"There was no fog yesterday."
Translation:Вчера не было тумана.
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- present: У меня нет котов (I have no cats), там нет людей (there are no people), там не бывает людей (there can be no people)
- future: у меня не будет котов (I'll have no cats), там не будет людей (there will be no people)
- past: у меня не было котов (I had no cats), там не было людей (there were no people)
As you see this impersonal negation is an invariable construction. So не было does not depend on туман. You can construct something like "вчера туман не был" where туман and не был are related but it sounds as unnatural as English "yesterday fog was not". I think this is something to get used to, not comprehend.
Thanks for your answer, but I realized that туман is not the subject of the sentence: they way I understand it is the subject is an unstated dummy noun (sort of like the word "it" in the sentence "it is raining"). So this sentence means something, a bit more liberally translated: like "Yesterday it was not of fog," where "it" is the dummy subject (which is apparently of the neuter gender).
Genitive is mandatory when you negate the existence of smth/smb ("Вчера не было тумана"), with abstract nouns (не терять времени), with pronouns ("не делать этого"), with the repetitive particle "ни" ("не знать ни боли, ни страха"). In some cases, you can choose between accusative and genitive with a little difference in the meaning: accusative makes the situation/object more definite.
Grammatically, in the very least I think you would still need тумана because the verb you're negating is still было, omitted or not. I doubt that it would be omitted in that case either. Also I don't think your sentence is acceptable as a translation of the English because it's adding extra content.