# Category In Which Not All Free Objects Exists

I am trying to think of a category in which not all free objects exists. I thought this might be the case in sets (I thought I might be able to violate the uniqueness ) but I couldn't get anywhere so I was looking for some examples of categories that don't have all free objects?

• If I understand you well then you are looking for a forgetful functor that has no left-adjoint. Have a look at mathoverflow.net/a/6383/40263 – drhab Apr 28 '14 at 10:54
• What do you mean "free objects"? Are you considering a category with some fixed "forgetful functor", and you're looking for a left adjoint? In this case you need to define what a forgetful functor is, because of course not all functors have a left adjoint. – Najib Idrissi Apr 28 '14 at 10:55
• @NajibIdrissi I am using this definition en.wikipedia.org/wiki/Free_object Is this not standard? – john smith Apr 28 '14 at 11:02
• That is standard, but you have not provided all the data! – Zhen Lin Apr 28 '14 at 12:13
• Categories are not the things that have free objects; the things that have free objects are concrete categories, namely categories $C$ equipped with (faithful) functors $F : C \to \text{Set}$. – Qiaochu Yuan Apr 28 '14 at 12:55

The concrete category $(\mathsf{Set},\operatorname{id}_{\mathsf{Set}} : \mathsf{Set} \to \mathsf{Set})$ has all free objects : for any set $S$, the set $S$ itself (with the canonical map $\operatorname{id}_S$) is free on $S$. So your attemp will not get you anywhere.
There is however a very easy (but dumb) example of a concrete category suiting your requirement : take the punctual category $\mathsf e$ (one object, one morphism) and make it concrete with $F \colon \mathsf e \to \mathsf{Set}$ sending the unique object of $\mathsf e$ to $\emptyset$. Then the only set $X$ admitting a free object is $\emptyset$ : indeed,if $X \neq \emptyset$ would admit a free object, it would be the only one of $\mathsf e$ and there should exists some map $i \colon X \to \emptyset$, which is of course absurd.