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Stephen has complained to Mulligan about their visitor, Haines, and Mulligan has threatened some violence against him if he acts up again. This has led Stephen to a sequence of thoughts about Mulligan’s real or imagined hazing of one Clive Kempthorpe, involving at least the threat of castration.
From here, Stephen’s mind has skipped to Mulligan’s cultural pretensions, of establishing a “new paganism” in the tower, setting a new cultural moment, with the tower as its “omphalos.” Poised on the knife-edge of Stephen’s analysis, Mulligan is revealed as a superficial intellectual with a violent bully not far beneath the surface. Stephen decides he can’t continue the ruse of being Mulligan’s friend.
Omphalos is a Greek word meaning navel or center, and it was used to refer to places like Delphi that were at the center of the world and a point at which the gods communicated with men. More particularly, it was a stone sculpture like this.
Which, of course, bears more than a passing resemblance to our Martello tower.