Locally class-presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory.

💡 Research Summary

The paper introduces a systematic generalisation of the well‑established notions of locally presentable and accessible categories by allowing the class of presentable objects to be a proper class rather than a set. In the classical setting a category K is λ‑accessible if it possesses λ‑filtered colimits and there exists a set A of λ‑presentable objects such that every object of K is a λ‑filtered colimit of objects from A. The authors drop the “set” requirement and define a class‑λ‑accessible category as one that has λ‑filtered colimits and a class A of λ‑presentable objects with the same colimit generation property. When such a category is also complete and cocomplete they call it class‑λ‑presentable (or class‑locally presentable).

The motivation stems from examples that fall outside the ordinary framework: the category of small presheaves on a large category, various Ind‑categories, and many constructions appearing in abstract homotopy theory. These examples naturally involve a proper class of generators, so the classical definition is too restrictive.

Key definitions and results:

1. Class‑λ‑accessible categories (Def. 2.1) – require λ‑filtered colimits and a class of λ‑presentable objects generating all objects via λ‑filtered colimits.

2. Class‑λ‑presentable categories – a class‑λ‑accessible category that is both complete and cocomplete.

3. Examples –

   • For any (possibly large) category A, the category P(A) of small presheaves (i.e., functors Aᵒᵖ → Set that are small colimits of representables) is class‑ω‑accessible. It is always cocomplete but may fail to be complete (e.g., when A is a large discrete category).
   • Indₗ(A), the full subcategory of P(A) consisting of small λ‑filtered colimits of representables, is class‑λ‑accessible. When A is λ‑co‑complete and “approximately complete” (every diagram has a weakly initial set of cones), Indₗ(A) becomes complete (Thm 2.4).
   • The category Top of topological spaces is not class‑locally presentable because only discrete spaces are presentable.

4. Characterisation via reflective subcategories (Thm 2.6) – a category K is class‑λ‑presentable iff it is equivalent to a full reflective subcategory of P(A) that is closed under λ‑filtered colimits, where A = presₗ(K) (the full subcategory of λ‑presentable objects). This mirrors the classical equivalence between accessible categories and reflective subcategories of presheaf categories, now lifted to the class level.

5. Class‑accessible functors (Def. 2.7) – a functor between class‑λ‑accessible categories preserving λ‑filtered colimits. A functor is strongly class‑λ‑accessible if it also preserves λ‑presentable objects. Unlike the set‑based case, strong accessibility does not automatically follow from accessibility; the paper gives a counterexample using a large discrete category.

6. Closure under limits – The central technical result (Prop. 3.1) shows that the pseudopullback of two strongly class‑λ‑accessible functors is again class‑λ⁺‑accessible, and the projection functors are strongly class‑λ⁺‑accessible. This extends the well‑known fact that accessible categories are closed under pseudopullbacks to the class‑level setting.

7. Additional observations – Class‑locally presentable categories need not be cowell‑powered (e.g., the class of all ordinals with a top element). The paper also sketches a theory of weak factorisation systems and injectivity in this broader context, indicating potential applications to model categories that are not combinatorial.

Overall, the authors develop a robust framework that retains the essential structural properties of accessible categories—such as closure under limits, reflective embeddings into presheaf categories, and a workable notion of presentability—while accommodating categories whose generators form a proper class. This opens the door to applying the machinery of accessible category theory to a wider array of homotopical and categorical constructions, notably those involving large indexing categories, Ind‑objects, and non‑combinatorial model structures.