nLab solid abelian group




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


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.


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


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