Talk:Fibred category

Latest comment: 2 years ago by Nmdwolf in topic Definition of fibred categories

Untitled

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Shouldn't we merge this with the fibration article? Samuel Mimram 13:28, 31 December 2005 (UTC)Reply

Well, no, that risks a technical mismatch. Charles Matthews 23:19, 1 January 2006 (UTC)Reply
Charles is right. Indeed, these are quite different topics. I'll begin work on fibred categories as soon as feasible, as background for Stack (descent theory). Stca74 15:30, 9 September 2007 (UTC)Reply

Different definitions of Cartesian

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I think I've corrected the definition. By my reckoning, and my from my references, Cartesian morphisms always compose. And it is not necessary that every morphism has an inverse. If I'm mistaken, or I am thinking of a different notion of fibred category, then let me know. Otherwise, I will fix the bugs in the bit about cleavages. Sam Staton 17:38, 24 October 2007 (UTC)Reply

I now see that the definition here is the one in SGA 1 and is right. I am more familiar with a different notion of cartesian morphism, which I understand is sometimes called "hypercartesian", that yields an equivalent definition of fibration. I intend to include the alternative definitions in this article in the near future because I think they are quite common. Sam Staton 20:26, 24 October 2007 (UTC)Reply
Yes, you're right: one can base the definition of a fibred category on hypercartesian morphisms as well (but not that of pre-fibred categories). More precisely:
  1. An equivalent definition of fibred categories emerges if one replaces the existence of a cartesian lifting with the existence of a hypercartesian lifting.
  2. If E/F is fibred, then all cartesian morphisms are hypercartesian.
I decided to limit my text to cartesian morphisms as I felt their definition is somewhat more immediately intuitive (?) and in order to avoid yet one more definition. However, if you find a gentle way to introduce hypercartesian morphisms here as well, that's great. A quick summary of the few definitions and lemmas can be found in Giraud (1964) pp. 1–2. Stca74 14:46, 25 October 2007 (UTC)Reply

2-category of E-categories

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Ryan Reich fixed the incorrect statement about the 2-cat of E-categories for fixed E being a subcat of the 2-cat of categories, replacing that with the correct statement that it is a subcat of the category of functors (understood in the "bivariant" way - see example 1 in the article). However, this corrected claim may not be that useful in this part of the article - indeed, it should be elaborated to give the definition of the "bivariant" categopry of functors to make sense immediately. Thus, I changed the text to be instead a slightly more explicit definition of the 2-cat structure itself. Stca74 (talk) 07:18, 4 June 2008 (UTC)Reply

2-category

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By definition a strict 2-category is one enriched over the monoidal category of locally small categories (i.e. over 1-Cat). This means, Hom(a,b) has to be a locally small category. But if for a given category E we consider the category Fib(E), are the categories of morphisms Hom_E(F,G) always locally small, i.e. given two cartesian E-functors f,g: F→G, is the class of natural transformations from f to g always a set? Saurav — Preceding unsigned comment added by 158.144.28.85 (talk) 02:32, 10 November 2010 (UTC)Reply

Requested move

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The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Not moved. (non-admin closure) Hot Stop talk-contribs 04:24, 7 March 2014 (UTC)Reply



Fibred categoryFibered category – Nothing wrong with the current title "fibred category", but the new title "fibered category" seems more common, at least nowadays (e.g., our own stack (mathematics).) Taku (talk) 12:52, 28 February 2014 (UTC)Reply

"Fibre" in India as well, I think. There is also preference to retain the existing spelling variants when already established in an article. To quote MOS:
In general, disputes over which English variety to use in an article are strongly discouraged. Such debates waste time and engender controversy, mostly without accomplishing anything positive.
Deltahedron (talk) 08:18, 1 March 2014 (UTC)Reply
The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Merge of Grothendieck construction

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This article is tagged with a suggestion that grothendieck construction be merged here. I don't think this is a good idea, since the Grothendieck construction arises in contexts that do not obviously involve all the machinery of fibered categories. — Preceding unsigned comment added by 50.169.30.99 (talk) 05:15, 25 March 2014 (UTC)Reply

Small Correction?

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In Examples, "The functor Ob : Cat→Set, sending a category to its set of objects" Shouldn't it refer to Small Categories? ---- — Preceding unsigned comment added by 173.64.34.166 (talk) 05:48, 11 November 2020 (UTC)Reply

Definition of fibred categories

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Shouldn't the definition of fibred categories state that for every morphism with codomain in the range of the projection AND every lift of the codomain there exists an inverse image with codomain the given lift? Right now it is only stated that an inverse image should exist. But there is no guarantee that this inverse image has a given lift as codomain. In particular, the current definition is not equivalent to that of a cloven category. Nmdwolf (talk) 08:49, 10 December 2021 (UTC)Reply