added pointer to:

- Jean Bénabou, §5.4.3 in:
*Introduction to Bicategories*, Lecture Notes in Mathematics**47**Springer (1967) 1-77 [doi:10.1007/BFb0074299]

added pointer to:

- Saunders MacLane, §XII.1 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

added pointer to:

- Bart Jacobs, Chapter 7 in:
*Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics**141**, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]

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?

]]>added pointer to

- Jean Bénabou, §5.1 of:
*Les distributeurs*, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, rapport**33**(1973) [pdf]

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 $p_1$ being the first of the morphisms and $p_2$ being the second in the composition (so $s\circ c=s\circ p_1$ 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.

]]>added pointer to:

- Enrico Ghiorzi,
*Complete internal categories*(arXiv:2004.08741)

added pointer to:

- Peter Johnstone, Chapter 2 of:
*Topos theory*, London Math. Soc. Monographs**10**, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

Oh, I see. Great.

]]>[deleted]

]]>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)

]]>[deleted]

]]>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!

]]>[deleted]

]]>And, interesting to note, that second, precursor, reference also has the Yoneda embedding, and the fact it preserves finite products!

]]>At long last, we have found the origin of the definition of internal categories (thanks to Dmitri here!):

- Alexander Grothendieck, p. 106 (9 of 21) of:
*Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients*, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

I have added that reference now, together with the precursor

- Alexander Grothendieck, p. 340 (3 of 23) in:
*Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf)

where the general definition of internalization is given.

So then, to the reference of Ehresmann’s “Catégories structurées” – which most authors cite as the origin of internal cateories – I have added the comment that

]]>the definition is not actually contained in there, certainly not in its simple and widely understood form due to Grothendieck61.