Accessible images revisited
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- by A. Brooke-Taylor and J. Rosický PDF
- Proc. Amer. Math. Soc. 145 (2017), 1317-1327 Request permission
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.References
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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
- MR Author ID: 870683
- Email: a.brooke-taylor@bristol.ac.uk
- J. Rosický
- Affiliation: Department of Mathematics and Statistics, Masaryk University, Faculty of Sciences, Kotlářská 2, 60000 Brno, Czech Republic
- MR Author ID: 150710
- Email: rosicky@math.muni.cz
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1317-1327
- MSC (2010): Primary 03E55, 18C35
- DOI: https://doi.org/10.1090/proc/13190
- MathSciNet review: 3589328