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In the book "Topos Theory" of Peter Johnstone (Topos Theory, LMS Monographs no. 10. Academic, 1977) one finds at page 41 in Chapter 2:

"For a detailed account of internal categories from a 2-categorical point of view, the reader is referred to the monumental work of JW Gray [168]."

The work of J.W. Gray has to be very important and significant (in those years Gray was one of the great masters of category theory) as well known and whose output is only the first part in Formal Category Theory: Adjointness for 2-Categories. Series: Lecture Notes in Mathematics, Vol 391.

Does anyone have news of such notes of J.W. Gray? Of course I would like to read these.

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    $\begingroup$ For everyone else: reference [168] from Johnstone is "Formal category theory II, To appear". @Buschi My advice is to look at work by Street, Kelly and Lack. I don't have proper references at the moment, I'm afraid. 'Tis late and I've got to get up early for prezzies :) At least look at Fibrations in bicategories by Street: archive.numdam.org/article/CTGDC_1980__21_2_111_0.pdf $\endgroup$
    – David Roberts ♦
    Commented Dec 24, 2010 at 11:20
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    $\begingroup$ Why don't you write Gray? His information is at math.uiuc.edu/~gray $\endgroup$
    – Todd Trimble
    Commented Dec 24, 2010 at 11:59
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    $\begingroup$ Did anything come of this, @Buschi? $\endgroup$
    – David Roberts ♦
    Commented Jun 24, 2020 at 2:27
  • $\begingroup$ I would like to second the question by @DavidRoberts (hopefully generating another notification). $\endgroup$
    – varkor
    Commented Oct 4, 2022 at 13:01
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    $\begingroup$ @Emily: sadly John Gray died in 2017, so it is no longer possible to ask him about this work directly. (I don't know why his website hasn't been updated accordingly.) $\endgroup$
    – varkor
    Commented Oct 4, 2022 at 23:45

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I asked Peter Johnstone about this citation, and he told me that he included the reference, despite not having a copy of the manuscript, since John Gray had been promising a book on the subject for many years, and he had expected that the book would appear soon. Unfortunately, it never did. It appears likely that any notes Gray might have made on the subject are now lost.

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    $\begingroup$ You can see why Johnstone decided on his hard "only actually published" policy for citations in The Elephant... $\endgroup$
    – David Roberts ♦
    Commented Oct 4, 2022 at 23:22

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