Abelian categories versus abelian 2-categories

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.

💡 Research Summary

The paper addresses a question raised by Dupont concerning the relationship between abelian categories and abelian 2‑categories. Dupont proved that in any abelian 2‑category 𝒞 the full sub‑categories consisting of discrete objects (those whose only 2‑morphisms are identities) and codiscrete, or connected, objects (those whose only 1‑morphisms are equivalences) are each abelian categories, and moreover these two abelian categories are equivalent. He then asked whether every abelian category can be realized in this way, i.e. as the category of discrete (or equivalently codiscrete) objects inside some abelian 2‑category.

The author gives an elementary affirmative answer under the additional hypothesis that the given abelian category 𝒜 possesses either enough projective objects or enough injective objects. The argument proceeds in three main steps.

First, the necessary background on abelian 2‑categories is recalled. An abelian 2‑category is a 2‑category equipped with a zero object, biproducts, kernels, cokernels, and exactness conditions that lift the usual abelian axioms to the 2‑dimensional setting. Within such a 𝒞, a discrete object is one for which every 2‑morphism is the identity, while a codiscrete (or connected) object is one for which every 1‑morphism is an equivalence. Dupont’s theorem guarantees that the full sub‑categories 𝒞_disc and 𝒞_codisc are abelian and equivalent.

Second, assuming 𝒜 has enough projectives, the author constructs an explicit abelian 2‑category 𝒞 whose objects are formal finite direct sums of projective objects of 𝒜. A 1‑morphism between two such sums is a matrix of morphisms in 𝒜, and a 2‑morphism is a matrix of homotopies (i.e., morphisms making the appropriate squares commute). The usual abelian 2‑category axioms are verified by using the projective nature of the summands: kernels and cokernels are computed component‑wise, and exactness follows from the exactness in 𝒜. In this construction the discrete objects are precisely the direct sums of projectives themselves; any object of 𝒜 appears as a cokernel of a morphism between such sums, so the category of discrete objects is equivalent to 𝒜. Dually, if 𝒜 has enough injectives, one builds a 2‑category whose objects are finite products of injectives; the codiscrete objects then recover 𝒜.

Third, the author checks the equivalence rigorously. The functor sending a projective sum P to its underlying object in 𝒜 is fully faithful because morphisms between projective sums are exactly the morphisms in 𝒜. Essential surjectivity follows from the existence of a projective cover: every X∈𝒜 fits into an exact sequence P→X→0 with P projective, so X is isomorphic to the cokernel of a morphism between discrete objects, i.e., it lies in the essential image of the inclusion of 𝒞_disc. The dual argument works for injectives. Consequently, the discrete (or codiscrete) sub‑category of the constructed 2‑category is equivalent to the original abelian category 𝒜.

The paper concludes by noting that the hypothesis “enough projectives or enough injectives” is sufficient but not necessary; the general problem of characterizing those abelian categories that arise as discrete (or codiscrete) sub‑categories of some abelian 2‑category remains open. Nonetheless, the result provides a clean and concrete method for embedding a large class of familiar abelian categories—such as module categories over rings, categories of sheaves on a site with enough injectives, or categories of representations of finite‑dimensional algebras—into the higher‑categorical framework introduced by Dupont. This bridges classical homological algebra with the emerging theory of higher abelian structures and suggests further avenues for research, for example, investigating whether weaker conditions (e.g., existence of enough flat objects) might also suffice, or exploring the interplay with derived and stable ∞‑categories.