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Let $X_1$, $X_2$, and $V$ be sets. Is there a standard name and a standard notation for the pre-composition operator $F$ that takes as input a function $\varphi:X_2^{X_1}$ and returns the operator $F_{\varphi}:V^{X_2}\rightarrow V^{X_1}$, which assigns for every function $f:V^{X_2}$ the function $F_{\varphi}(f)\in V^{X_1}$ defined by $$ F_{\varphi}(f) := f\circ\varphi $$

Is this what is meant by the term pullback in category theory? And, again, is there a standard notation for designating $F$, given $X_1$, $X_2$, and $V$?

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    $\begingroup$ $V^f$ and $\varphi^*$ are common names for $F_\varphi$. Occasionally $\varphi^{-1}$ too. $\endgroup$ – Hurkyl Jun 22 '18 at 8:57
  • $\begingroup$ @Hurkyl: Thanks. What is it called? $\endgroup$ – Evan Aad Jun 22 '18 at 8:59
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    $\begingroup$ I would write it as $(\circ \varphi)$ - this notation is pretty self-explanatory $\endgroup$ – lisyarus Jun 22 '18 at 9:00
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In category theory, this would be denoted $\mathrm{Hom}_\mathbf{Set}(-,V)$, or simply $\mathbf{Set}(-,V)$. It's called the "contravariant hom-functor." That makes sense; given a function $$f : X_1 \rightarrow X_2,$$ we get a corresponding function $$\mathbf{Set}(f,V) : \mathbf{Set}(X_2,V) \rightarrow \mathbf{Set}(X_1,V)$$ going "the other way", defined by $$\mathbf{Set}(f,V)(g) = g \circ f.$$

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  • $\begingroup$ Note that this is the same idea as the $V^f$ suggested by Hurkyl in the comments. $\endgroup$ – Arnaud D. Jun 22 '18 at 9:06
  • $\begingroup$ @ArnaudD., good point. $\endgroup$ – goblin Jun 22 '18 at 9:07
  • $\begingroup$ @ArnaudD.: In what field is the notation $V^f$ used? If I wish to look it up in a textbook, what kind of textbook should I seek? $\endgroup$ – Evan Aad Jun 22 '18 at 9:08
  • $\begingroup$ @EvanAad It's also widely used in category theory. In general category theorists are happy to denote the action of any functor by plugging a morphism symbol in place of an object. Any book on category theory will have examples similar to this one. $\endgroup$ – Kevin Carlson Jun 22 '18 at 18:42

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