nLab solid abelian group

Idea

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

See also

• condensed abelian group
• solidification functor
• solid spectrum?

References

• Peter Scholze, Lectures on condensed mathematics, pdf