Reflections in locally presentable categories

Question

In this paper, -4th line in the first paragraph on the (first) page 89,

then each full subcategory of $\cal H$ closed under limits ...

should or should not the word reflective be present:

then each full reflective subcategory of H closed under limits ... ?

I think that the word "reflective" is omitted by mistake, but I'm not sure.

It seems strange to me that if $\cal H$ is a locally presentable category then each full (but possibly non-reflective) subcategory of $\cal H$ closed under limits is closed under $\alpha$-filtered colimits for some regular cardinal $\alpha$.

Answer 1

The word "reflective" can be left out, since it is implied by the other properties. We are talking here about a statement that is equivalent to Vopěnka's principle, which is a very strong large cardinal axiom. It can be stated in multiple equivalent ways (see e.g. nLab and Wikipedia for a few quick examples). On of these ways is the sentence you refer to in the paper.

So why can we leave out "reflective" in this particular statement? There is also a weak Vopěnka's principle, which is of course implied by Vopěnka's principle. The weak Vopěnka's principle is equivalent to:

For every locally presentable category $\mathcal{C}$, every full subcategory $\mathcal{D} \hookrightarrow \mathcal{C}$ which is closed under limits is a reflective subcategory.

This is Theorem 2.7 on the nLab page I linked before. It can also be found as Theorem 6.22 in the book Locally Presentable and Accessible Categories by J. Adámek and J. Rosický (well, the direction we are interested in, the other direction is Example 6.23). The entire chapter 6 in that book is dedicated to Vopěnka's principle, so it might be worth a look.