nLab hyperbolic functor




In the context of quadratic/Hermitian forms and in particular of Hermitian K-theory, by the hyperbolic functor one refers to the construction which sends a module 𝒱\mathscr{V} over a given (star-)ground ring RR to its direct sum

𝒱H(𝒱)𝒱𝒱 * \mathscr{V} \;\mapsto\; H(\mathscr{V}) \;\coloneqq\; \mathscr{V} \oplus \mathscr{V}^\ast

with its (anti-)linear dual and then equipped with the (sesqui- or) bilinear form

H(𝒱)H(𝒱) R \array{ H(\mathscr{V}) \otimes H(\mathscr{V}) &\longrightarrow& R }

which on the mixed summands is (the conjugation of) the evaluation map

𝒱𝒱 *R \mathscr{V} \otimes \mathscr{V}^\ast \longrightarrow R

and vanishing on the homogeneous summands.

This construction extends to functor from the core groupoid of R R -modules to that of inner product spaces with isometries between them, by sending an isomorphism ϕ:𝒱𝒲\phi \colon \mathscr{V} \overset{\sim}{\to} \mathscr{W} to its direct sum with the dual linear map of its inverse:

ϕϕ(ϕ 1) *. \phi \,\mapsto\, \phi \oplus (\phi^{-1})^\ast \,.

Regarded with suitable /2×/2\mathbb{Z}/2 \times \mathbb{Z}/2-equivariance, the hyperbolic functor establishes an equivalence between KR-theory and topological Hermitian K-theory (see the references there).



In the simple special case that the ground ring is the real numbers \mathbb{R} and 𝒱\mathscr{V} is also \mathbb{R}, regared as the 1-dimensional vector space over itself, and finally identifying *\mathbb{R}^\ast \simeq \mathbb{R}, canonically, the hyperbolic construction on \mathbb{R} is the plane 2=\mathbb{R}^2 = \mathbb{R} \oplus \mathbb{R} equipped with its standard Minkowski metric.

Here the subspaces of fixed non-vanishing norm are the usual hyperbolas. This gives the name “hyperbolic form” to all modules of the above form 𝒱𝒱 *\mathscr{V} \oplus \mathscr{V}^\ast.


The term “hyperbolic functor” seems to originate with:

  • Hyman Bass (notes by Amit Roy), §5.2 in: Lectures on Topics in Algebraic K-Theory, Tata Institute of Fundamental Research (1965) [pdf]

See also:

Further discussion in the context of Hermitian K-theory:

See also:

Last revised on November 16, 2023 at 06:59:05. See the history of this page for a list of all contributions to it.