nLab hyperbolic functor

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]

See also:

• C. T. C. Wall (ed. Andrew Ranicki), p. 177 of: Surgery on compact Manifolds, Math. Surveys and Monographs 69 (1969) [pdf]

Further discussion in the context of Hermitian K-theory:

• Max Karoubi, Orlando Villamayor, p. 60 of K-théorie algébrique et K-théorie topologique II, Math. Scand. 32 (1973) 57-86 [jstor:24490565]

• Max Karoubi, pp. 307 (7 of 111) in: Périodicité de la K-théorie hermitienne, in: Hyman Bass (ed.), Algebraic K-Theory III – Hermitian K-Theory and Geometric Applications, Lecture Notes in Mathematics 343 (1973) 301-411 [doi:10.1007/BFb0061366]

• Anthony Bak, §3 in: Grothendieck Groups of Modules and Forms Over Commutative Orders, American Journal of Mathematics, 99 1 (1977) 107-120 [jstor:2374010, doi:10.2307/2374010]

• Max Karoubi, §1.10 in: Le théorème de périodicité en K-théorie hermitienne, in Quanta of Maths, Clay Mathematics Proceesings 11, AMS and Clay Math Institute Publications (2010) 257-282 [arXiv:0810.4707, pdf]

See also:

• Richard Elman, Nikita Karpenko, Alexander Merkurjev, p. 4 of: Algebraic and Geometric Theory of Quadratic Forms, Colloquium Publication 56, AMS (2008) [ams:coll-56, pdf]

• Matthew B. Young, p. 15 of: Self-Dual Hall modules, PhD thesis, Stony Brook (2013) [pdf, pdf]

• Marco Schlichting, p. 8 of: Higher $K$-Theory of Forms I. From Rings to Exact Categories, Journal of the Institute of Mathematics of Jussieu 20 4 (2021) 1205-1273 [doi:10.1017/S1474748019000410]