nLab condensed mathematics




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



Condensed mathematics aims to provide a more convenient framework in which to treat algebraic objects which are equipped with a topology, such as topological abelian groups or topological vector spaces. Related aims are to turn functional analysis into a branch of commutative algebra, and various types of analytic geometry into algebraic geometry.

For instance, the category of topological abelian groups is not abelian, which means for instance that it lacks a good theory of exact sequences. But the category of condensed abelian groups is abelian and it contains topological abelian groups as a full subcategory (under appropriate conditions, see ScholzeLCM, Proposition 1.7).

According to Peter Scholze in this comment on the nCafé, current expositions of condensed mathematics rely heavily on the axiom of choice and choice-like axioms such as the presentation axiom, so it is unknown how much they still hold in the context of more constructive mathematics without choice.


Condensed objects

Solid objects

Liquid objects


Higher condensed objects


The theory of solid modules was used to formulate the correct theory of \ell-adic sheaves for the geometrization of the local Langlands correspondence by Fargues and Scholze (FarguesScholze21). It has also been used by Mann to formalize the six operations for rigid analytic geometry (Mann22).

Condensed mathematics has also been used in ClausenScholze22 to provide new “analysis-free” proofs of the finiteness of coherent cohomology, Serre duality, GAGA, and the Hirzebruch-Riemann-Roch theorem in complex analysis.


Formalization/verification in proof assistants (Lean):

Applications of condensed mathematics include the following:

On condensed fractured \infty -topos-structure (cf. condensed local contractibility):

  • Qi Zhu, Fractured Structure on Condensed Anima, MSc thesis, Bonn (2023) [pdf, pdf]

with exposition in

  • Qi Zhu, Fractured structure on condensed spaces, talk notes (2023) [pdf, pdf]

  • Nima Rasekh, What is a topological structure?, talk notes (April 2023) [pdf, pdf]

Last revised on November 4, 2023 at 08:36:18. See the history of this page for a list of all contributions to it.