# Slice category and free objects

*The slice category or over category $$C/c$$ of a category $$C$$ over an object $$c∈C$$ has

objects that are all arrows $$f∈C$$ such that $$cod(f)=c$$,

and

morphisms $$g:X→X'∈C$$ from $$f:X→c$$ to $$f':X'→c$$ such that $$f'∘g=f$$.

*There is a forgetful functor $$U_{c}: C/c→C$$ which maps an object $$f: X→c$$ to its domain $$X$$ and a morphism $$g:X→X'∈C/c$$ (from $$f:X→c$$ to $$f':X'→c$$ such that $$f'∘g=f$$) to the morphism $$g: X→X'$$.

Please i need to know how one can define free objects using the free functor in that category.

• It is not very clear what you mean by free objects in this context (you are not working over a concrete category). Do you mean to ask whether your forgetful functor has a left adjoint ? Commented Nov 13, 2019 at 20:57
• Yes that what i actually mean Commented Nov 13, 2019 at 21:06
• Then in general it won't exist. Try to work out a simple example, like $C$ is the category of sets, and $c=\{0,1\}$. Commented Nov 13, 2019 at 21:21

## 1 Answer

The forgetful functor does not have a left adjoint in general, because the identity on $$c$$ is the terminal object of $$C/c$$, even if $$U_c(id_c)=c$$ is not a terminal object in $$C$$. In fact, if $$c$$ is terminal then $$U_c$$ is an isomorphism, so $$U_c$$ has a left adjoint if and only if it has an inverse.

On the other hand, if $$C$$ has products then $$U_c$$ has a right adjoint, which takes an object $$X$$ to the projection $$X\times c\to c$$.