In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms
Grothendieck’s dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I: [1]
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, Lawvere argues in his 1975 paper [2] 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.
In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms
Grothendieck’s dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I: [1]
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, Lawvere argues in his 1975 paper [2] 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.