Moved content to enrichment versus internalisation.
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Marta Bunge, Robert Paré, Stacks and equivalence of indexed categories, Cahiers de Top. et Géom. Diff. Catég 20 4 (1979) 373-399 [numdam:CTGDC_1979__20_4_373_0]
Marta Bunge, Stack completions and Morita equivalence for categories in a topos, Cahiers de Top. et Géom. Diff. Catég 20 4, (1979) 401-436 [numdam, MR558106]
André Joyal, Myles Tierney, Strong stacks and classifying spaces, in: Category Theory (Como, 1990), Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [doi:10.1007/BFb0084222]
Tomas Everaert, Rudger W. Kieboom, Tim Van der Linden, Model structures for homotopy of internal categories, Theory and Applications of Categories, 15 3 (CT2004) 66-94 [tac:15-03]
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Well, I think there may be many places where someone would want to link directly to internal profunctor. It’s a different concept, so why not have a different page for it? The page internal category is already quite long.
]]>We have a page for internal profunctor, but it would seem reasonable to me to collapse that page into the internal category entry. There doesn’t seem to be an advantage to having two different pages. Would anyone object if I made this change?
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for the definition of internal profunctors (to readers who already know all about internal categories?).
]]>Internal profunctors and 2-cells between them are already present in §5.1 of Bénabou’s Les distributeurs (1973), which must be the earliest definition for those. I don’t see a definition of internal functor or natural transformation there, though I would imagine it to be known earlier.
]]>For what it’s worth, Johnstone’s “Topos theory” (1977) considers internal functors in section 2.1 and internal profunctors in section 2.4. That seems to be the earliest mentioning of these concepts among the references already collected in the entry (here), though I have no idea if there is an earlier one.
]]>The definition of internal category is due to Grothendieck. However, what’s the earliest reference for internal functors, natural transformations, and profunctors?
]]>Fixed links for English translation of FGA.
]]>Make the s-t swaps (hopefully corrections!) as described
Julian Gilbey
]]>On further thought, I’m pretty sure I’m right so I’ll make the changes. Feel free to undo them if I’m wrong!
]]>I think the s and t in the 2nd-5th pullback diagrams in the Internal categories section are inconsistent with those in the first pullback diagram and the laws specifying the source and target of composite morphisms; the earlier diagrams have being the first of the morphisms and being the second in the composition (so for example), whereas the 2nd-5th pullback diagrams seem to have them the other way round. But I may be wrong, so I am hesitant about making this edit.
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Oh, I see. Great.
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]]>Thanks. But checking out the web version on my system, it appears broken: Most of the pages I see there appear empty except for a section headline, and those that are not empty break off in the middle of a sentence after a few lines. (using Firefox 89.0.1 (64-bit) on Windows 10)
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]]>I have added pointers to Hosgood’s translations and also added pointer to FGA where more information can be found (and can be added, such as pointer to Hosgood’s TeX sources, if that is felt to be relevant)
]]>Absolutely!
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