Category In Which Not All Free Objects Exists

Question

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?

Answer 1

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.

There must be more natural and less ad hoc examples in the nature, but I don't have the time to think further for now.

Answer 2

An example that shows up "in nature" is the (concrete) category of fields, equipped with the usual forgetful functor to sets. There isn't a free field on any number of objects, including zero.