Divorce decree states he pay maintenance until I remarry. I am not remarried, nor do I have plans to do so. He is saying that because I have been "cohabitating" with someone for a year that spousal support is normally dropped.
Death or your remarriage are the standard conditions upon which a spousal maintenance obligation would end (short of a defined time period). Some parties negotiate for additional conditions on the maintenance, such as cohabitation.
If your decree does not set forth such a terminating condition then you should seek to enforce the decree, perhaps by hiring an attorney and seeking an award of fees for having to enforce the decree. This is especially the case if his basis for ceasing payment is your alleged "cohabitation".
Divorce is the process of formally ending a marriage. Divorces may be jointly agreed upon, resolved by negotiation, or decided in court.
Alimony is the regular monetary allowance one ex-spouse may be required to pay the other after a divorce.