In mathematics, a generalized space is a generalization of a
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 points ...
. Impetuses for such a generalization comes at least in two forms:
# A desire to apply concepts like
cohomology
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 viewe ...
for objects that are not traditionally viewed as spaces. For example, a
topos was originally introduced for this reason.
# A practical need to remedy the deficiencies that some naturally-occurring categories of spaces (e.g., ones in
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 space#Definition, inner product, Norm (mathematics)#Defini ...
) tend not to be
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, a standard requirement to do homological algebra.
Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I:
However,
William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself."
A generalized space should not be confused with a geometric object that can substitute the role of spaces. For example, a
stack
Stack may refer to:
Places
* Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group
* Blue Stack Mountains, in Co. Donegal, Ireland
People
* Stack (surname) (including a list of people ...
is typically ''not'' viewed as a space but as a geometric object with a richer structure.
Examples
*A
locale is a sort of a space but perhaps not with enough points. The topos theory is sometimes said to be the theory of generalized locales.
*
Jean Giraud's
gros topos,
Peter Johnstone's topological topos,
or more recent incarnations such as
condensed set
Condensed mathematics is a theory developed by and Peter Scholze which, according to some, aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.
Idea
The fundamental idea in the develop ...
s or
pyknotic sets. These attempt to embed the category of (certain) topological spaces into a larger category of generalized spaces, in a way philosophically if not technically similar to the way one generalizes a function to a
generalized function. (Note these constructions are more precise than various
completions of the category of topological spaces.)
References
*
* Lawvere
Categories of spaces may not be generalized spaces as exemplified by directed graphs*
*
{{topology-stub
Mathematical terminology
Topological spaces