Are all subcategories of locally finitely presentable categories that are closed under limits and $\lambda$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case $\lambda = \aleph_0$ the answer is affirmative also for all iso-full subcategories, i. e., those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from $\aleph_0$ to an arbitrary $\lambda$.
Keywords: locally presentable category, reflective subcategory, elementary equivalence
2010 MSC: 18A40, 18C35
Theory and Applications of Categories, Vol. 30, 2015, No. 41, pp 1306-18.