Talk:Fibred category

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

Well, no, that risks a technical mismatch. Charles Matthews 23:19, 1 January 2006 (UTC)

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)

Different definitions of Cartesian

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)

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)

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)

2-category of E-categories

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)

2-category

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)

Requested move

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)

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Fibred category → Fibered 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)

• comment. Personally, I spell it fibred, and feel that WP:RETAIN might apply. But I don't really have strong opinions on the matter. Sławomir Biały (talk) 14:34, 28 February 2014 (UTC)
• Oppose. Google Books gives 607 hits for "Fibred category" and 494 for "Fibered category". There seems no pressing need to make the change. Deltahedron (talk) 17:46, 28 February 2014 (UTC)
• The spelling "fibre" is used in Britain, Canada, Australia, and New Zealand, and "fiber" is used in the USA. There is an official preference from one spelling over the other only when the article's topic is associated with one of those geographic regions. Michael Hardy (talk) 23:24, 28 February 2014 (UTC)

"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)

• Strong oppose as per WP:ENGVAR, specifically WP:RETAIN. Red Slash 16:50, 3 March 2014 (UTC)

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

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)

Small Correction?

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)

Definition of fibred categories

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)