Abstract elementary classes and accessible categories

Abstract

We investigate properties of accessible categories with directed colimits and their relationship with categories arising from Shelah's Abstract Elementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.cardinals λ such that L is λ-categorical and, at the same time, a proper class of cardinals λ such that L is not λ-categorical.If this K has directed colimits then the categoricity conjecture fails for the class (4). By 4.12, however, this simple trick does not help for categories in class (3) of our hierarchy.Remark 6.4. Shelah's conjecture seems to be unknown even for finitely accessible categories, lying at level (1) of the hierarchy. It would be interesting to understand whether the exquisite Galois-theoretic machinery of [3] can be brought to bear implications in this setting.

Internal sizes in μ-abstract elementary classes

“…This weakens the GCH hypothesis in [BR12, 2.3(5)] and also shows (Remark 2.3) that sufficiently large presentability ranks are always successors if there is an ω 1 -strongly compact cardinal. Note that we could have obtained the result directly by carefully examining the proof in [BR12], but Theorem 3.10 is new and can be used even in situations where SCH does not hold (see Corollary 5.5).…”

Elementary equivalences and accessible functors

“…(7) Finally, for any Abstract Elementary Class in Shelah's sense, the class of structures and strong embeddings form an accessible category. The relation between AEC's and accessible categories has been investigated by several articles; see, for example, Beke-Rosický [6].…”

Universal abstract elementary classes and locally multipresentable categories

“…Later Makkai and Paré [MP89] independently rediscovered these results and showed that they say essentially the same thing in different languages. The connection between AECs and accessible categories was then studied more closely by the first author [Lie11] and independently by Beke and the second author [BR12]. These works characterized AECs as special kinds of accessible categories with all directed colimits and whose morphisms are monomorphisms.…”

“…This conjecture has profoundly shaped the development of the field (see the introduction of [Vas17a] for a survey and history of the conjecture). Note that Shelah's eventual categoricity conjecture can be rendered as a purely category-theoretic statement, with cardinalities replaced by presentability ranks-see [BR12,§6])-and hence can be posed in relation to general accessible categories.…”