The category of pointed sets with point-preserving functions is not a topos, since it doesn't have power objects: if PX were the power object of the pointed set X, and 1 denotes the pointed singleton, then there is only one point-preserving function r\colon 1\to PX, but the relations in 1\times X are as numerous as the pointed subsets of X. The category of abelian groups is also not a topos, for a similar reason: every group homomorphism must map 0 to 0.
