[Vuejs]-Vue: How to use v-model in post ApI

Your v-model=posts.AddCategory won’t work as you are hitting one of the object caveats as defined in vue https://v2.vuejs.org/v2/guide/reactivity.html#For-Objects.
What you can do is first in your Add method you could to so:

Add(category,index){
     this.$set(this.post,'AddCategory',category);
     .... 
}