The geometry of the cyclotomic trace

We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\mathrm{\infty }$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X)\to TC(X)$ 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.