# Non-small objects in categories

An object $$c$$ in a category is called small, if there exists some regular cardinal $$\kappa$$ such that $$Hom(c,-)$$ preserves $$\kappa$$-filtered colimits.

Is there an example of a (locally small) category $$C$$ and an object $$c$$ of $$C$$, such that $$c$$ is not small, i.e. such that $$Hom(c,-)$$ doesn't preserve all $$\kappa$$-filtered colimits for any $$\kappa$$ whatsoever?

• Another good example of categories with lots of objects that are not small, is pro-categories. For more, see here: mathoverflow.net/questions/209734/… Commented Aug 19, 2019 at 18:39

Indeed, if $$A$$ is a set with at least two elements, consider functions $$f:\{0,1\}^\kappa\to A$$ where $$\kappa$$ is some infinite cardinal. If $$\lambda<\kappa$$ then $$\{0,1\}^\kappa$$ may be viewed as a $$\lambda$$-cofiltered limit of all products of at most $$\lambda$$ of the copies of $$\{0,1\}$$. For $$A$$ to be $$\lambda$$-small in $$\mathrm{Set}^{\mathrm{op}}$$, we would have to be able to guarantee that $$f$$ depends on at most $$\lambda$$ coordinates in the domain.
In the category $$\mathsf{Top}$$ of topological spaces and continuous maps the only $$\lambda$$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.