Example: Accessible category without colimits

Question

I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch.

Bonus points if the sketch and/or the colimit diagram is finite.

Answer 1

• The category Hil of Hilbert spaces, considered as a full subcategory of Ban is $\aleph_1$-accessible but not locally presentable, in fact it is self dual.

• The category Lin of linear orders and strictly increasing maps is finitely accessible.

• The category of Sets and one-to-one functions is finitely accessible and not locally presentable.

• The category Fld of fields is accessible but not locally presentable.

All these examples and some others can be found in Locally Presentable and Accessible Categories, 2.3.

Answer 2

Here's a general way to get an accessible category that likely will not have many colimits: let $C$ be a locally presentable category, and then let $C^{mono}$ be the category with the same objects as $C$, but just the monomorphisms of $C$ for monomorphisms. Then $C^{mono}$ is accessible. But $C^{mono}$ will typically fail to have products, for example -- it's unlikely that projection maps for a product will be monos.

In a similar vein but different direction, start with a locally presentable category, and pass to a subcategory of objects defined by some lifting property. For example, the category of injective abelian groups. For a finite example, how about the category of $p$-divisible abelian groups for a fixed prime $p$ -- to sketch this in a finite way, start with the usual sketch $S$ for abelian groups, and let $Z$ be the object of the sketch such that $Hom(Z,-) = \mathbb Z$. Then there is a map $p: Z \to Z$ given by multiplication by $p$. Put in the colimit condition saying that for $F: S \to Set$ in our category, the map $p: F(Z) \to F(Z)$ is an epimorphism -- i.e. the diagram

$\require{AMScd} \begin{CD} Z @>1>> Z\\ @V1VV @VpVV\\ Z @>p>> Z \end{CD}$

is in the colimit part of the sketch.

Another point is that an accessible category has all colimits iff it has all limits, iff it is locally presentable, iff it is the category of models for a limit sketch. So any category sketched by a (limit,colimit) sketch which actually uses the colimit part is unlikely to have all colimits (if it does have all colimits, then there's an alternative way to sketch it without using the colimit part of the sketch).

Another class of examples is this: every small category with split idempotents is accessible. A small category is never complete / cocomplete, except maybe if it's a poset.

Another class of examples would be the category of models and elementary embeddings for any first-order theory $T$ -- this is always an accessible category, but never (co)complete except in trivial cases.