Playing vanilla Civ 4. Captured Beijing. It seems like only buildings that I built generate culture. Is that how it's supposed to work? The city has an Academy, Great Lighthouse, Parthenon and Notre Dame and none of them are generating any culture.

Behaving as designed:
Cultural buildings are destroyed upon capture (theater, library, monument).
However, since Wonders and Great Person Buildings cannot be destroyed they instead lose their cultural production.

Once a captured city comes out of revolt the only culture it can have is from artists, build culture, or religion. Of course, you can build culture buildings of your own and they will produce culture normally.

And if the Chinease ever recapture Beijing, the academy and those wonders will start generating culture again (assuming the Chinease controlled Beijing when those buildings were compelteed).

The added effects of the wonders do still apply, right?
I captured the city that had the Great Wall in it and never had Barb trouble

Correct, only culture output is affected; the primary benefits still apply.

Ah i get it, so for example, in Terminator 2 movie when the Terminator took the barkeepers sunglasses, the stupid terminator did not look more cool because he cannot get any cultural benefit from the captured sunglasses.... yea makes perfect sense!