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Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$

For my purposes, which are a little too complicated to really describe here, it would be useful to have a name for this process, at least for use in the privacy of my own mind and/or notebook. Furthermore, since this process turns a mere value or element of $X$ into an 'operator' in its own right, I'm tempted to call this process 'operationalization.' So in particular, $\tilde{x}$ is the operationalization of $x$.

Question. Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

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  • $\begingroup$ I'm not sure if this has a name, but it happens all the time in functional analysis. In that context, you would be referring to an element in the dual. $\endgroup$
    – parsiad
    Commented Jun 27, 2016 at 16:54
  • $\begingroup$ @par, true. So perhaps calling this the 'dual' of $x$ is okay. Its a bit shorter than 'operationalization', at least :) $\endgroup$ Commented Jun 27, 2016 at 16:55
  • $\begingroup$ I know that in ring theory we have a substitution operator denoted by $\varphi _a$, which does exactly that: $\varphi _a (f) = f(a)$. I have seen this only when $f$ is a polynomial but I'm sure you can expand the definition. $\endgroup$
    – Noam
    Commented Jun 27, 2016 at 16:57

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It's called the canonical map into the double dual. It does not have another name that I know of. In a monoidal category with duals (I'm being vague), the uncurried version of it, $\operatorname{ev} : X \otimes X^* \to I$, is called evaluation.

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In the context of functional analysis, the map $$ \Phi: A \to (\Bbb F)^{\Bbb F^A}\\ \Phi:x \mapsto (f \mapsto f(x)) $$ is called the "evaluation map".

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  • $\begingroup$ Isn't $\Phi(x)$ called the evaluation map (at $x$), rather than $\Phi$ itself? (I'm not an analyst so maybe conventions are different there.) $\endgroup$ Commented Jun 27, 2016 at 17:00
  • $\begingroup$ @NajibIdrissi bases on the language of the Wiki page, $\Phi$ itself is referred to as the evaluation map $\endgroup$ Commented Jun 27, 2016 at 17:02
  • $\begingroup$ A related notion is that of the Gelfand transform and the more general Pontryagin duality. $\endgroup$ Commented Jun 27, 2016 at 17:03
  • $\begingroup$ Well in the context of rigid monoidal categories the evaluation map is $\operatorname{ev} : X \otimes X^* \to I$, in any case. So this isn't universal terminology. (It looks more natural to me, but again, not an analyst.) $\endgroup$ Commented Jun 27, 2016 at 17:03
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    $\begingroup$ @goblin how about "the evaluation at $x$"? I don't know if it's used, but I think it's clear enough to be understood. $\endgroup$ Commented Jun 27, 2016 at 17:05
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In linear algebra the double-dual comes to mind, but the general term that's most relevant is "evaluation map". One somewhat common notation is $\mathrm{ev}_{x}(f)=f(x)$. Either ev (your ~) is the evaluation map, or each $\mathrm{ev}_{x}$ is the evaluation map at $x$.

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Looking at $x \in X$ as a map $x \colon 1 \to X$, the operator $\tilde x$ is precisely the natural transformation given by co-Yoneda's embedding $$ \mathsf{Set}(X,-) \stackrel{\mathsf{Set}(x,-)} \longrightarrow \mathsf{Set}(1,-) \simeq \mathrm{id}_{\mathsf{Set}} $$

So you could call it the restriction along $x$, or precomposition by $x$, and denote it $x^\ast$.

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