Practice problem

Categories and Functors

Which of the following objects is NOT universal (in the sense of universal properties) with respect to the given property?

Choose one answer

The localization $A\to S^{-1}A$ of a ring $A$ with respect to a multiplicative set $S$ of $A$ .

Property: an initial object among rings homomorphisms $A\to R$ such that the image of $S$ is invertible in $R$ .

The quotient $G\to G/N$ of a group $G$ modulo a normal subgroup $N$ .

Property: an initial object among group homomorphisms $G\to H$ whose kernel contains $N$ .

The kernel of an abelian group homomorphism $A\to B$ .

Property: a terminal object among abelian group homomorphisms $C\to A$ such that the composition $C\to A\to B$ is zero.

The algebraic closure $k\to \overline{k}$ of a field $k$ .

Property: an initial object among field homomorphisms $k\to K$ such that every polynomial in $k[x]$ has a root in $K$ .