In mathematics, a '''generalized space''' is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: # A desire to apply concepts like cohomology 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) tend not to be abelian, 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:<ref>{{harvnb|Grothendieck|Verdier|1972}}</ref> {{blockquote|On peut done dire que la notion de topos, dérivé naturel du point de vue faisceautique en Topologie, constitue à son tour un élargissement substantiel de la notion d'espace topologique, un grand nombre de situations qui autrefois n'étaient pas considérées comme relevant de intuition topologique}}

However, William Lawvere argues in his 1975 paper{{sfn|ps=|Lawvere|1975}} 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 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.<ref>{{cite web |title=Locales as geometric objects |url=https://mathoverflow.net/questions/250263/locales-as-geometric-objects/250307#250307 |access-date=2024-07-22 |website=MathOverflow |language=en }}</ref> The topos theory is sometimes said to be the theory of generalized locales.{{sfn|ps=|Johnstone|1985}} * Jean Giraud's gros topos, Peter Johnstone's '''topological topos''',<ref>{{cite web |url=https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html |title=On a Topological Topos at The n-Category Café |website=golem.ph.utexas.edu }}</ref> or more recent incarnations such as condensed sets 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.)

== Citations == {{reflist}}

== Bibliography == * {{cite book |doi=10.1016/S0049-237X(08)71947-5 |chapter=Continuously Variable Sets; Algebraic Geometry = Geometric Logic |title=Logic Colloquium '73, Proceedings of the Logic Colloquium |series=Studies in Logic and the Foundations of Mathematics |date=1975 |last1=Lawvere |first1=F. William |volume=80 |pages=135–156 |isbn=978-0-444-10642-1 }} * {{citation |last1=Lawvere |first1=F. William |url=http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf |title=Categories of spaces may not be generalized spaces as exemplified by directed graphs |journal=Reprints in Theory and Applications of Categories |number=9 |date=2005 |pages=1–7 }} * {{cite book |doi=10.1017/CBO9781107359925.004 |chapter=How general is a generalized space? |title=Aspects of Topology |date=1985 |last1=Johnstone |first1=Peter T. |pages=77–112 |isbn=978-0-521-27815-7 }} * {{citation |last1=Grothendieck| first1 = A. | last2 = Verdier | first2 = J. L. |author-link2=Jean-Louis Verdier | chapter = Topos | title = Théorie des Topos et Cohomologie Etale des Schémas | series = Lecture Notes in Mathematics | volume = 269 | date = 1972 | pages = 299–518 | publisher = Springer | doi = 10.1007/BFb0081555| isbn = 978-3-540-05896-0 }}

{{topology-stub}} Category:Mathematical terminology Category:Topological spaces