Accessible images revisited

Brooke-Taylor, A.; Rosický, J.

doi:10.1090/proc/13190

Accessible images revisited
HTML articles powered by AMS MathViewer

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

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, DOI 10.1017/CBO9780511600579
Arthur W. Apter, Measurability and degrees of strong compactness, J. Symbolic Logic 46 (1981), no. 2, 249–254. MR 613279, DOI 10.2307/2273618
Joan Bagaria and Andrew Brooke-Taylor, On colimits and elementary embeddings, J. Symbolic Logic 78 (2013), no. 2, 562–578. MR 3145196, DOI 10.2178/jsl.7802120
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, DOI 10.4171/JEMS/511
Joan Bagaria and Menachem Magidor, Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857–1877. MR 3152715, DOI 10.1090/S0002-9947-2013-05871-0
Joan Bagaria and Menachem Magidor, On $\omega _1$-strongly compact cardinals, J. Symb. Log. 79 (2014), no. 1, 266–278. MR 3226024, DOI 10.1017/jsl.2013.12
Will Boney, Tameness from large cardinal axioms, J. Symb. Log. 79 (2014), no. 4, 1092–1119. MR 3343531, DOI 10.1017/jsl.2014.30
W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Preprint. arXiv:1509.01191.
B. Chorny and J. Rosický, Class-locally presentable and class-accessible categories, J. Pure Appl. Algebra 216 (2012), no. 10, 2113–2125. MR 2925805, DOI 10.1016/j.jpaa.2012.01.015
Katsuya Eda and Yoshihiro Abe, Compact cardinals and abelian groups, Tsukuba J. Math. 11 (1987), no. 2, 353–360. MR 926461, DOI 10.21099/tkbjm/1496160585
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
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, DOI 10.1142/S0219061306000554
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
M. Lieberman and J. Rosický, Classification theory for accessible categories, J. Symb. Log. 81 (2016), no. 1, 151–165. MR 3471133, DOI 10.1017/jsl.2014.85
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, DOI 10.1090/conm/104
Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327–359. MR 357121, DOI 10.1016/0003-4843(75)90009-1
Mike Prest, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. MR 2530988, DOI 10.1017/CBO9781139644242
G. Raptis and J. Rosický, The accessibility rank of weak equivalences, Theory Appl. Categ. 30 (2015), Paper No. 19, 687–703. MR 3356340
Jiří Rosický, More on directed colimits of models, Appl. Categ. Structures 2 (1994), no. 1, 71–76. MR 1283214, DOI 10.1007/BF00878503
J. Rosický, On combinatorial model categories, Appl. Categ. Structures 17 (2009), no. 3, 303–316. MR 2506258, DOI 10.1007/s10485-008-9171-2
S. Shelah, Maximal failures of sequence locality in a.e.c., arXiv:0903.3614v3.
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, DOI 10.1016/0168-0072(90)90016-U