In the sentence,
There is a full moon in the sky.
... will the sky be treated as the object in the sentence?
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"Sky" is the object of the preposition "in." There is no direct object in the sentence. The verb "to be" is copulative and doesn't take objects. "Moon" is the subject.
The sentence type "There is + noun/ There are + noun plural" contains no object. "There is a man at the door" is the same as "A man is at the door". There are various views about the sentence parts "there" and "man". Some say "there" is an adverb and "man" the subject with a special position, some say "there" is the subject. Actually such questions are of minor importance as the sentence structure is simple and clear: There is someone/something somewhere. In any case "man" is a nominative and can't be an object.