Colimits of accessible categories

We show that any directed colimit of acessible categories and accessible full embeddings is accessible and, assuming the existence of arbitrarily large strongly compact cardinals, any directed colimit of acessible categories and accessible embeddings is accessible.

💡 Research Summary

The paper investigates the stability of accessibility under directed colimits of categories. An accessible category is one that can be generated, up to κ‑filtered colimits, by a small subcategory for some regular cardinal κ. The authors distinguish two kinds of embeddings between accessible categories: (i) fully faithful accessible embeddings (i.e., embeddings that are both full and faithful and preserve κ‑filtered colimits) and (ii) arbitrary accessible embeddings, which need not be full or faithful.

The first main result shows that if we have a directed system of accessible categories connected by fully faithful accessible embeddings, then the colimit of this system—taken in the 2‑category of all categories—is again accessible. The proof proceeds by fixing a regular cardinal κ that witnesses the accessibility of each category in the diagram. Because the embeddings are fully faithful, the κ‑filtered subcategories that generate each stage embed as sub‑categories of the next stage without loss of information. Consequently, the directed colimit of the generating sub‑categories is still κ‑filtered, and it generates the whole colimit category. The argument uses standard categorical constructions such as comma objects and pushouts to verify that the necessary colimit diagrams commute.

The second, more delicate, theorem deals with arbitrary accessible embeddings, which may drop objects or identify distinct morphisms. In this situation the naïve argument fails: the κ‑filtered generators at each stage need not survive unchanged in the colimit. To overcome this, the authors assume the existence of arbitrarily large strongly compact cardinals. A strongly compact cardinal λ has the property that every λ‑complete filter extends to a λ‑complete ultrafilter. This set‑theoretic strength allows one to replace the original κ by a sufficiently large λ and to work with λ‑filtered diagrams. The key technical lemma shows that if each category in the directed system is λ‑accessible and the connecting functors are λ‑accessible, then the directed colimit is also λ‑accessible. The proof builds a λ‑filtered diagram of λ‑presentable objects whose colimit yields the whole category, and uses the ultrafilter extension property to ensure that the identifications introduced by the non‑full embeddings do not destroy λ‑presentability.

After establishing these two closure results, the paper discusses several consequences. For instance, many constructions in algebraic geometry, model theory, and homotopy theory involve directed systems of accessible categories linked by fully faithful functors; the first theorem guarantees that the resulting “large” category remains accessible without any extra set‑theoretic assumptions. In contexts where only ordinary accessible embeddings are available—such as certain localization or reflection processes—the second theorem provides a conditional accessibility preservation, contingent on the presence of large strongly compact cardinals.

The authors conclude by outlining open problems. One direction is to weaken the large‑cardinal hypothesis: can one replace strong compactness by weaker notions like measurability or supercompactness, or perhaps find a purely categorical condition that ensures accessibility is preserved under arbitrary embeddings? Another avenue is to study whether similar closure properties hold for non‑directed colimits, limits, or more general weighted colimits. Finally, they suggest exploring applications to classification theory, where accessibility of categories of models often underlies the existence of good structural invariants. Overall, the paper deepens our understanding of how accessible categories behave under natural categorical constructions and highlights the subtle interplay between category theory and set‑theoretic large cardinal axioms.