Definition in category theory.

Question

In category theory, which is the definition of "contravariant involution"? and this is different of contravariant functor?

Answer 1

Let me first recall the context of the paper in which the phrase occurs.

• A relation is understood to be a triple $(A_-, A_+, A)$ where $A_-, A_+$ are arbitrary sets (of challenges and responses, respectively) and $A\subseteq A_-\times A_+$.

• A morphism $\varphi$ from ${\bf A}$ to ${\bf B}$ is a pair of functions $$\varphi_-: B_-\rightarrow A_-,\quad \varphi_+: A_+\rightarrow B_+$$ such that for all $y\in B_-$ and $x\in A_+$ we have $\varphi_-(y)Ax\implies yB\varphi_+(x)$. This gives the class of relations the structure of a category, call it $\mathcal{R}$.

Now, the dual of a relation ${\bf A}=(A_-, A_+, A)$ is the relation ${\bf A^\perp}=(A_-, A_+, \neg \check{A})$, where $\neg$ denotes complementation and $\check\cdot$ denotes the converse; we have $(x, y)\in\neg\check{A}$ iff $(y, x)\not\in A$. A good motivating example is "dominates" versus "escapes" in the context where $A_-=A_+=\mathbb{N}^\mathbb{N}$.

There is also a natural notion of the dual $\varphi^\perp$ of a morphism $\varphi$: namely, swap $\varphi_-$ and $\varphi_+$. Blass doesn't actually use the term "dual" here, but the superscript $^\perp$ is used in both cases. At this point it's easy to check that the duality map - acting as ${\bf A}\mapsto{\bf A^\perp}$ on objects and $\varphi\mapsto \varphi^\perp$ on morphisms - is a contravariant functor from a category to itself whose square is the identity. So it seems fairly clear that this is what "contravariant involution" means.

If you are still not satisfied, you can always email the author of the paper (Andreas Blass).