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**Related concepts**

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

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.

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 $f = g \circ m$.

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.

- condensed abelian group
- solidification functor
- solid spectrum?

- Peter Scholze,
*Lectures on condensed mathematics*, pdf

Last revised on April 14, 2023 at 21:14:44. See the history of this page for a list of all contributions to it.