Recovering unitary calculus from calculus with reality

TL;DR

This paper demonstrates that unitary functor calculus can be fully derived from its version with reality, paralleling how complex K-theory is recovered from K-theory with reality by ignoring a C2-action.

Contribution

It establishes a method to recover unitary functor calculus from the calculus with reality, extending the analogy with complex K-theory and its real counterpart.

Findings

Unitary functor calculus is recoverable from calculus with reality.

The analogy with complex K-theory and K-theory with reality is formalized.

The paper provides a conceptual framework linking these calculi.

Abstract

By analogy with complex $K$--theory and $K$--theory with reality, there are theories of unitary functor calculus and unitary functor calculus with reality, both of which are generalisations of Weiss' orthogonal calculus. In this paper we show that unitary functor calculus can be completely recovered from the unitary functor calculus with reality, in analogy to how complex topological $K$--theory is completely recovered from $K$--theory with reality via forgetting the $C_2$--action.