Dependency Injection With Unity in MVC 5 without repository and unit of work
Naming  


Based on the naming guidelines methods should be named using PascalCasing casing.
addNewCategory -> AddNewCategory
Based on the same guideline classes should be named using PascalCasing casing.
dataBaseContext -> DataBaseContext and therefor IdataBaseContext -> IDataBaseContext  


General  

You are injecting an interface in the constructor, but then you cast it to the concrete implementation. If you don't need any methods/properties of the concrete implementation, you should stick to the interface.  

private IDataBaseContext _context;

public ProductCatagoryController(IDataBaseContext context)
{
    this._context = context;
}


This will make the ProductCatagoryController class independent of the concrete DatabaseContext object.  

For the case that ModelState.IsValid is false, how will the consumer of this method see, that no category is added ?
If an exception occurs, the consumer of this method will only see it, if the log file is parsed. Returning an Exception view would do better.