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One of the most famous and unifying theorems in category theory is that right adjoints preserve limits. I wonder: Who was the first one to prove this fact?

The notion of adjoint functors is, of course, due to Daniel Kan. But I couldn't find the mentioned fact in his paper Adjoint Functors.

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  • $\begingroup$ Peter Freyd both for left and right adjoints in his book abelian Categories, an Introduction to the Theory of Functors its called Freyd's adjoint functor theorem $\endgroup$
    – user177829
    Mar 29, 2021 at 10:59
  • $\begingroup$ I think the adjoint functor theorems are rather a sort of converse to the statement I'm talking about. $\endgroup$
    – user907616
    Mar 29, 2021 at 11:59
  • $\begingroup$ Agreed. This does not answer the question. But, welcome to mathoverflow! $\endgroup$
    – David White
    Mar 29, 2021 at 12:47
  • $\begingroup$ Whether the citation of Abelian Categories is correct, I don't know, but as I understand it, Peter Freyd is definitely the person who first recognised the unifying notion of "limit". After that, the fact that they are (preseved by) right adjoints is trivial. So @rft34, as a "new contributor", deserves the "correct answer" bonus. $\endgroup$
    – Paul Taylor
    Mar 29, 2021 at 14:52
  • $\begingroup$ @PaulTaylor: Daniel M. Kan defined limits and colimits in his 1956 paper “Adjoint functors” and proved that the (co)limit functor is left/right adjoint to the diagonal functor. Freyd's earliest paper (his Ph.D. thesis) is from 1960, there is no way he could be credited for (co)limits. $\endgroup$
    – Dmitri Pavlov
    Mar 29, 2021 at 16:07

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Daniel M. Kan defined adjoint functors in his paper Adjoint functors (written in 1956).

In Chapter II he defines limits and colimits of arbitrary small diagrams and proves that the limit and colimit functors are right and left adjoints to the diagonal functor in Theorems 7.8 and 8.6.

In Chapter III, he defines the notion of a limit-preserving functor in Definition 13.1. In Theorem 13.8 and 13.8* he proves that left/right adjoints preserve (co)limits.

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    $\begingroup$ The implication seems immediate to us now. But did Kan observe it then? $\endgroup$
    – Mike Shulman
    Mar 29, 2021 at 16:08
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    $\begingroup$ @MikeShulman: This is observed explicitly in Chapter III, I added a reference. $\endgroup$
    – Dmitri Pavlov
    Mar 29, 2021 at 16:23
  • $\begingroup$ The specific references you give are very helpful/useful for those of us interested in the historical record. Small quibble: your answer currently says 1956 but it seems the paper is from 1958. $\endgroup$
    – Yemon Choi
    Mar 30, 2021 at 16:15
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    $\begingroup$ @YemonChoi: The paper states on the first page that it was received by the editors on September 20, 1956, so it is indeed a 1956 paper. That it took 2 years for the AMS to print it is AMS's problem, not Kan's. On the other hand, after 65 years, the backlog at AMS journals is even worse now, despite all the improvements in technology. $\endgroup$
    – Dmitri Pavlov
    Mar 30, 2021 at 21:47
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    $\begingroup$ @DmitriPavlov I saw that, so we are just interpreting or allocating dates in different ways:) For me, the year of a paper is the year of its publication. (I am happy to say "this is a theorem of Kan from 1956".) As for publishing, I note once again that unlike most very online mathematicians I am at heart a (copy-)editor not a programmer. $\endgroup$
    – Yemon Choi
    Mar 30, 2021 at 23:37

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