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)
[edit] 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:
- An equivalent definition of fibred categories emerges if one replaces the existence of a cartesian lifting with the existence of a hypercartesian lifting.
- 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)
- 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:
[edit] 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)
