I am not sure I completely understand what you are trying to ask. If you are curious how to obtain sole custody of your child - you will have to show that your child's mother is "unfit." Most of the time, courts are reluctant to award sole custody. Typically, you will see courts award the parents with joint custody. However, this does not mean you cannot obtain sole custody.
One thing to keep in mind is that just because a parent is awarded sole custody, that does not mean the other parent is precluded from seeing the child. The other parent can still be given possession periods. The parent whom is awarded sole custody gets certain rights and decision making powers that the other parent does not. So, in your scenario, if the father agrees to allow the mother to have sole custody, the father can still have "visitation rights" to the child. Before you make any decisions, you need to speak to an attorney. Best of luck.