A Categorical Perspective on Gluing

Abstract

This paper introduces the concept of gluing in a general category, enabling us to define categories that admit glued-up objects. To achieve this, we introduce the notion of a gluing index category. Subsequently, we provide an entirely abstract definition of a gluing data functor requiring only the given category to admit pushouts. We explore various characterizations of cones and limits over these functors. We introduce the concept of refined gluing, which in turn enables us to combine different gluing data effectively. Furthermore, we demonstrate that several categories of topological spaces admit glued-up objects. This, in turn, allows us to establish a concept of gluing covering and to prove that the collection of those coverings forms a Grothendieck topology.

Figures (14)

• Figure 1: Diagram representation of a gluing index category with three elements $i, j, k$. 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 (39)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Lemma 1.6
• proof
• Theorem 1.7
• proof
• Remark 1.8