# "Operations" in category theory that are not defined for arrows

Functors in category theory are defined for both objects and arrows. Depending on how they treat arrows, functors are characterized as either covariant or contravariant. Some "operations" however, are not considered functors, because they are neither covariant nor contravariant.

For example, if we define in the $${\bf CPO}$$ category the "operation" $$F(D,E) = [D \rightarrow E]$$, this is not a functor in $${\bf CPO} \times {\bf CPO} \rightarrow {\bf CPO}$$ because it is contravariant in the first argument and covariant in the second. Overall, $$F$$ is neither covariant nor contravariant. It is usually written that $$F$$ is covariant in $${\bf CPO}^{op} \times {\bf CPO} \rightarrow {\bf CPO}$$.

However, we can not do the same trick for $$F(D) = [D \rightarrow D]$$, which is neither covariant in its unique argument nor contravariant. Do such "operations" have a special name in category theory?

• – D.W.
Nov 5, 2022 at 6:55

There are several ways to formalize this idea in category theory that I have seen in the literature.

An overview is that you can either separate the two occurrences of $$D$$ into the positive and negative occurrences or you can change the morphisms of $$CPO$$ so that $$F$$ is covariant.

1. As you said you can think of it as $$F(D^-,D^+) = [D^- \to D^+]$$ as being a functor of two different objects, i.e. a functor $$CPO^o\times CPO \to CPO$$. One name for this would be a $$CPO$$-enriched profunctor. Thinking of both occurrences of $$D$$ as the "same" comes up when you take ends and coends.
2. You can think of $$F : |CPO| \to CPO$$ where $$|CPO|$$ is the discrete category whose objects are CPOs but the only morphisms are identities. This might be called an "action on objects".
3. You can think of $$F : CPO^{ep} \to CPO^{ep}$$ where $$CPO^{ep}$$ has CPOs as objects and embedding-projection (ep) pairs as morphisms. An ep pair from $$A$$ to $$B$$ is a pair of continuous morphisms $$e : A \to B$$ and $$p : B \to A$$ such that $$p \circ e = \text{id}_A$$ and $$e \circ p \sqsubseteq \text{id}_B$$. $$F$$ ends up being covariant because in the contravariant case you simply swap the role of embedding and projection. I don't know a name for this exactly but it is a very famous way to approach this and is how Dana Scott showed how to solve domain equations by "making F be functorial".
4. There are many similar ideas to the previous one. You might have instead of ep pairs, arbitrary continuous galois connections, or arbitrary retractions or arbitrary opposing pairs of morphisms. More interestingly, you can use notions of admissible relation or if you have algebraic CPOs, basis embeddings.
• Thank you for your detailed answer! However, my question was intended to be slightly more general, ie., applicable to other categories that just ${\bf CPO}$. Let me rephrase the final question: in category theory, covariant functors correspond to what we usually call in mathematics $monotone$ functions and contravariant functors to $antimonotone$ ones. So my question can be rephrased: is there a notion in category theory that corresponds to $non−monotone$ functions?
– LaR
May 25, 2022 at 9:14
• To continue the question: given an operation $F: {\bf C} \rightarrow {\bf C}$ on objects, can you always express it as a functor $F': {\bf C}^{op} \times {\bf C} \rightarrow {\bf C}$?
– LaR
May 25, 2022 at 9:23
• The general version would be #2 using the discrete category. I usually just see this called a "function on objects". May 25, 2022 at 14:35