TITLE: What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locally presentable?
QUESTION [16 upvotes]: Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$
Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less than $\kappa$.  A $\kappa$-directed category is a poset in which every $\kappa$-small set has an upper bound.  A $\kappa$-directed colimit is the colimit of a diagram for which the indexing category $\kappa$-directed.  An object $a$ of a category is $\kappa$-small if $\Hom(a,-)$ preserves $\kappa$-directed colimits. A category is $\kappa$-locally presentable if it is (locally small and) cocomplete and there exists a set of objects, all of which are $\kappa$-small, such that the cocompletion of (the full subcategory on) this set in the category is the entire category.
It is a fact that every $\kappa$-locally presentable category is also $\lambda$-locally presentable for every $\lambda > \kappa$.
In a current research project, we have some constructions that work naturally for $\kappa$-locally presentable categories for arbitrary regular cardinals $\kappa$.  But all of our applications seem to be to $\aleph_0$-locally presentable categories.  For example, for any ring $R$, I'm pretty sure that the category of $R$-modules is $\aleph_0$-locally presentable.  (Every module has a presentation; any particular element or equation in the module is determined by some finite subpresentation.)  The category of groups is $\aleph_0$-locally presentable (by the same argument).  I'm told that every topos is locally presentable for some $\kappa$; is a topos necessarily $\aleph_0$-locally presentable?
Indeed, although I know of some categories that are not $\aleph_0$-locally presentable but are $\kappa$-locally presentable for some larger $\kappa$; the example, apparently, is the poset of ordinals strictly less than $\kappa$ for some large regular cardinal $\kappa$.  But this is not a category I have ever encountered "in nature"; it's more of a zoo specimen.  Hence my somewhat ill-defined question:

Question: Do there exist "in nature" (or, "used by working mathematicians") categories that are $\kappa$-locally presentable for some $\kappa > \aleph_0$ but that are not $\aleph_0$-locally presentable?

Put another way, is there any use to having a construction that works for all $\kappa$?
So far, I haven't thought much about general topoi, and we do want our project to include those, so I would certainly accept as an answer "yes, this particular topos".  But there might be other "representation theoretic" categories, or other things.

REPLY [5 votes]: The category of $\Delta$-generated spaces (the full category of the category of general topological spaces which are colimits of simplices) is locally $\lambda$-presentable for a regular cardinal $\lambda \geq 2^{\aleph_0}$. See http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html. This category contains all realizations of simplicial sets, all CW-complexes, all spheres, etc... i.e. all topological spaces used in algebraic topology. It is moreover cartesian closed.