If "mate" refers to a companion or friend:
* Companionship
* Friendship
* Camaraderie
If "mate" refers to a spouse or partner:
* Marriage
* Partnership
* Relationship
If "mate" refers to a person or animal that is sexually compatible:
* Reproduction
* Pairing
* Mating
If "mate" refers to a person who shares the same occupation or interests:
* Colleagueship
* Fellowship
* Shared interest
The best abstract noun will depend on the specific meaning of "mate" in the sentence.