $U_1$, $U_2$, and $U_3$, which are mapped via the limit maps to $Q_1$, $Q_2$, and $Q_3$ respectively, forming a covering of $Q$. Moreover, $U_{12}$ is glued with $U_{21}$ and mapped to $Q_1\cap Q_2$, $U_{13}$ is glued with $U_{31}$ and mapped to $Q_1\cap Q_3$, and $U_{32}$ is glued with $U_{23}$ and mapped to $Q_2\cap Q_3$. Furthermore, all the double intersections, namely $U_{12}\cap U_{13}$, $U_{21}\cap U_{23}$, and $U_{31}\cap U_{32}$, are all mapped to the triple intersection $Q_1\cap Q_2 \cap Q_3$.
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Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Theorem 2.8
• proof