The geometry of the cyclotomic trace

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We provide a new construction of the topological cyclic homology TC(C) of any spectrally-enriched ∞-category C, which affords a precise algebro-geometric interpretation of the cyclotomic trace map K(X) → TC(X) from algebraic K-theory to topological cyclic homology for any scheme X. This construction rests on a new identification of the cyclotomic structure on THH(C), which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization of more primitive data.