Talk:Subobject classifier

Introductory example

This sentence "Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow: true: {0} → {0, 1} which maps 0 to 1. " has to be modified, no logical link. Moreover where are now explicitly in the category of sets. Noix07 (talk) 14:21, 13 October 2014 (UTC)

Oh, I got it now, it means that the object "true" (here {0}) is identified with the arrow from 0 to 1 — Preceding unsigned comment added by Noix07 (talk • contribs) 15:05, 13 October 2014 (UTC)

don't have time to correct this.... what is that axiom business?? — Preceding unsigned comment added by Noix07 (talk • contribs) 15:25, 13 October 2014 (UTC)

?

What is the map 1 -> Ω used in the pullback diagram?

It intuitively represents the "true" point in the subobject classifier [StefanLjungstrand 84.217.41.47 20:51, 1 February 2006 (UTC)]

More examples?

Anybody think there should be more examples (at the very least, the 'classical' subobject classifier in Set, and maybe one nonclassical example with no topology?) BenetD 21:00, 20 February 2006 (UTC)

Yes, especially since many subobject classifiers have structure beyond the set {true,false}. As it sits, I think the article is a bit misleading, at least for people who can't follow the topology example. But unfortunately, my topos theory isn't quite good enough to improve this article yet. -emk (talk) 18:20, 20 June 2008 (UTC)

Within a topos, every pair of subobject classifiers are categorically isomorphic.

Some other examples are Z2 (the cyclic ab. group) in AbGrp, the Be(1/2) (this is the coin) in the Prob category or a simple two vertex connected graph for FinGraph category. All of them have their own internal structure. — Preceding unsigned comment added by 80.58.250.87 (talk) 20:58, 1 April 2015 (UTC)

Example Wrong?

I don't think the definition of the subobject classifier of sheaves of sets on a space X is correct. Under the definition given, Ω would have only one global section, but the map of global sections from a sheaf A to the subobject classifier should provide information about where restrictions of the global sections of A belong to the subsheaf. For instance, for the one point set, Ω would be the one point set, not the two point set. I think the correct definition should be the sheaf that assigns an open set to it's set of open subsets, restricting by taking intersections.

For instance, to get the map which represents the subsheaf H^0(X) of constant functions inside C(X), we would take each function to the subset on which it is constant. Right? I'll change the page and add this example if no one objects. Holomorphic (talk) 07:27, 28 May 2009 (UTC)

You are quite right. I went ahead and made a correction. You might want to give a fuller description and add your example. But it will need to say `locally constant.' —Preceding unsigned comment added by Colin McLarty (talk • contribs) 17:37, 18 June 2009 (UTC)

Definition

The references Mac Lane (1998) p.105 and Pedicchio & Tholen (2004) p.330 both define a subobject classifier not as just a special object Ω but as a monomorphism t:1 → Ω where 1 is a terminal object (here t is interpreted as the "truth" map), such that every mono is a pullback of t. Deltahedron (talk) 20:55, 5 April 2013 (UTC)