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Let $F: A \to B$ and $G: B \to A$ be adjoint functors, with $F \dashv G$. There is a full subcategory $A'$ of $A$ consisting of those objects $a$ for which the unit map $a \to GF(a)$ is an isomorphism, and there is a dually-defined full subcategory $B'$ of $B$. It is an elementary exercise to show that $F$ and $G$ restrict to an equivalence $A' \simeq B'$.

Either of the equivalent categories $A'$ and $B'$ is called the invariant part or fixed category of the adjunction. There are other names too; the terminology hasn't settled down.

Q. Where did this general construction first appear in print?

Adjoint functors were introduced by Kan in 1958. I don't see this construction in his paper. But I guess someone must have mentioned or used it quite soon thereafter. I want to know who I should cite.

Let me make clear that I'm not asking about particular instances of this construction. It's the general construction I'm after. I'm also interested in early references in which the general construction plays a significant part, even if they're not the original source.

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  • $\begingroup$ I think we had a question about this before. Of course the subcategories $A'$ and $B'$ are likely to be empty. Someone of your eminence must have some deeper reason for asking about this: please let on. $\endgroup$
    – Paul Taylor
    Commented May 23, 2021 at 11:33
  • $\begingroup$ Thanks for the flattery, Paul :-) My reason is very boring: a referee has asked if I could add a citation for the construction, so I would like to add one. As for previous questions, I see now that there was one on MSE back in 2013 asking something similar but not quite identical; they wanted an extensive treatment rather than the first treatment/mention (math.stackexchange.com/questions/562947/…). But no one answered. $\endgroup$
    – Tom Leinster
    Commented May 23, 2021 at 11:40
  • $\begingroup$ Who firrst wrote down the symmetrical "triangle laws" for the unit and counit? Daniel Kan? They would either have commented on these equivalent categories or considered the point too trivial to mention. Either way, if you have to give such a citation, this would be it. $\endgroup$
    – Paul Taylor
    Commented May 23, 2021 at 11:46
  • $\begingroup$ Tom, are you also interested in non-original, but still old references? $\endgroup$
    – Martin Brandenburg
    Commented May 23, 2021 at 12:30
  • $\begingroup$ @MartinBrandenburg Yes, absolutely! $\endgroup$
    – Tom Leinster
    Commented May 23, 2021 at 12:32

1 Answer 1

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The best reference I know of so far is this:

J. Lambek and B. A. Rattray. Localization and duality in additive categories. Houston Journal of Mathematics 1 (1975), 87-100.

The result I mentioned appears as part of Theorem 1.1.

1975 seems late for this to be the first place where this result is stated explicitly, but I'm not aware of anything earlier.

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