# Clutching and gluing in tropical and logarithmic geometry

## TL;DR

This paper introduces sub-logarithmic morphisms to extend clutching and gluing maps in tropical and logarithmic geometry, enabling these maps to be compatible with tropicalization and enriching the structure of moduli stacks.

## Contribution

It develops a new sub-logarithmic framework that makes classical clutching and gluing maps compatible with logarithmic and tropical geometry.

## Key findings

- Clutching and gluing maps are shown to be sub-logarithmic.
- The framework ensures compatibility with tropicalization.
- Enables new stack-theoretic constructions in tropical geometry.

## Abstract

The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in the tropical world and show that the resulting maps naturally commute with the process of tropicalization.