Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 14 of 14
Reading through the construction of the generalized universal bundle in the section category of fibrant objects if have a problem with the existence.
Accordng to the definition, In a category of fibrant objects it is not assumed that every pullback exists, but only those of fibrations. But then given a morphism in a category of fibrant objects, the generalized universal bundles is defined by a pullback diagram as in Definition 3 of category of fibrant objects.
Is this an ordinary (not homotopy) pullback? Then I would say, this does not necessarily exist, since is not assumed to be a fibration.
Then concerning Definition (computing the homotopy fiber in terms of certain pullbacks) again, those pullbacks in the limit do not necessarily exist, do they? Because the proof of Lemma 10 requires the existence of pullbacks, but in a category of fibered objects pullbacks need only exist for fibrations. So if in the proof or are not fibrations, the construction does not work, I think.
Am I seeing that wrong?
And by the way, the link to the book of Kenneth S. Brown gives an empty pdf on my computer.
A working link with open access is here: http://www.ams.org/journals/tran/1973-186-00/S0002-9947-1973-0341469-9/S0002-9947-1973-0341469-9.pdf
Reading through the book of K.S. Brown the question remains, how the definition in the book is equivalent to the definition in category of fibrant objects ?
That entry if kicking! On Firefox I get error messages:
XML Parsing Error: no element found Location: http://ncatlab.org/nlab/show/category+of+fibrant+objects Line Number 1331, Column 1585:
On Safari some of the page loads but the end does not, so presumably there is something wrong after that.
(EDIT: the problem has cleared.)
On Chromium, I face some math typesetting errors for days all over the nLab, too.
Loading it on Safari is patchy. I tried shortly after my previous message and it loaded. (The pdf file seems dead as you said, Mirco.)
Has there been an update of some of the Mathml related programs?
Yes but the new link I gave above
http://www.ams.org/journals/tran/1973-186-00/S0002-9947-1973-0341469-9/
or directly http://www.ams.org/journals/tran/1973-186-00/S0002-9947-1973-0341469-9/S0002-9947-1973-0341469-9.pdf
works for me.
Anyway, my problem boils down to the following:
1.) In the nLab entry the axiom says: finite products and pullbacks of fibrations exists.
2.) In the paper of brown ther axiom (c) saying, that for a diagram
where is a fibration, the pullback exist, and the projection is a fibration.
So what I don’t see is how the nLab def is equivalent to that of Brown, I.e how we can follow axiom (C) from the nLab definition. Maybe this is obvious, but I would say it is worth, at least, to mention how this works. Axiom (C) is of central importance for the computation of the generalized universal bundles and the homotopy fiber product as in the entry.
Or is it just that “pullbacks of fibrations exist” just means exactly axiom (C), that is “pullbacks of fibrations exist” means only one arrow in the index diagram needs to be a fibation?
Ok I’m not an English native, but then “pullbacks of fibrations exist” should be called “pullbacks of a fibration exist” or something. Don’t? For me the first one sounds like “the pullback exists, if BOTH arrows are fibrations.”
I would read it as ’a pullback of any fibration along a morphism to its base exists’. I think the point is that the traditional / classical view of ’change of base’ is influencing the wording in the original paper. The sense you had read was perhaps the result of the many years from KSB’s article to the present.
Now it makes sense. However not Browns definition sounds ambiguous. His ’axiom (C)’ is very clear, but the one in the nLab entry leaves room for misinterpretation.
Yes, as Tim says, “pullback of XYZ” means along any morphism. Pullback is something that one does to one thing and along a base morphism. Otherwise, when both morphisms are regarded on the same footing, then one rather speaks of their fiber product. But to clarify I have added to the entry “(along any morphism)”.
Please fix the link to the pdf where you see the need!
All right then.
I would propose condensing the first three axioms
has finite products and pullbacks of fibrations (along all morphisms);
has a terminal object ;
fibrations are preserved under pullback;
into
has finite products, and in particular a terminal object ;
the pullback of a fibration along an arbitrary morphism exists, and is again a fibration;
Fine with me. Please feel invited to polish the entry.
OK, done.
1 to 14 of 14