Generalized space explained

In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms:

1. 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.
2. 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:

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 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.^[1] The topos theory is sometimes said to be the theory of generalized locales.
• Jean Giraud's gros topos, Peter Johnstone's topological topos,^[2] 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.)

References

• Book: 10.1016/S0049-237X(08)71947-5 . Continuously Variable Sets; Algebraic Geometry = Geometric Logic . Logic Colloquium '73, Proceedings of the Logic Colloquium . Studies in Logic and the Foundations of Mathematics . 1975 . Lawvere . F. William . 80 . 135–156 . 978-0-444-10642-1 .
• Lawvere, Categories of spaces may not be generalized spaces as exemplified by directed graphs
• Book: 10.1017/CBO9781107359925.004 . How general is a generalized space? . Aspects of Topology . 1985 . Johnstone . Peter T. . 77–112 . 978-0-521-27815-7 .

Notes and References

1. Web site: Locales as geometric objects . 2024-07-22 . MathOverflow . en.
2. Web site: On a Topological Topos at The n-Category Café. golem.ph.utexas.edu.