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Conformal nets are a classical topic in quantum field theory: they assign operator algebras to one-dimensional manifolds, and have close connections with one-dimensional topological field theories. It seems to be well-known that the usual axioms for these constructions imply close relations between the action of the projective group on the line, and Connes' intrinsic flow on $C^*$-algebras. This note attempts to pin down this specific fact, in terms of a category of (noncommutative) algebras and equivalence classes, under inner automorphisms, of homomorphisms between them. That category may be of independent interest.