Metric Abstract Elementary Classes as Accessible Categories

Abstract

We show that metric abstract elementary classes (mAECs) are, in the sense of [16], coherent accessible categories with directed colimits, with concrete ℵ 1directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC-an AEC-like category in which only the κ-directed colimits need be concrete-and develop the theory of such categories, beginning with a categorytheoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [16] yield a proof that any categorical mAEC is µ-d-stable in many cardinals below the categoricity cardinal. 2 M. LIEBERMAN AND J. ROSICKÝ(4) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, where "coherence" is a property of U corresponding to the coherence axiom for AECs. (5) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, and satisfies the iso-fullness condition described in Remark 3.5 in [16]-such a category is equivalent to an AEC.Certain essential results from the theory of AECs are shown to hold at greater levels of generality: categories of the form (2) satisfy a presentation theorem generalizing that of Shelah and, if large, admit a robust EM-functor. Categories of the form (3) allow the development of Galois types and satisfy a generalization of Boney's theorem on tameness under the assumption of a proper class of strongly compact cardinals (see [7]). Categories of the form (4) satisfy the essential technical condition that Galois saturation corresponds to, in AEC terms, model-homogeneity, and support the development of a fragment of classification theory. We show in Section 3 that any mAEC K is an accessible category with directed colimits, and note that, if we take U : K → Set to be the usual underlying set functor, K is coherent with concrete monomorphisms. As is well known, though, directed colimits in K need not be concrete: when taking the colimit of a chain of structures in K, we must, in general, take the completion of the union of the underlying sets. This would seem to place us, at best, in type (2) above, which is already sufficient to give a presentation theorem and guarantee the existence of an EM-functor for a general mAEC-this is in itself a generalization of [10], the results of which hold only in the homogenous case. As we note in Remark 2.9, however, mAECs do have concrete ℵ 1 -directed colimits, which suggests that we may benefit from a generalization of the hierarchy of [16], considering categories with concrete κ-directed colimits for some κ. In case a category of this form has (not necessarily concrete) directed colimits, is coherent, has concrete monomorphisms, and is suitably replete and iso-full-the conditions of (4) above-we call it a κ-concrete AEC, or κ-CAEC for short. Incidentally, there is an alternative option already being pursued in, e.g., [8], namely to consider classes of structu...

Are Banach spaces monadic?

“…Then Met ∞ is complete and cocomplete and monoidal closed, with the internal hom providing the hom-set Met ∞ (X, Y ) with the sup-metric d(f, g) = sup{d(f x, gx) | x ∈ X}. Moreover, Met ∞ is locally ℵ 1 -presentable (see [11] 4.5 (3)). The category CMet ∞ of complete generalized metric spaces is locally ℵ 1 -presentable too (see [4] 2.3(2)).…”

Approximate Injectivity

“…Changing all distances d(f x, gx) to ε gives a distance function that satisfies all axioms of a generalized metric but the triangle inequality. Such structures are called semimetrics and, following [14] 4.5(3), Met ∞ is reflective in the corresponding category SMet ∞ : the reflector provides a semimetric space (X, d) with the metric d given by…”

Tameness in generalized metric structures

“…This is easy to verify, with the argument paralleling the one for complete metric spaces in [LR17b,3.5].…”

“…(b) The category cMet is ℵ 1 -accessible, but not finitely accessible, in part by 2.2(1)(b) and 2.2(2)(b). It is less clear that a space can be obtained as an ℵ 1 -directed colimit of its separable subspaces: an alternative argument can be given (see [LR17b,4.5(3)]). (c) If K is an AEC, closure under unions of chains-hence directed colimits, per 2.2(1)(a)-and the Löwenheim-Skolem property ensure that K is λ-accessible for all regular λ > LS(K), where LS(K) is the Löwenheim-Skolem number of K. The same holds for any mAEC, by the metric variants of the aforementioned axioms.…”

“…• The class need not be closed under unions of chains-the union of a chain of complete metric spaces need not be complete-but does contain the completion of such unions. Put another way, an mAEC has directed colimits, but they need not be concrete ([LR17b]). • In the Löwenheim-Skolem axiom, cardinality is replaced by density character.…”