While the topic of topos subobject classifier is current, how far back can the historical and conceptual origins of the use of "$\Omega$" as the symbol denoting such be traced? Regards, Keith Harbaugh 30-Aug-2001 07:15:05 -0300,1069;000000000000-0000001f
While the topic of topos subobject classifier is current, how far back can the historical and conceptual origins of the use of "$\Omega$" as the symbol denoting such be traced?
Regards, Keith Harbaugh
If a topos E is a generalized topological space, then Omega is the internal version of its (ungeneralized) topology, its lattice of opens: for the global elements 1 -> Omega are in bijection with the continuous maps [geometric morphisms] from E to [sheaves over] Sierpinski space. (This extends to generalized elements too. Suppose U is an object of E, a sheaf over E, and let U' -> E be the corresponding local homeomorphism. Then the morphisms U -> Omega correspond with the opens of U'.) Hence Omega can be read as an abbreviation for Open. Is this how the notation actually arose, or is it just a pleasant rationalization of my own? If X is an ordinary topological space, then Omega X is one notation used for its topology. But though I use it myself, I don't know anything about its history. Putting these together you see that when you take E as your base category of sets then Omega 1, the topology of the 1-point space, is just Omega. Steve Vickers Omega University
On Mon, 27 Aug 2001, Keith Harbaugh wrote:
While the topic of topos subobject classifier is current, how far back can the historical and conceptual origins of the use of "$\Omega$" as the symbol denoting such be traced?
Regards, Keith Harbaugh
I seem to recall being told that it occurs somewhere in the original (mimeographed) version of SGA4 (as an interesting example of a sheaf on a site), and that Lawvere and Tierney borrowed the notation that Grothendieck et al. had used for it. However, I've never been able to find it there myself; I'm pretty sure it's not in the revised version published in Springer Lecture Notes. Peter Johnstone
[note from moderator: apologies to Colin for delayed posting...Bob] "Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk> wrote of $\Omega$ as subobject classifier:
I seem to recall being told that it occurs somewhere in the original (mimeographed) version of SGA4 (as an interesting example of a sheaf on a site), and that Lawvere and Tierney borrowed the notation that Grothendieck et al. had used for it. However, I've never been able to find it there myself; I'm pretty sure it's not in the revised version published in Springer Lecture Notes.
I have heard that too. And I am sure it is not in the published SGA4. I have one mimeographed version in my office. Omega does not occur there the sections on topologies, but next week I will look to see if it occurs as an example of a sheaf--unless someone who knows writes in sooner. best, Colin _________________________________________ Dialectic, the purest part of philosophy, hovers attentively over mathematics, encompasses its whole development, and of itself contributes to the special sciences their various perfecting, critical, and intellective powers--the powers, I mean, of analysis, division, definition, and demonstration. --Proclus, ca. 460 AD, A COMMENTARY ON THE FIRST BOOK OF EUCLID'S ELEMENTS
[note from moderator: apologies to Jack for the delayed posting...Bob]
On Mon, 27 Aug 2001, Keith Harbaugh wrote:
While the topic of topos subobject classifier is current, how far back can the historical and conceptual origins of the use of "$\Omega$" as the symbol denoting such be traced?
Regards, Keith Harbaugh
I seem to recall being told that it occurs somewhere in the original (mimeographed) version of SGA4 (as an interesting example of a sheaf on a site), and that Lawvere and Tierney borrowed the notation that Grothendieck et al. had used for it. However, I've never been able to find it there myself; I'm pretty sure it's not in the revised version published in Springer Lecture Notes.
Peter Johnstone
-- I just got out my old original mimeographed copy of SGA 4 Fasicule 1 (by Verdier) and I'm afraid that I can't find it there either. Quite consistently, $\Omege$ is used there only to denote a generic topological space: " Soient $\Omega$ un espace topologique, $\Omega^{\tilde}$ le topos des faisceaux d'ensembles sur $\Omega$...." etc. Also I seem to recall hearing that when Grothendieck saw the Lawvere-Tierney subobject classifier $\Omega$ he was amazed that they could have missed the centrality of such a powerful notion in Topos Theory. He always subsequently referred to it technically as "the Lawvere element"! But to settle this, at least partly, why don't we just ask Bill or Myles to tell us where they got the $\Omega$ notation? Jack Duskin
Sorry I didn't reply sooner to this; I was enroute from Perugia to Buffalo. As I recall, the original omega was the sheaf of CLOSED sets on a topological space, in a paragraph devoted to applications to notions like sections with supports. Having recognized the importance of the subobject classifier, making the observation that isomorphic things probably would not remain notationally distinct forever, and especialy wanting not to change an "established"(!) symbol, we replaced the T (for truth) (which was used in my IAM 69 talk and ICM 70 paper). Thus the mega oh is a coincidence. Unfortunately I do not recall in which part of the prepublished SGA4 that paragraph occurs .
participants (6)
-
Colin McLarty -
Dr. P.T. Johnstone -
F. William Lawvere -
John Duskin -
Keith Harbaugh -
S Vickers