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Keith Elliott Peterson (Feb 16 2021 at 20:37):I don't see this mentioned on the nlab, but in Robert Goldblatt's 'Topoi: The Categorial Analysis of Logic, 2nd Edition', Chp 4, pg 85, example 3 gives , the category of finite ordinals, as an example of a topos (source screenshot).
Robert never specifies how the two topoi are related, but I assume is a subtopos of in some specific way.
If so, then by rights, shouldn't also be a subtopos of ?
And are there other categories of ordinals that are also topoi?
Fawzi Hreiki (Feb 16 2021 at 20:39):This depends on what you mean by 'the category of sets'. If your meta-theory is ZFC then you can actually isolate the 'category of ordinals' from the category of ZFC sets and it will be an equivalent category.
Fawzi Hreiki (Feb 16 2021 at 20:40):On the other hand, if you axiomatise the category of sets directly in topos theoretic language (as a well-pointed topos with NNO and choice), then 'ordinal' doesn't really mean anything at all.
Morgan Rogers (he/him) (Feb 16 2021 at 20:43):Sure it does, it's an object equipped with a well-ordering.
Morgan Rogers (he/him) (Feb 16 2021 at 20:44):@Keith Peterson does Goldblatt mention what the morphisms in are meant to be?
Fawzi Hreiki (Feb 16 2021 at 20:44):Ok but then that's not what Goldblatt meant in that paragraph
Fawzi Hreiki (Feb 16 2021 at 20:44):He just means the full subcategory of sets on the actual ZFC-ordinals but still with arbitrary functions as maps (not monotone maps)
Morgan Rogers (he/him) (Feb 16 2021 at 20:46):What are "the" ZFC ordinals?
Morgan Rogers (he/him) (Feb 16 2021 at 20:47):Fawzi Hreiki said:
He just means the full subcategory of sets on the actual ZFC-ordinals but still with arbitrary functions as maps (not monotone maps)
In that case is equivalent to , so it's not as exciting an example as you might hope.
Fawzi Hreiki (Feb 16 2021 at 20:47):Yeah I think this example is just explaining the notion of equivalence
Fawzi Hreiki (Feb 16 2021 at 20:49):Morgan Rogers (he/him) said:
What are "the" ZFC ordinals?
{}, {{}}, {{}, {{}}}, etc..
Fawzi Hreiki (Feb 16 2021 at 20:51):The transitive pure sets which are well-ordered by
John Baez (Feb 16 2021 at 20:53):Keith wrote:
Robert never specifies how the two topoi are related, but I assume FinOrd is a subtopos of FinSet in some specific way.
FinOrd is a skeleton of FinSet. Any skeleton of some category is equivalent to that category. A category equivalent to a topos is again a topos. FinSet is a topos. So, FinOrd is also a topos.
John Baez (Feb 16 2021 at 20:55):By the way, being a topos is a property of a category, not an extra structure. So a subcategory of a topos either is or is not a topos: it's a yes-or-no question. So, we don't have to say a "specific way" in which FinOrd is a subtopos of FinSet. It's just a subcategory of a topos which happens to be a topos itself.
John Baez (Feb 16 2021 at 21:03):(Goldblatt is being a bit peculiar is defining FinOrd to be the category of finite ordinals
0 = {}
1 = {0}
2 = {0,1}
etc.
and all functions between these, instead of all order-preserving maps - but he must being doing that, because he says it's a topos. Anyway, this skeleton of FinSet deserves to have some name, since it's widely used. I suggest using Fnord. :upside_down:)
Fawzi Hreiki (Feb 16 2021 at 21:07):Usually its FinCard and Card which are the go to examples in category theory books when defining skeletons
John Baez (Feb 16 2021 at 21:28):Yes, that's better; Goldblatt should have said FinCard instead of FinOrd (though he's allowed to use his own definitions), and to answer Keith's other question: Card is a good name for the subtopos of Set where the objects are just the cardinals and the morphisms are all functions between these.
David Michael Roberts (Feb 16 2021 at 21:30):Presumably he did this so as to get a small topos, rather than an essentially small one.
Oscar Cunningham (Feb 16 2021 at 21:31):So if we don't assume Choice, is the full subcategory of on the well-orderable sets a topos?
John Baez (Feb 16 2021 at 21:40):David Michael Roberts said:
Presumably he did this so as to get a small topos, rather than an essentially small one.
Personally I think Goldblatt is just giving an example to show people how you can take a skeleton of a topos and still get a topos. But yes, that's one reason you might want to do it.
Keith Elliott Peterson (Feb 17 2021 at 00:56):Morgan Rogers (he/him) said:
Keith Peterson does Goldblatt mention what the morphisms in are meant to be?
Maybe. He does describe an exponential object, . So I would think an element of that object ought to be a morphism.
John Baez (Feb 17 2021 at 02:10):Yes.
John Baez (Feb 17 2021 at 02:11):But normally when you describe a category, the first thing you do is specify the objects, the morphisms and (if there's any uncertainty about this) how to compose the morphisms.
John Baez (Feb 17 2021 at 02:12):If Goldblatt doesn't do this, he's being naughty. Lots of category theorists like to complain about Goldblatt's book; this would be one little reason for doing so.
David Michael Roberts (Feb 17 2021 at 03:33):@Oscar Cunningham P(N) can be non-well-orderable in some models of ZF, so I'm not sure that category has powerobjects
Keith Elliott Peterson (Feb 17 2021 at 04:49):I was going to ask how do we differentiate between the subcategory from the subcategory , but with a little thought, has, as objects, the downsets of .
For example, a countable ordinal can be identified with a downward closed set in the poset .
Edit: Wait, I might be jumping the gun here.