<p>Let <strong>C</strong> be a category and let <span class="math-container">$F$</span> be a mapping on <strong>C</strong>. Assume that <span class="math-container">$F$</span> is defined for all objects of <strong>C</strong>. However, on the arrows of <strong>C</strong> the mapping <span class="math-container">$F$</span> does not always behave like a (covariant) functor. For example, it may be the case that for some arrow <span class="math-container">$f: x \rightarrow y$</span> in <strong>C</strong>, the mapping <span class="math-container">$F$</span> returns an arrow <span class="math-container">$F(f): F(x)\rightarrow F(y)$</span> in <strong>C</strong>. For another arrow <span class="math-container">$g:a \rightarrow b$</span> in <strong>C</strong>, the mapping <span class="math-container">$F$</span> may return an arrow <span class="math-container">$F(g): F(b)\rightarrow F(a)$</span> in <span class="math-container">${\bf C}^{op}$</span>. And possibly for other arrows of <strong>C</strong>, the mapping <span class="math-container">$F$</span> may not even be defined. Such a mapping is neither covariant not contravariant and it may also be partial. Is there a special name for such mappings?</p>

CC BY-SA 4.0
2022-12-16T19:13:33.460Z
What's the example you have in mind?
varkor
CC BY-SA 4.0
2022-12-17T14:50:02.943Z
Take for example a partially ordered set $(P,\sqsubseteq)$, which gives rise to a category. Consider a function $F: P \rightarrow P$. Such functions can be monotone, anti-monotone, or neither. It seems that in category theory we are mainly interested in covariant functors (ie., monotone functions) and contravariant functors (ie., anti-monotone ones).  Is there a more general notion than just covariant/contravariant? Or is this just called ``a mapping''? I am new to category theory, so I apologize if my question sounds pedantic.
TetraDex
CC BY-SA 4.0
2022-12-17T14:53:27.683Z
This simply corresponds to a function between the object-classes of the two categories.
varkor
CC BY-SA 4.0
2022-12-17T14:59:14.690Z
But what if we are also interested in the effect that such a function has on arrows (and only on objects)?
TetraDex
CC BY-SA 4.0
2022-12-17T15:06:22.087Z
Do you have a concrete example in mind that is not an example of a covariant functor, a contravariant functor, or a function between object-classes?
varkor
CC BY-SA 4.0
2022-12-17T15:13:10.593Z
Not something specific. But, for example, functions from sets to sets are not always either monotone or anti-monotone.
TetraDex
CC BY-SA 4.0
2022-12-17T15:43:04.000Z
But in these cases, the posetal structure is irrelevant. It's difficult to suggest a definition without any motivating examples.
varkor
<ct.category-theory>
Mappings that are not functors in category theory
TetraDex
# Mappings that are not functors in category theory

(Asked by: username_0 on 2022-12-16)

Let **C** be a category and let $F$ be a mapping on **C**. Assume that $F$ is defined for all objects of **C**. However, on the arrows of **C** the mapping $F$ does not always behave like a (covariant) functor. For example, it may be the case that for some arrow $f: x \rightarrow y$ in **C**, the mapping $F$ returns an arrow $F(f): F(x)\rightarrow F(y)$ in **C**. For another arrow $g:a \rightarrow b$ in **C**, the mapping $F$ may return an arrow $F(g): F(b)\rightarrow F(a)$ in ${\bf C}^{op}$. And possibly for other arrows of **C**, the mapping $F$ may not even be defined. Such a mapping is neither covariant not contravariant and it may also be partial. Is there a special name for such mappings?

**Tags:** ct.category-theory

> **Comments:**
> **username_1**: What's the example you have in mind?  
> **username_0**: Take for example a partially ordered set $(P,\sqsubseteq)$, which gives rise to a category. Consider a function $F: P \rightarrow P$. Such functions can be monotone, anti-monotone, or neither. It seems that in category theory we are mainly interested in covariant functors (ie., monotone functions) and contravariant functors (ie., anti-monotone ones).  Is there a more general notion than just covariant/contravariant? Or is this just called ``a mapping''? I am new to category theory, so I apologize if my question sounds pedantic.  
> **username_1**: This simply corresponds to a function between the object-classes of the two categories.  
> **username_0**: But what if we are also interested in the effect that such a function has on arrows (and only on objects)?  
> **username_1**: Do you have a concrete example in mind that is not an example of a covariant functor, a contravariant functor, or a function between object-classes?  
> **username_0**: Not something specific. But, for example, functions from sets to sets are not always either monotone or anti-monotone.  
> **username_1**: But in these cases, the posetal structure is irrelevant. It's difficult to suggest a definition without any motivating examples.  

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## 0 answers

