We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

我们构建了 Weiss 开发的正交微积分的酉类比，利用模型类别对所涉及的等方差和同伦理论中的复杂性给出了清晰的描述。真实几何和复杂几何之间的细微差别导致了正交微积分和酉微积分之间的细微差别。为了解决这些差异，我们构建了酉谱（正交谱的一种变体）作为稳定同伦类别的模型。我们通过 Quillen 等价的锯齿形表示，具有第n个酉群作用的酉谱对酉微积分的齐次部分进行了建模。我们通过引入弱多项式函子来解决泰勒塔的收敛问题，弱多项式函子类似于 Goodwillie 的弱解析函子，但计算上更容易处理。