Doesn't want to breakup

So my boyfriend and I had not had sex since having our baby ( nine months old ) despite me saying that we need to be closer, try to make things work. Found out he was cheating, he apologised and said we'll work things out. Two months later nothing has changed, we still haven't had sex, he spends no time with either me or the kids and I realised that I no longer trust him. I told him that I think we should break up and coparent but he said that's selfish as we're still working things out together... But he's making no effort whatsoever?? Once the trust is gone the relationship is dead right?