Does anyone know why in books on category theory the notation for the subobject classifier is almost everywhere the capital greek letter $\Omega$?
Gérard Lang

According to Mac Lane [Concepts and categories in perspectives], this was the notation used in "the initial IHES edition of SGA IV, where it is noted that the set $\Omega (X)$ of all subobjects of an object $X$ in a Grothendieck topos $\mathbb{E}$ defines a sheaf for the canonical topology on $\mathbb{E}$ and so, by Giraud's theorem, is representable by some object $\Omega$." That answers the question of where the convention originates, but not why...
– Zhen LinFeb 7 '13 at 17:55

5

Maybe it's from the case of sheaves on a topological space, where Ω is the sheaf of open subsets (i.e., Ω(U) is the set of open subsets of U). Open starts with the letter O, which in greek is Ω.
– Omar Antolín-CamarenaFeb 7 '13 at 18:12

1

Maybe they had the christian 'alpha & omega' in mind. And the characteristic functions, as well as the true arrow end in omega.
– Stephan MüllerFeb 7 '13 at 18:54

Concepts and categories in perspectives], this was the notation used in "the initial IHES edition of SGA IV, where it is noted that the set $\Omega (X)$ of all subobjects of an object $X$ in a Grothendieck topos $\mathbb{E}$ defines a sheaf for the canonical topology on $\mathbb{E}$ and so, by Giraud's theorem, is representable by some object $\Omega$." That answers the question of where the convention originates, but not why... – Zhen Lin Feb 7 '13 at 17:55