Abstract
AbstractWe construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.
Highlights
The orthogonal and unitary calculi [27, 24] systematically study J -spaces, where J is the category of finite-dimensional real inner product spaces or complex inner product spaces, respectively
An initial step in a much larger project to understand equivariant orthogonal calculus is the following calculus with reality. This is unitary calculus, constructed to take into account the C2-action on the category of complex inner product spaces given by complex conjugation
In [24, 25], we explained the strong analogy of orthogonal and unitary calculi with real and complex topological K-theory. This analogy was the motivation behind the comparisons of [25]
Summary
The orthogonal and unitary calculi [27, 24] systematically study J -spaces, where J is the category of finite-dimensional real inner product spaces or complex inner product spaces, respectively. We produce a model structure on the category of functors with reality, which captures the homotopy theory of n-homogeneous functors, in particular, the n-th layer of the Taylor tower, see Proposition 4.2. In the orthogonal calculus, Barnes and Oman [3] constructed a zigzag of Quillen equivalences using only one intermediate step between their n-homogeneous model structure and orthogonal spectra with an action of O(n). We further strengthen the idea of calculus with reality being in analogy with the KR-theory of Atiyah, by giving an equivalence of categories between the category C2 E1R which features in our zigzag and the Real spectra (see Definition 6.3) of Schwede, see Proposition 6.4. We achieve a Taylor tower of the following form: F
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