<p>There are several ways to formalize this idea in category theory that I have seen in the literature.</p>
<p>An overview is that you can either separate the two occurrences of <span class="math-container">$D$</span> into the positive and negative occurrences or you can change the morphisms of <span class="math-container">$CPO$</span> so that <span class="math-container">$F$</span> is covariant.</p>
<ol>
<li>As you said you can think of it as <span class="math-container">$F(D^-,D^+) = [D^- \to D^+]$</span> as being a functor of two different objects, i.e. a functor <span class="math-container">$CPO^o\times CPO \to CPO$</span>. One name for this would be a <span class="math-container">$CPO$</span>-enriched profunctor. Thinking of both occurrences of <span class="math-container">$D$</span> as the &quot;same&quot; comes up when you take <a href="https://ncatlab.org/nlab/show/end" rel="nofollow noreferrer">ends and coends</a>.</li>
<li>You can think of <span class="math-container">$F : |CPO| \to CPO$</span> where <span class="math-container">$|CPO|$</span> is the <em>discrete</em> category whose objects are CPOs but the only morphisms are identities. This might be called an &quot;action on objects&quot;.</li>
<li>You can think of <span class="math-container">$F : CPO^{ep} \to CPO^{ep}$</span> where <span class="math-container">$CPO^{ep}$</span> has CPOs as objects and <em>embedding-projection (ep) pairs</em> as morphisms. An ep pair from <span class="math-container">$A$</span> to <span class="math-container">$B$</span> is a pair of continuous morphisms <span class="math-container">$e : A \to B$</span> and <span class="math-container">$p : B \to A$</span> such that <span class="math-container">$p \circ e = \text{id}_A$</span> and <span class="math-container">$e \circ p \sqsubseteq \text{id}_B$</span>. <span class="math-container">$F$</span> 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 &quot;making F be functorial&quot;.</li>
<li>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.</li>
</ol>

CC BY-SA 4.0
2022-05-25T09:14:13.713Z
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
CC BY-SA 4.0
2022-05-25T09:23:28.513Z
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
CC BY-SA 4.0
2022-05-25T14:35:04.247Z
The general version would be #2 using the discrete category. I usually just see this called a "function on objects".
Max New
CC BY-SA 4.0
2022-05-24T13:33:53.427Z
2022-05-24T13:33:53.427Z
Max New
<p>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 &quot;operations&quot; however, are not considered functors, because they are neither covariant nor contravariant.</p>
<p>For example, if we define in the <span class="math-container">${\bf CPO}$</span> category the &quot;operation&quot; <span class="math-container">$F(D,E) = [D \rightarrow E]$</span>, this is not a functor in <span class="math-container">${\bf CPO} \times {\bf CPO} \rightarrow {\bf CPO}$</span> because it is contravariant in the first argument and covariant in the second. Overall, <span class="math-container">$F$</span> is neither covariant nor contravariant. It is usually written that <span class="math-container">$F$</span> is covariant in <span class="math-container">${\bf CPO}^{op} \times {\bf CPO} \rightarrow {\bf CPO}$</span>.</p>
<p>However, we can not do the same trick for <span class="math-container">$F(D) = [D \rightarrow D]$</span>, which is neither covariant in its unique argument nor contravariant.
Do such &quot;operations&quot; have a special name in category theory?</p>

CC BY-SA 4.0
2022-11-05T06:55:04.280Z
Cross-posted: https://cstheory.stackexchange.com/q/51501/5038, https://math.stackexchange.com/q/4456756/14578.  Please [do not post the same question on multiple sites](https://meta.stackexchange.com/q/64068).
D.W.
<semantics>
"Operations" in category theory that are not defined for arrows
LaR
# "Operations" in category theory that are not defined for arrows

(Asked by: username_0 on 2022-05-23)

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?

**Tags:** semantics

> **Comments:**
> **username_1**: Cross-posted: https://cstheory.stackexchange.com/q/51501/5038, https://math.stackexchange.com/q/4456756/14578.  Please [do not post the same question on multiple sites](https://meta.stackexchange.com/q/64068).  

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## 1 answers

### Answer by username_2 on 2022-05-24. Score: 4 | Accepted answer

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](https://ncatlab.org/nlab/show/end).
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.

> **Comments:**
> **username_0**: 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?  
> **username_0**: To continue the question: given an operation $F: {\bf C} \rightarrow {\bf C}$ on objects, can you always express it as a functor $F': {\bf C}^{op} \times {\bf C} \rightarrow {\bf C}$?  
> **username_2**: The general version would be #2 using the discrete category. I usually just see this called a "function on objects".  

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