Exercise 1

Let {$F < E$} be a field extension and let {$X \subseteq E$} be a set of algebraically independent elements over {$F$}. Let {$U: \mathbf{Field} \Rightarrow \mathbf{Set}$} be the forgetful functor. Find a universal pair for {$X$} and {$U$}.

Exercise 5

Show that the polynomial ring {$F[x]$} is a universal object.

Exercise 7

Let {$U:\mathbf{Vect}_k\Rightarrow\mathbf{Set}$} be the underlying set functor. Which sets {$S$} have couniversal pairs?

Exercise 9

Let {$G:\mathbf{D}\Rightarrow\mathbf{C}$} and let {$C\in\mathbf{C}$} and {$U\in\mathbf{D}$}. Let

{$$\{τ_{C,D}\}_{\mathbf{C},\mathbf{D}}:\textrm{hom}(C,GD)\leftrightarrow\textrm{hom}(U,D)$$}

be a family of bijections and

{$$u=τ^{-1}_{C,U}(1_U)$$}

Show that the following are equivalent.

a) {$\{τ_{C,D}\}$} is natural in {$D$}.

b) {$\{τ_{C,D}\}_D$} satisfy the formula

{$$τ^{-1}_{C,D'}(g\circ h)=Gg\circ τ^{-1}_{C,D}(h)$$}

for all {$h:U\rightarrow D$} and {$g:D\rightarrow D'$}.

c)

{$$τ^{-1}_{C,D'}(g)=Gg\circ τ^{-1}_{C,D}(1_U)$$}

for all {$g:D\rightarrow D'$}.

Exercise 11

Let {$F:D\Rightarrow C$} and let {$C\in C$}. Prove that the functor {$\textrm{hom}_\mathbf{C}(C,F\cdot):\mathbf{D}\rightarrow\mathbf{Set}$} is representable if and only if there is a universal pair for {$(C, F)$}.