nLab orthogonal calculus

Idea

The orthogonal calculus is a calculus of functors from vector spaces to homotopy types, similar to but different from the Goodwillie calculus of (infinity,1)-functors.

The prime example is the assignment $V \mapsto B O(V)$ of the classifying space of the orthogonal group $O(V)$ to any inner product vector space $V$. In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of Stiefel-Whitney classes. Similarly, second derivatives in the orthogonal calculus reproduce and generalize much of the theory of Pontryagin classes. (see Weiss 95)

The complex analog of the orthogonal calculus is known as the unitary calculus.

References

• Michael Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347 (1995), 3743-3796, pdf

• David Barnes, Rosona Eldred, Comparing the orthogonal and homotopy functor calculi, Journal of Pure and Applied Algebra 220 11 (2016) 3650-3675 [arXiv:1505.05458, doi:10.1016/j.jpaa.2016.05.005]