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

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.

• Marta Bunge is still active at McGill, so you could ask her, but my guess would be maybe a decade earlier than this. Jul 2 at 15:10
• I would look at early papers which deal with Kan extensions (since the construction is the Kan extension along the Yoneda embedding). Maybe it is even in the last sections of Kan's Adjoint functors from 1958 (where he defines limits, adjoints and, well, Kan extensions - it all started there...), which I skimmed through, but I doubt it. Jul 3 at 0:48
• I have sent an email to Marta Bunge and will update this question accordingly soon. Jul 5 at 13:49

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.

• Thanks, this is a good suggestion. I took a look through and couldn't find a reference for this result, but it's a useful upper bound on when the result was known. Jul 2 at 16:13
• (offtopic) Did any other author except for Lambek use the terms "supremum" and "infimum" for "colimit" and "limit"? Jul 2 at 21:05
• Lambek doesn't mention or prove the universal property in that book. Also because he was mainly interested in cocompletions which preserve existing colimits. Jul 2 at 21:07
• @MartinBrandenburg OK, never mind then. I unthinkingly assumed that the word "completion" in Lambek's title referred to a universal property, but that's a 2021 view of 1966. E.g. Isbell's 1966 paper "Structure of categories" contains this sentence (first paragraph): "By a completion of [a category] $A$ is meant a complete caegory in which $A$ is fully embedded so that no complete full proper subcategory contains it". Jul 2 at 22:09
• I like the terminology limit=infimum and colimit=supremum very much. For many diagrams limits or colimits are very poor approximations and only give very little information. This is very similar to infima and suprema in partially ordered sets. Oct 12 at 13:13

The earliest reference I can find to the universal property of the presheaf construction is Remark 2.29 of Ulmer's Properties of Dense and Relative Adjoint Functors (1968). 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. Note that Ulmer actually considers the universal property for arbitrary (possibly large) categories, by taking small presheaves rather than arbitrary presheaves.

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.

• This is cool! I always wondered where the statement of Lemma 1.1 appeared in the literature (I have used this from time to time). But Pitts doesn't prove that the two 2-functors are inverse to each other. What a pitty. Jul 5 at 21:11
• My comment refered to the 1st version of the answer and thus to Pitts' paper. Regarding your edit: Okay, so it's not my personal fault that I never know if one of my results is new or not. It happened all the time, even to the pioneers in the field. Jul 6 at 15:30
• The result is much older than Pitts’ 1985 paper and was absolutely part of the categorical folklore before then. For example, it is Theorem 4.51 in Kelly’s 1982 book tac.mta.ca/tac/reprints/articles/10/tr10abs.html, and it is surely in Gabriel-Ulmer’s 1971 SLN volume, and also probably in SGA4. (I don’t have the last two references handy so can’t confirm at the moment.) Jul 6 at 22:04
• @AlexanderCampbell: thanks for that, I can't believe I didn't check Basic concepts. It seems it was part of the categorical folklore in some places, but perhaps not others, then. Jul 6 at 22:08
• In "Limits of small functors" (2007) Day and Lack credit "V-enriched small presheaves giving the free small-cocompletion of large V-categories" as due to Lindner. As I understand it they refer to theorem 2.11 of Lindner's "Morita equivalences of enriched categories" (1974). Lindner cites Ulmer's paper for the unenriched version of this result. Oct 12 at 12:12