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# Contents

## Idea

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 $\mathscr{V}$ over a given (star-)ground ring $R$ to its direct sum

$\mathscr{V} \;\mapsto\; H(\mathscr{V}) \;\coloneqq\; \mathscr{V} \oplus \mathscr{V}^\ast$

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

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

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

$\mathscr{V} \otimes \mathscr{V}^\ast \longrightarrow R$

and vanishing on the homogeneous summands.

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

$\phi \,\mapsto\, \phi \oplus (\phi^{-1})^\ast \,.$

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

## Example

###### Example

In the simple special case that the ground ring is the real numbers $\mathbb{R}$ and $\mathscr{V}$ 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 $\mathbb{R}^2 = \mathbb{R} \oplus \mathbb{R}$ 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 $\mathscr{V} \oplus \mathscr{V}^\ast$.

## References

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]