  We combine the theory of traces in homotopical algebra with sheaf theory in
derived algebraic geometry to deduce general fixed point and character
formulas. The formalism of dimension (or Hochschild homology) of a dualizable
object in the context of higher algebra provides a unifying framework for
classical notions such as Euler characteristics, Chern characters, and
characters of group representations. Moreover, the simple functoriality
properties of dimensions clarify celebrated identities and extend them to new
contexts.
  We observe that it is advantageous to calculate dimensions, traces and their
functoriality directly in the nonlinear geometric setting of correspondence
categories, where they are directly identified with (derived versions of) loop
spaces, fixed point loci and loop maps, respectively. This results in universal
nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz
trace formulas, and Frobenius-Weyl character formulas. On the one hand, we can
then linearize by applying sheaf theories, such as the theories of coherent
sheaves and D-modules, developed by Gaitsgory and Rozenblyum, as functors out
of correspondence categories (in the spirit of topological field theory). This
recovers the familiar classical identities, in families and without any
smoothness or transversality assumptions. On the other hand, the formalism also
applies to higher categorical settings not captured within a linear framework,
such as characters of group actions on categories.
