nLab H-star-category

Idea

An $H^*$-category is a horizontal categorification of the notion of an $H^\ast$-algebra.

Definition

Definition

An $H^*$-category is a $\mathrm{Hilb}$-category with a $\dagger$-structure that defines an antinatural transformation from $\operatorname{hom}(x,y)$ to $\operatorname{hom}(y,x)^{\mathrm{conj}}$, where $\operatorname{hom}(y,x)^{\mathrm{conj}}$ is the conjugate hom-Hilbert space.

(Baez 1997 Def. 2)

The data of such an antinatural transformation is equivalent to the existence of compatible involutory antilinear maps between the hom-Hilbert spaces.

Proposition

An $H^*$-category is equivalently a $\mathrm{Hilb}$-category with antilinear maps $\ast : \operatorname{hom}(x,y) \to \operatorname{hom}(y,x)$ for all objects $x$ and $y$, such that

• $f^{**} = f$,

• $(g \circ f)^* = f^* \circ g^*$,

• $\langle g \circ f, h \rangle = \langle g, h \circ f^* \rangle$,

• and $\langle g \circ f , h \rangle = \langle f, g^* \circ h\rangle$

for all $f \in \operatorname{hom}(x,y)$, $g \in \operatorname{hom}(y,z)$, and $h \in \operatorname{hom}(x,z)$.

(Baez 1997 Prop. 3)

Related concepts

• H-star-algebra

• 2-Hilbert space

References

Introduced in:

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997) 125-189 [arXiv:q-alg/9609018]