Quick and dirty illustration of @Property declared (but not synthesized) in a category.

CategoryPropertyDemo.m

#import <Foundation/Foundation.h>

@interface MyClass : NSObject
@end

@implementation MyClass
@end

@interface MyClass (Properties)

@property (nonatomic, copy) NSString *string;

@end

@implementation MyClass (Properties)

// @synthesize string = _string; // Won't work
- (NSString *)string
{
    NSString *result = @"It's a string!"; // Get string from somewhere (derived from other properties, associated object, etc.)
    return result;
}

- (void)setString:(NSString *)string
{
    // Set string somehow (set other properties derived from this value, create associated object, etc.)
    // NSLog to show that this will be called
    NSLog(@"Got a string: %@", string);
}

@end

int main(int argc, char *argv[]) {
    @autoreleasepool {

        MyClass *object = [[MyClass alloc] init];
        NSLog(@"object.string: %@", object.string);
        object.string = @"Hello";

    }
}