Beyond orthogonal calculus: the unitary and real cases

These calculi produce Taylor towers approximating a functor, and we show through a zig-zag of Quillen equivalences (in both versions of the calculi) that the layers of these towers are classified by spectra with an action of either U(n) in the unitary case, or the semidirect product of C_2 with U(n) in the calculus with reality, where C_2 acts on U(n) by term-wise complex conjugation of the matrices.<br><br>From the complexification-realification adjunction between real and complex vector spaces we construct functors between the orthogonal and unitary calculi, allowing for movement between these two versions of calculus, and direct comparisons of the Taylor towers.

We introduce a class of functors, which we call ``weakly polynomial'' and we show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.