Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of $\mathcal{K}$) such that if $\mathcal{K}$ is categorical in some $\lambda \geq \mu$ (i.e. $\mathcal{K}$ has exactly one model of size $\lambda$ up to isomorphism), then $\mathcal{K}$ is categorical in $\theta$ for all $\theta \ge \mu$.

Due to a result of Will Boney it is known that a weaker form of this conjecture is consistent up to consistency of proper class many strongly compact cardinals. My question is about the inverse direction.

Question: Does Shelah's categoricity conjecture have large cardinal strength? If there is no known result in this direction, what are some known notable connections or theorems that suggest possible existence of some large cardinal strength for this conjecture or an approach for proving such a statement?

The only connection that I'm aware of and seems related to the problem of the consistency strength of Shelah's categoricity conjecture is a connection that exists between abstract elementary classes and accessible categories on one hand and accessible categories and Vopěnka's principle on the other hand. The latter appears in the following characterizations of Vopěnka's principle in terms of category theory (see here):

Theorem: The followings are equivalent:

(1) The Vopěnka's principle.

(2) Every discrete full subcategory of a locally presentable category is small.

(3) For $C$ a locally presentable category, every full subcategory $D↪C$ which is closed under colimits is a coreflective subcategory.

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    $\begingroup$ See LARGE CARDINAL AXIOMS FROM TAMENESS IN AECS. In this paper it is shown that to get tameness, some large cardinal assumptions are needed. Of course it does not imply if we need large cardinals for Shelah's conjecture. On the other hand Vasey has proved Shelah's conjecture for universal classes without the use of any large cardinals. See Shelah's eventual categoricity conjecture in universal classes. $\endgroup$ – Mohammad Golshani Nov 26 '15 at 8:59
  • $\begingroup$ @MohammadGolshani (+1) Very nice references. Thanks a lot! $\endgroup$ – user82740 Nov 26 '15 at 16:53
  • $\begingroup$ In the opposite of the direction you're interested in, note that Boney's result has been improved slightly by Brooke-Taylor and Rosicky. See these slides, which I believe refer to this paper. $\endgroup$ – Tim Campion Jul 7 '16 at 16:17

Please note that the paper by Boney & Unger establishes (with Bone's earlier result) that tameness is a large cardinal axiom. It does not shade light on the original question.

Under the assumptions of tameness and the amalgamation property, strong versions of Shelah's categoricity conjectures are ZFC theorems. See several papers by Sebastien Vasey and others. There is much evidence that an exciting classification theory can be developed for such AECs (tame and AP).

While some people believe that tameness and the AP could be derived from categoricity above a certain Hanf number (I was the first to make such conjectures). This is not clear at all. I can imagine that a couterexample to Shelah's categoricty conjecture exists.

  • $\begingroup$ (+1) Very eye-opening answer. Thanks. $\endgroup$ – user82740 Nov 28 '15 at 18:35

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