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For a set $A$, do $A\to 2^{A}$ or its elements have canonical names?

I know the term "set-valued", but I want to express specifically that the same set appears on both sides.

I'd also be interested in domain specific terminology, e.g., when $A$ is a topological space or lattice or when specific conditions like $f(a)\ni a$ or $f$ mapping to open sets or $f$ being continuous are satisfied.

Bananach
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  • There should be something topos theoretic about this. Morally it's adjunct to the equality morphism $A\times A\to 2$ that returns true iff. the two factors are equal. – FShrike Mar 11 '24 at 23:32
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    I don't understand the morally part, but there's plenty I don't know when it comes right down to it. – A rural reader Mar 11 '24 at 23:38
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    @Aruralreader I too would like to understand what it means to be mathematically moral... – H. sapiens rex Mar 11 '24 at 23:40
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    You could call it the currying of a binary relation on $ A $, although I'm not sure how many people would understand that. If you think of a relation as a subset of $ A \times A $, think of a subset as a map to $ 2 $, and think of a map from $ X \times Y $ to $ Z $ as a map from $ X $ to $ Z ^ Y $ (this is currying), then a binary relation on $ A $ is equivalent to a map from $ A $ to $ 2 ^ A $. For example, the currying of the equality relation on $ A $ is the map $ x \mapsto { x } $ (which I think is what @FShrike was getting at). – Toby Bartels Mar 11 '24 at 23:54
  • Does the notation $A\to2^A$ denote the set of all mappings from $A$ to $2^A$? And does the notation $2^A$ denote the set of all mappings from $A$ to $2$? – user14111 Mar 11 '24 at 23:56
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    @H.sapiensrex: to say that something is "morally true" is a not uncommon way of saying that it is intuitively right and useful but that it needs some work on fine details to make it rigorous (in the case of FShrike's comment, via the natural correspondences between $A \times A \to 2$ and $A \to A \to 2$ and between $A \to 2$ and $2^A$). – Rob Arthan Mar 12 '24 at 00:00
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    @user14111: yes (morally speaking $\ddot{\smile}$). I.e., you can certainly choose to define $2^A$ to be $2 \to A$, although many people prefer to think of $2^A$ as the powerset, $\Bbb{P}(A)$, of $A$, so this definition is a natural equivalence for them. – Rob Arthan Mar 12 '24 at 00:02
  • It's adjunct to the map $A\times A\to 2$ which classifies the diagonal subobject $\Delta:A\hookrightarrow A\times A$. That probably has some kind of internal semantics interpretation, but I have no topos logic experience – FShrike Mar 12 '24 at 00:03
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    @FShrike: many people think topos theory is downright immoral - but I'm with you so far! – Rob Arthan Mar 12 '24 at 00:05
  • The term "set mapping" has been used for maps $f:A\to\mathcal P(A)$ such that $x\in f(x)$ for all $x$. – user14111 Mar 12 '24 at 00:37

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