Condensed mathematics is a theory developed by and

Peter Scholze Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. He ha ...

which, according to some, aims to unify various

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

subfields, including

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and co ...

, and algebraic geometry.

Idea

The fundamental idea in the development of the theory is given by replacing

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

s by ''condensed sets'', defined below. The

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of condensed sets, as well as related categories such as that of condensed abelian groups, are much better behaved than the category of topological spaces. In particular, unlike the category of topological abelian groups, the category of condensed abelian groups is an

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...

, which allows for the use of tools from

homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...

in the study of those structures. The framework of condensed mathematics turns out to be general enough that, by considering various "spaces" with sheaves valued in condensed algebras, one is able to incorporate algebraic geometry, p-adic analytic geometry and

complex analytic geometry In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced or complex analytic space is a general ...

Definition

A ''condensed set'' is a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper s ...

of sets on the

site Site most often refers to: * Archaeological site * Campsite, a place used for overnight stay in an outdoor area * Construction site * Location, a point or an area on the Earth's surface or elsewhere * Website, a set of related web pages, typi ...

profinite set In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in t ...

s, with the

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...

given by finite, jointly surjective collections of maps. Similarly, a ''condensed group'', ''condensed ring'', etc. is defined as a sheaf of groups, rings etc. on this site. To any topological space

X

one can associate a condensed set, customarily denoted

\underline X

, which to any profinite set

S

associates the set of continuous maps

S\to X

. If

X

is a topological group or ring, then

\underline X

is a condensed group or ring.

History

In 2013, Bhargav Bhatt and

introduced a general notion of ''pro- étale site'' associated to an arbitrary

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

. In 2018, together with Dustin Clausen, they arrived at the conclusion that already the pro-étale site of a single point, which is isomorphic to the site of profinite sets introduced above, has rich enough structure to realize large classes of topological spaces as sheaves on it. Further developments have led to a theory of condensed sets and ''solid abelian groups'', through which one is able to incorporate

non-Archimedean geometry In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properti ...

into the theory. In 2020 Scholze completed a proof of a result which would enable the incorporation of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

as well as complex geometry into the condensed mathematics framework, using the notion of '' liquid vector spaces''. The argument has turned out to be quite subtle, and to get rid of any doubts about the validity of the result, he asked other mathematicians to provide a formalized and verified proof. Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the

proof assistant In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor ...

Lean Lean, leaning or LEAN may refer to: Business practices * Lean thinking, a business methodology adopted in various fields ** Lean construction, an adaption of lean manufacturing principles to the design and construction process ** Lean governme ...

. As of 14 July 2022, the proof has been completed. Coincidentally, in 2019 Barwick and Haine introduced a very similar theory of '' pyknotic objects''. This theory is very closely related to that of condensed sets, with the main differences being set-theoretic in nature: pyknotic theory depends on a choice of

Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' ...

s, whereas condensed mathematics can be developed strictly within ZFC.

References

Firther reading

* https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential

External links