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.

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    $\begingroup$ The category of Banach spaces (over the reals or complex numbers) and linear contractions is locally $\aleph_{1}$-presentable but not $\aleph_{0}$-presentable. It is easily seen that the ground field $k$ is a strong generator but its represented functor $Ban(k,-)$ only commutes with $\aleph_{1}$-filtered colimits. $\endgroup$ – Theo Buehler Dec 12 '10 at 21:20
  • $\begingroup$ A proof can be found in Borceux' handbook. And there certainly is a case that Banach spaces with contractions have been used by quite a few mathematicians. However, I don't know about topoi. $\endgroup$ – Theo Buehler Dec 12 '10 at 21:22
  • $\begingroup$ @other Theo: Oh, great. You should leave this as an answer, and I'll accept it. Banach spaces is definitely a useful category to working mathematicians. $\endgroup$ – Theo Johnson-Freyd Dec 12 '10 at 21:23
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    $\begingroup$ Any time you have infinitary operations, you'll have to use a higher $\kappa$ -- you have to use a regular cardinal bigger than the arity of your operations. That's why the Banach space example works -- you have infinite sums. $\endgroup$ – arsmath Dec 12 '10 at 21:36
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    $\begingroup$ Re: topoi, there are topoi of arbitrarily large local presentability. A topos is locally κ-presentable just when it has a site of definition where the covering sieves are generated by covering families of cardinality <κ. (Here the "infinitary operation" is gluing along a covering family.) So, for instance, the topos of sheaves on a non-compact space will be locally presentable but not locally $\aleph_0$-presentable. $\endgroup$ – Mike Shulman Dec 22 '10 at 5:22

The category of Banach spaces and contractions (over the reals or any other complete normed field, I think) is an example of an $\aleph_{1}$-presentable category which is not $\aleph_{0}$-presentable. The point is that the ground field is a strong generator and its represented functor $Ban(k,-)$ commutes with $\aleph_{1}$-filtered colimits. It essentially boils down to the fact about infinitary operations that arsmath pointed out in the above remark, details can be found in Borceux, Handbook of categorical algebra, vol. 2, Example 5.2.2 (e).

To see that the "unit ball functor" $Ban(k,-)$ does not commute with ordinary filtered colimits, you can take for example the identity $\varinjlim_{n < \omega} \ell^{1}(n) \cong \ell^{1}(\omega)$, where the $n$ are the finite ordinals and the maps the obvious inclusions. The set $\lim_{n < \omega} Ban(k,\ell^{1}(n))$ only consists of sequences with finitely many non-zero entries, while the set $Ban(k,\ell^{1}(\omega))$ has all summable sequences of norm $\leq 1$.

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Here are examples appearing in algebraic topology.

The category of $\mathrm{Ext}$-$p$-complete abelian groups, as discussed in Homotopy limits, completions and localizations (Chapter VI, Sections 2-4) is locally $\aleph_1$-presentable but not locally $\aleph_0$-presentable. This follows from the fact that the $p$-adic integers $\mathbb{Z}_p$ are an object that is $\aleph_1$-presentable but not $\aleph_0$-presentable in that category.

Likewise, the category of $L$-complete modules, as discussed in Morava $K$-theories and localisation (Appendix A) is locally $\aleph_1$-presentable but not locally $\aleph_0$-presentable. More details can be found in this preprint (Appendix A.2).

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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.

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