Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Accessible images revisited


Authors: A. Brooke-Taylor and J. Rosický
Journal: Proc. Amer. Math. Soc. 145 (2017), 1317-1327
MSC (2010): Primary 03E55, 18C35
DOI: https://doi.org/10.1090/proc/13190
Published electronically: November 18, 2016
MathSciNet review: 3589328
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136
  • [2] Arthur W. Apter, Measurability and degrees of strong compactness, J. Symbolic Logic 46 (1981), no. 2, 249-254. MR 613279, https://doi.org/10.2307/2273618
  • [3] Joan Bagaria and Andrew Brooke-Taylor, On colimits and elementary embeddings, J. Symbolic Logic 78 (2013), no. 2, 562-578. MR 3145196
  • [4] Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, and Jiří Rosický, Definable orthogonality classes in accessible categories are small, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 3, 549-589. MR 3323199, https://doi.org/10.4171/JEMS/511
  • [5] Joan Bagaria and Menachem Magidor, Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857-1877. MR 3152715, https://doi.org/10.1090/S0002-9947-2013-05871-0
  • [6] Joan Bagaria and Menachem Magidor, On $ \omega _1$-strongly compact cardinals, J. Symb. Log. 79 (2014), no. 1, 266-278. MR 3226024, https://doi.org/10.1017/jsl.2013.12
  • [7] Will Boney, Tameness from large cardinal axioms, J. Symb. Log. 79 (2014), no. 4, 1092-1119. MR 3343531, https://doi.org/10.1017/jsl.2014.30
  • [8] W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Preprint. arXiv:1509.01191.
  • [9] B. Chorny and J. Rosický, Class-locally presentable and class-accessible categories, J. Pure Appl. Algebra 216 (2012), no. 10, 2113-2125. MR 2925805, https://doi.org/10.1016/j.jpaa.2012.01.015
  • [10] Katsuya Eda and Yoshihiro Abe, Compact cardinals and abelian groups, Tsukuba J. Math. 11 (1987), no. 2, 353-360. MR 926461
  • [11] Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
  • [12] Rami Grossberg and Monica VanDieren, Categoricity from one successor cardinal in tame abstract elementary classes, J. Math. Log. 6 (2006), no. 2, 181-201. MR 2317426, https://doi.org/10.1142/S0219061306000554
  • [13] Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings. MR 1994835
  • [14] M. Lieberman and J. Rosický, Classification theory for accessible categories, J. Symb. Log. 81 (2016), no. 1, 151-165. MR 3471133, https://doi.org/10.1017/jsl.2014.85
  • [15] Michael Makkai and Robert Paré, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics, vol. 104, American Mathematical Society, Providence, RI, 1989. MR 1031717
  • [16] Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327-359. MR 0357121
  • [17] Mike Prest, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. MR 2530988
  • [18] G. Raptis and J. Rosický, The accessibility rank of weak equivalences, Theory Appl. Categ. 30 (2015), No. 19, 687-703. MR 3356340
  • [19] Jiří Rosický, More on directed colimits of models, Appl. Categ. Structures 2 (1994), no. 1, 71-76. MR 1283214, https://doi.org/10.1007/BF00878503
  • [20] J. Rosický, On combinatorial model categories, Appl. Categ. Structures 17 (2009), no. 3, 303-316. MR 2506258, https://doi.org/10.1007/s10485-008-9171-2
  • [21] S. Shelah, Maximal failures of sequence locality in a.e.c., arXiv:0903.3614v3.
  • [22] Saharon Shelah and Michael Makkai, Categoricity of theories in $ L_{\kappa \omega },$ with $ \kappa $ a compact cardinal, Ann. Pure Appl. Logic 47 (1990), no. 1, 41-97. MR 1050561, https://doi.org/10.1016/0168-0072(90)90016-U

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E55, 18C35

Retrieve articles in all journals with MSC (2010): 03E55, 18C35


Additional Information

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

DOI: https://doi.org/10.1090/proc/13190
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

American Mathematical Society