I highly doubt it. Please keep in mind I am a huge fan of the katana. It is arguably the best made sword throughout history. They were incredibly sharp. I do not believe however that it would have been sharper than a straight razor. The sword is much thicker and simple geometry would tell you that the angle of the edge would be greater than that of a straight razor. It would have been hard to get a "razor's edge" on a weapon like that.
All that said, the katana would cut deeper and longer and straighter than a straight razor. Give and take, what we lose in "sharpness" compared to a razor make up for in many other places. A katana is not the sharpest cutting instrument, it is simply the sharpest sword.
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