topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
basic constructions:
strong axioms
further
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 module?
solid spectrum?
liquid module?
nuclear module?
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.
Peter Scholze, Lectures on condensed mathematics, pdf
Peter Scholze, Lectures on analytic geometry, pdf
Dustin Clausen, Peter Scholze, Condensed Mathematics and Complex Geometry, pdf
Dustin Clausen, Peter Scholze, Masterclass in condensed mathematics, YouTube playlist, website (including pdf notes)
Formalization/verification in proof assistants (Lean):
Peter Scholze, Liquid tensor experiment, December 2020
Peter Scholze, Half a year of Liquid Tensor Experiment: Amazing developments, June 2021
Applications of condensed mathematics include the following:
Laurent Fargues, Peter Scholze, Geometrization of the local Langlands correspondence (arXiv:2102.13459)
Lucas Mann, A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry (arXiv:2206.02022)
On condensed fractured $\infty$-topos-structure (cf. condensed local contractibility):
with exposition in
Last revised on November 4, 2023 at 08:36:18. See the history of this page for a list of all contributions to it.