Locally presentable vs compactly presentable categories

Question

I am struggling to appreciate the definition of locally presentable categories. Can someone please give a naturally occurring example (in geometry or algebra) of a category which is locally presentable (i.e. locally k-presentable for some cardinality k) which is not compactly presentable (i.e. locally finitely presentable)?

For example, according to Presheaf category is locally finitely presentable presheaf categories are locally finitely presentable.

Answer 1

Any algebraic structure with operations of more than finite arity will give you good examples, such as posets admitting $\kappa$-ary suprema or small categories admitting limits or colimits of size $\kappa.$ A trickier, but finally similar, example is the category of Banach spaces and continuous linear maps of norm no greater than $1$; the infinitary operations here are sums of convergent series, or equivalently limits of convergent sequences.

Any category along these lines will be $\kappa$-presentable for the least $\kappa$ larger than the arity of all the operations, but generally for no smaller $\kappa.$ (So $\kappa=\aleph_1$ for Banach spaces.) For the examples above, there are strong generators formed by, respectively, the one-point poset, the one-point category together with the arrow category, and the base field, say $\mathbb R.$ None of these objects are finitely presentable in their respective categories, since a countable filtered colimit of objects of any of these categories has "extra points" coming from the need to freely complete under the $\kappa$-ary operation. However, these categories are all closed in their parent categories (all posets, all small categories, or all topological vector spaces) under $\kappa$-filtered colimits, so these strong generators are formed of $\kappa$-presentable. And any nontrivial (nonempty, nonzero) object in any of these categories admits an epimorphism to the terminal object in the strong generator, so no such object is finitely presentable, which shows these categories cannot be locally finitely presentable.

Answer 2

Here are a few examples.

• The category $\omega\text{-}\mathbf{CPO}$ of $\omega$-complete partial orders is locally countably presentable (i.e. $\kappa = \aleph_1$), but not locally finitely presentable (Example 1.18(2) of Locally Presentable and Accessible Categories).
• The category $\mathbf{Ban}$ of Banach spaces and linear contractions is locally countably presentable but not locally finitely presentable (Example 1.48 of Locally Presentable and Accessible Categories).
• The category $\mathbf{MS}$ of metric spaces with distances in $[0, 1]$ and nonexpanding functions is locally countably presentable but not locally finitely presentable (Example 2.2(5) and Example 2.6(3) of On Finitary Functors and Their Presentations).
• More generally, a Grothendieck topos is locally $\kappa$-presentable for some $\kappa$, but in general $\kappa \neq \aleph_0$ (Proposition 3.4.16 of Handbook of Categorical Algebra (Volume 3)).