Representation of categories

One of the earliest theorems in category theory stated that an abelian category could be represented faithfully by exact functors into the category Ab of abelian groups [Freyd, 1964], [Lubkin, 1961] and [Heron, unpublished]. Then Mitchell [1965] showed that every such category had a full exact embedding into a module category. An equivalent formulation is that every abelian category into a category of additive functors into Ab or even into a Set-valued functor category. Mitchell’s argument was based on what was essentially the earliest theorem in category theory: Grothendieck’s theorem that every AB5 category with a generator had an injective cogenerator [Grothendieck 1957]. Continuing in this vein, I showed in [Barr, 1971] that every regular category had a full, regular embedding into a category of set-valued functors. In doing this, I first tried to mimic Grothendieck’s argument. Unfortunately, I never succeeded in demonstrating a non-abelian version of Grothendieck’s theorem. There is a very good reason for that: it is false, see Corollary 12, below. Instead, the proof was based on showing that the obvious non-abelian adaptation of Lubkin’s argument [Lubkin, 1960] not only continued to give a family of embeddings, but when the functors were put together into a category (with all natural transformations between them), the embedding was even full. The proof was difficult, to say the least (it has been described as ‘hermetic’), and the theorem has apparently had little impact although at least one better proof has been published since [Makkai, 1980]. Here we give yet another proof (Corollary 15). Surprisingly, it is based on Grothendieck’s argument. It turns out that a weaker condition than injectivity is sufficient to make the proof work and the non-abelian version of Grothendieck’s argument is sufficient to give that weaker condition. This argument ultimately goes back to Baer’s proof that divisible abelian groups are injective. Here is an outline of the new proof.