Original reference for categories of presheaves as free cocompletions of small categories

Question

It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A \to \widehat{\mathbf A}$ exhibits $\widehat{\mathbf A}$ as the cocompletion of $\mathbf A$ under small colimits.

Where was this first observed? If an observation first appears independently of a proof, I would be interested in knowing where a proof first appears too.

Bunge's 1966 thesis Categories of Set-Valued Functors seems a likely candidate, as it is concerned with properties of presheaf categories. However, I have been unable to obtain a copy of the thesis, so I do not know whether it appears here, and it is not mentioned in Bunge's summary of thesis, Regular categories.

Answer 1

The earliest reference I can find to the universal property of the presheaf construction is Proposition 9.1 of André's Categories of Functors and Adjoint Functors (1966).

There is an earlier reference for the universal property of the Ind-completion (i.e. cocompletion under filtered colimits) in Proposition 8.7.3 of SGA4 (dated 1963–1964, but published in 1972), which suggests the universal property of free cocompletion was also known as this time, though an explicit statement does not appear.

For the more general case of the small presheaf construction as the free cocompletion of a not-necessarily-small category, Remark 2.29 of Ulmer's Properties of Dense and Relative Adjoint Functors (1968) appears to be the earliest reference. However, the proof is only lightly sketched, and in the introduction Ulmer states:

As an application of relative adjoints we will show in a subsequent paper that every category $\mathbf M'$ admits a free right complete category.

As far as I can tell, this paper never appeared.

In the enriched context, the universal property first appears as Theorem 2.11 of Lindner's Morita equivalences of enriched categories (1974), where the Lindner attributes the unenriched result to Ulmer.

Answer 2

This isn't a direct answer to your question, but you might find the answer by looking in here:

Joachim Lambek, Completions of Categories. Springer Lecture Notes in Mathematics 24, 1966.

The acknowledgement says that the notes began life as a graduate course at McGill in 1965.

I gather that Lambek was generally careful to give accurate references. All I can see at a first glance is something a bit vague: in the second paragraph of the introduction (p.2), he writes:

Can every small category $\mathbf{A}$ be embedded as a (full) sup-dense subcategory into a sup-complete category $\mathbf{A}'$? The answer, also noted by others, is yes: Take $\mathbf{A}'$ to be the category of functors from $\mathbf{A}^\circ$, the opposite category of $\mathbf{A}$, to $\mathrm{Ens}$, the category of sets.

(Lambek used "sup" for what is now called colimit.)

This introductory paragraph mentions neither the original sources nor the universal property of $[\mathbf{A}^\circ, \mathrm{Ens}] = [\mathbf{A}^\textrm{op}, \mathbf{Set}]$. But it's only an introduction --- maybe he goes into detail somewhere in the main text.