nLab solid abelian group

Contents

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

A solid abelian group is a condensed abelian group AA such that for all profinite sets SS and all maps f:SAf:S \to A, there is a unique map g:[S] Ag:\mathbb{Z}[S]^\square \to A such that f=gmf = 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

References

Last revised on May 30, 2022 at 08:57:03. See the history of this page for a list of all contributions to it.