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?
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1$\begingroup$ What's the example you have in mind? $\endgroup$– varkorDec 16, 2022 at 19:13
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$\begingroup$ 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. $\endgroup$– TetraDexDec 17, 2022 at 14:50
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$\begingroup$ This simply corresponds to a function between the object-classes of the two categories. $\endgroup$– varkorDec 17, 2022 at 14:53
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$\begingroup$ But what if we are also interested in the effect that such a function has on arrows (and only on objects)? $\endgroup$– TetraDexDec 17, 2022 at 14:59
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3$\begingroup$ But in these cases, the posetal structure is irrelevant. It's difficult to suggest a definition without any motivating examples. $\endgroup$– varkorDec 17, 2022 at 15:43
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