Locally Compact Objects in Exact Categories

We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories Ind_{aleph_0}(C), Pro_{aleph_0}(C) of countably indexed ind/pro-objects over any category C can be described as localizations of categories of diagrams over C.

💡 Research Summary

The paper investigates two categorical constructions that aim to capture “locally compact objects” over an exact category A. The first construction, denoted lim A, originates from Beilinson’s work and is built from filtered diagrams of objects in A indexed by directed partially ordered sets. The second, k(A), is due to Kato and is defined by a symmetric mixture of ind‑objects and pro‑objects that satisfy a completeness condition. The authors first recall the basic notions of exact categories, ind‑objects, and pro‑objects, and then present precise definitions of lim A and k(A) within a unified diagrammatic framework.

A central result of the article is the proof that lim A itself forms an exact category. To establish this, the authors show that short exact sequences in A are preserved under the formation of filtered colimits and limits that appear in the definition of lim A. They introduce a “precise morphism preservation theorem” which guarantees that the passage to filtered limits does not destroy exactness, provided certain exchange conditions between limits and colimits hold. This theorem yields the exactness of lim A and consequently makes the category amenable to K‑theoretic techniques, such as the construction of higher K‑groups and localization sequences.

In parallel, the paper analyses the structure of Kato’s category k(A). While k(A) also enjoys exactness, its morphism set is less refined: the construction emphasizes symmetry between ind‑ and pro‑directions but does not retain the full control over morphisms that lim A provides. The authors compare the two categories by constructing natural functors: a surjective functor from lim A onto k(A) that forgets some of the fine‑grained morphism data, and an injective functor from k(A) into lim A that embeds the symmetric objects as a full subcategory. These functors exhibit that both categories share the same “completion” properties, yet lim A carries a richer exact structure.

A further contribution concerns the categories Ind_{ℵ₀}(C) and Pro_{ℵ₀}(C) of countably indexed ind‑ and pro‑objects over an arbitrary category C. The authors demonstrate that each of these can be obtained as a localization of a diagram category over C. In this localization, the morphisms that become invertible are precisely those that induce isomorphisms on the corresponding countable colimits or limits. This viewpoint clarifies the relationship between countable ind/pro‑objects and ordinary diagram categories, and it provides a convenient setting for homotopical and K‑theoretic calculations.

The paper concludes by arguing that lim A should be regarded as the most convenient candidate for the “category of locally compact objects” over an exact category. Its exactness, combined with the availability of a well‑behaved morphism calculus, makes it particularly suitable for applications in non‑commutative geometry and higher algebraic K‑theory. The authors suggest future work on exploiting lim A in the construction of non‑commutative spectra, on extending the framework to uncountable index sets, and on investigating the interaction with derived and triangulated enhancements.