I am interested in finding the terminal objects of a presheaf topos, and in particular, ones constructed via Yoneda embedding $\hat{C} = \mathrm{Set}^{C^{\mathrm{op}}}$ from a category $C$.
Starting from category $C$, recall that the Yoneda embedding $Y \colon C^{\mathrm{op}} \to \mathrm{Set}$ is given by $Y(c) \colon x \mapsto \mathrm{Hom}(x, c)$.
Now let’s say we know the terminal object in $C$ is $1$ and we want to find the terminal object $\hat{1}$ in $\hat{C}$. Then consider the presheaf $Y(1) \colon x \mapsto \mathrm{Hom}(x, 1)$. Since $1$ is terminal in $C$, we know that $\mathrm{Hom}(x, 1)$ is a singleton, which basically means $Y(1)$ is constant. Then $\hat{1} = Y(1)$.
Is the above reasoning correct?
What if $C$ doesn't have a terminal object, then how would you find the terminal in $\hat C$?
Moreover, how can we find the subobjects of $\hat{1}$?
Thanks!
Best Answer
Limits and colimits in functor categories are computed pointwise. In particular the terminal presheaf always exists and is just the presheaf with constant value the terminal object $1$ in $\text{Set}$, whether or not $C$ has a terminal object. ($C$ has a terminal object iff this presheaf is representable.)
It's also true that in functor categories a natural transformation is a monomorphism resp. epimorphism iff it is pointwise a monomorphism resp. an epimorphism. So the subobjects of the terminal presheaf $1$ are exactly the presheaves $F$ such that the unique map $F \to 1$ is a monomorphism, meaning it is a pointwise monomorphism, meaning that $F(c)$ has either one element or is empty for every $c \in C$.