My ex-spouse betrayed me, but we have to raise our children together. How?

You can be a cooperative co-parent without needing to trust or like your ex-spouse as before. Separate personal wounds from parenting matters: one is about your heart, the other about the children's needs. Limit interaction to matters about the children, through a brief channel that does not stir up old emotions. You do not have to pretend everything is fine, only be mature enough not to make the children victims. Pour out the pain of betrayal to a counsellor or a trusted person, not to the children or in every conversation with your ex. Forgiving for your own peace is different from forgetting. With time, calm cooperation becomes easier. See counselling.