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For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given by set-indexed families $(A_i)_{i \in I}$ of objects of $\mathbf C$, and morphisms $(A_i)_{i \in I} \to (B_j)_{j \in J}$ are given by functions $f : I \to J$ together with families of morphisms $(f_i : A_i \to B_{f(i)})_{i \in I}$.

The $\mathrm{Fam}$-construction is well known in category theory. However, it is difficult to track down the original reference, as it appears to have been well known for a long time, and is often introduced without citing an earlier text. I would like to know where this construction and its universal property first appears in print (or elsewhere).

The two earliest references I have found so far are:

  • Diers's Catégories localisables (1977), where the notation $\mathbb F(\mathbf C)$ is used; the universal property does not appear to be given there.
  • Bénabou's Fibered Categories and the Foundations of Naive Category Theory (1985), where the notation $\mathrm{Fam}(\mathbf C)$ appears and its universal property is given.

Considering that the same name for the construction is given in both papers, but Bénabou does not cite Diers, it seems plausible that there is an earlier reference for the construction and it was already well-known in 1985. I would be happy to receive information otherwise (though I appreciate is it difficult to prove there is not an earlier reference).

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    $\begingroup$ It might also have been folklore before 1977. Early category theory had a lot of folklore results. $\endgroup$
    – Simon Henry
    Commented May 28, 2021 at 3:46
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    $\begingroup$ I think this is a good question. But questions like these are really hard to answer. Oftentimes people reinvent things without knowing it. That some older paper is not cited may also happen just because the author was not aware of it. Sometimes also people at different locations get the same idea, maybe even at the same time, the ideas circulate within seminars, and it is almost impossible to track down the first written or spoken appearance of that idea. And maybe it's also not that important? It is of course nice to cite the original paper of an idea, but I don't see any other reason. $\endgroup$
    – Martin Brandenburg
    Commented May 28, 2021 at 22:59
  • $\begingroup$ For the specific question, my guess is that this has been known already in the 60s, because it is a very basic construction - perhaps not using the notation $\mathrm{Fam}(\mathcal{C})$ though. $\endgroup$
    – Martin Brandenburg
    Commented May 28, 2021 at 23:01
  • $\begingroup$ Related question: what is an early reference for the cocompletion of a category? I am pretty sure that the authors of such a paper must have also known the coproduct cocompletion. $\endgroup$
    – Martin Brandenburg
    Commented May 28, 2021 at 23:09
  • $\begingroup$ @MartinBrandenburg: it is true that the free coproduct completion may have been discovered independently many times. However, the consistent terminology in these early references (namely, "the category of families") suggests there is a common ancestor, as I don't think this terminology is inevitable. It could be that there isn't a good earliest reference. However, I would rather cite an original reference if one exists. $\endgroup$
    – varkor
    Commented May 29, 2021 at 1:07

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