Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which replaces a topological space by a certain sheaf of sets, in order to solve some technical problems of doing homological algebra on topological groups.
According to some,[ who? ] the theory aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.[ citation needed ]
The fundamental idea in the development of the theory is given by replacing topological spaces by condensed sets, defined below. The category 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, which allows for the use of tools from homological algebra 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 might expect to be able to incorporate algebraic geometry, p-adic analytic geometry and complex analytic geometry. [1]
In condensed mathematics, liquid vector spaces are alternatives to topological vector spaces. [2] [3]
A condensed set is a sheaf of sets on the site of profinite sets, with the Grothendieck topology 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 one can associate a condensed set, customarily denoted , which to any profinite set associates the set of continuous maps . If is a topological group or ring, then is a condensed group or ring.
In 2013, Bhargav Bhatt and Peter Scholze introduced a general notion of pro-étale site associated to an arbitrary scheme. In 2018, Dustin Clausen and Scholze arrived at the conclusion that the pro-étale site of a single point, which is isomorphic to the site of profinite sets introduced above, already 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 into the theory. [4]
In 2020 Scholze completed a proof of their results which would enable the incorporation of functional analysis 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. [5] [6] Over a 6-month period, a group led by Johan Commelin verified the central part of the proof using the proof assistant Lean. [7] [6] As of 14 July 2022, the proof has been completed. [8]
Coincidentally, in 2019 Barwick and Haine introduced a 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 universes, whereas condensed mathematics can be developed strictly within ZFC. [9]
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 called a site.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
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 and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets.
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed ; it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning.
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras.
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and spaces can be generalized.
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular.
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another structure.
In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of -modules in the étale topology and inducing .
In mathematics, especially in topology, a pyknotic set is a sheaf of sets on the site of compact Hausdorff spaces. The notion was introduced by Barwick and Haine to provide a convenient setting for homological algebra. The term pyknotic comes from the Greek πυκνός, meaning dense, compact or thick. The notion can be compared to other approaches of introducing generalized spaces for the purpose of homological algebra such as Clausen and Scholze‘s condensed sets or Johnstone‘s topological topos.
