Objectivism is a philosophy, or set of beliefs, created by the philosopher Ayn Rand. Objectivism says that every person should be free to pursue their own happiness, as long as it doesn't hurt anyone else. Objectivism also says that reality and truth are knowable, and that morality and values are based on reason and facts.
When it comes to homosexuality, Objectivism would say that people should be free to have relationships with whoever they want, regardless of gender. In other words, Objectivists believe that everyone should have the same rights and freedom when it comes to love and relationships. It also believes that people should be judged based on what they think and do, not their sexual orientation or gender identity.