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Proceedings of the American Mathematical Society

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Accessible images revisited

Authors: A. Brooke-Taylor and J. Rosický
Journal: Proc. Amer. Math. Soc. 145 (2017), 1317-1327
MSC (2010): Primary 03E55, 18C35
Published electronically: November 18, 2016
MathSciNet review: 3589328
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Abstract: We extend and improve the result of Makkai and Paré (1989) that the powerful image of any accessible functor $ F$ is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of $ L_{\mu ,\omega }$-compact cardinals for sufficiently large $ \mu $, and also show that under this assumption the $ \lambda $-pure powerful image of $ F$ is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result -- one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the set-theoretic universe.

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Additional Information

A. Brooke-Taylor
Affiliation: School of Mathematics, University of Bristol, Howard House, Queen’s Avenue, Bristol, BS8 1SN, United Kingdom

J. Rosický
Affiliation: Department of Mathematics and Statistics, Masaryk University, Faculty of Sciences, Kotlářská 2, 60000 Brno, Czech Republic

Received by editor(s): June 5, 2015
Received by editor(s) in revised form: September 26, 2015, and February 26, 2016
Published electronically: November 18, 2016
Additional Notes: The first author was supported by the UK EPSRC Early Career Fellowship EP/K035703/1, “Bringing set theory and algebraic topology together”.
The second author was supported by the Grant agency of the Czech Republic under the grant P201/12/G028.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society