nLab antinatural transformation

Idea

Antinatural transformations provide a way to define maps between covariant and contravariant functors in the setting of dagger category theory. The point is that composing with a dagger (which is contravariant) makes a covariant functor contravariant and a contravariant functor covariant.

Definition

Definition

Given a dagger category $(C,\dagger)$ and a category $D$ with a covariant functor $F: C \to D$ and a contravariant functor $G: C^{\mathrm{op}} \to D$, an antinatural transformation $\alpha : F \Rightarrow G$ between them is a natural transformation from $F$ to $G \circ \dagger$.

Similarly, an antinatural transformation $\beta: G \Rightarrow F$ is a natural transformation from $G$ to $F \circ \dagger$.

Remark

A contravariant functor from $C$ to $D$ can either be interpreted as a functor $C^{\mathrm{op}} \to D$ or as a functor $C \to D^{\mathrm{op}}$. Thus, a dagger structure on $C$ is both a functor $\dagger: C \to C^{\mathrm{op}}$ and a functor $\dagger: C^{\mathrm{op}} \to C$, by abuse of notation. It thus makes sense to precompose both $F$ and $G$ with $\dagger$.

Likewise, if we instead wrote $G$ in the definition above as a functor $G:C \to D^{\mathrm{op}}$, we would need $D$ to have a dagger structure to recover the analogous definition.

Related entries

• natural transformation

• H-star-category

References

Introduced in:

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