Morphisms of generalized affine buildings

TL;DR

This paper introduces a new framework for morphisms between generalized affine buildings, extending existing concepts, and explores conditions for their existence and applications in various geometric and algebraic contexts.

Contribution

It defines a notion of morphism for generalized affine buildings, broadening the theoretical foundation and connecting different types of these structures.

Findings

Established sufficient conditions for morphism existence

Analyzed morphisms between buildings from different literature sources

Demonstrated functoriality in non-standard symmetric spaces

Abstract

We define a notion of morphism for generalized affine buildings, also known as affine $\Lambda$-buildings, extending existing definitions and giving rise to a category of generalized affine buildings. For affine $\Lambda$-buildings equipped with a transitive group action, we provide sufficient conditions for the existence of morphisms between them. As an application, we investigate under which conditions morphisms or isomorphisms between various generalized affine buildings from the literature (defined via lattices, norms, non-standard symmetric spaces, or \`a la Bruhat-Tits) can be defined. For generalized affine buildings coming from non-standard symmetric spaces we further show functoriality for subgroups and under change of valued field.