Exponential Notation

Let’s enjoy some familiar concepts in sets and be happy with the exponential notation of function space.

Normally, a function from set X to set Y is represented as f:X \rightarrow Y. Sometimes, people say that it is the exponential object in the category of Set, but what does it mean? Is it similar to the exponentiation of numbers, e.g. 2^3?

Indeed, yes.

We can view the disjoint union and intersection as addition and multiplication respectively. For example, we have

\displaystyle{ (\mathbf{2} \rightarrow A) \cong A \times A}

where \mathbf{2} denotes the two-pointed set. In face, the isomorphism is given by f \mapsto (f(0), f(1)) and (x, y) \mapsto g where g(0) = x and g(1) = y. In general, it means that

\displaystyle{(B \rightarrow A) \cong \underbrace{A \times A \times \dots \times A}_{| B |}}.

Using exponential notation, it reads that A^{B} \cong \prod_{i = 1}^{|B|} A. It satisfies familiar equalities we learned, e.g.

  1. A^B \times A^C \cong A^{B + C}.
  2. (A^{B})^{C} \cong A^{B \times C}.
  3. A^{0} \cong 1 where 0 is the empty set and 1 is the one-pointed set.
  4. 0^{A} \cong 0 for any non-empty set A.

With this notation, we can drop the subset notation \subseteq since S \subseteq X is equivalent to S \in \mathbf{2}^{X} where S becomes the characteristic function on S. That is,

\chi_{S}(x) = 1 if x \in S or \chi_{S}(x) = 0 otherwise.

Moreover, the belonging relation \in can be viewed as a function from one-pointed set to X, e.g. (* \rightarrow A) \equiv A^{*}. Note that x \in X is a proposition in set theory but a function x : * \rightarrow A is a proposition in the meta-language.

Looking at the above discussion more carefully, we may suspect whether we can talk about set theory purely in terms of objects, morphisms, hom-set and so on. Well, indeed! Basically it is called topos theory and maybe I can say something about topos theory after I fully understand it …


4 thoughts on “Exponential Notation

  1. And when will we learn that 2+2 = 4?

    Do you mean Peano axioms? or, the coproduct of 2 and 2 is 4 in \mathbf{Finord}?

    Although I wrote that I would talk about topos, but it’s still far from the goal … Any idea to introduce the exponential object, sub-object and sub-object classifier?

  2. 難怪這個 2+2=4 好眼熟。XD

    Okay, let me explain it to myself and write down what I have been explained!

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