Contents

group theory

# Contents

## Idea

Solid abelian groups are objects which form the basis of non-archimedean analysis in certain approaches to mathematics, such as condensed mathematics.

## Definition

### As a functor

A solid abelian group is an additive functor from the category of free abelian groups to the category of abelian groups.

This definition is due to Dustin Clausen here.

### In terms of condensed abelian groups

Let $S = \underset{\leftarrow}{\lim}_i S_i$ be a profinite set, and define the condensed abelian group $\mathbb{Z}[S]^\square \coloneqq \underset{\leftarrow}{\lim}_i \mathbb{Z}[S_i]$, where $\mathbb{Z}[T]$ is the free abelian group on the set $T$. There is a natural map $m:S \to \mathbb{Z}[S]^\square$ which induces a map $\mathbb{Z}[S] \to \mathbb{Z}[S]^\square$.

A solid abelian group is a condensed abelian group $A$ such that for all profinite sets $S$ and all maps $f:S \to A$, there is a unique map $g:\mathbb{Z}[S]^\square \to A$ such that $f = g \circ m$.

## Properties

According to Peter Scholze in this comment on the nCafé in the absense of the presentation axiom, the category of solid abelian groups is not a condensed category.