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 . 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. ?

Indeed, yes.

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

where denotes the two-pointed set. In face, the isomorphism is given by and where and . In general, it means that

.

Using exponential notation, it reads that . It satisfies familiar equalities we learned, e.g.

- .
- .
- where is the empty set and is the one-pointed set.
- for any non-empty set .

With this notation, we can drop the subset notation since is equivalent to where becomes the characteristic function on . That is,

if or otherwise.

Moreover, the belonging relation can be viewed as a function from one-pointed set to , e.g. . Note that is a proposition in set theory but a function 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 …

### Like this:

Like Loading...

*Related*

Interesting. I’d be interested learning something about topos.

And when will we learn that 2+2 = 4? 🙂

Do you mean Peano axioms? or, the coproduct of 2 and 2 is 4 in ?

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?

欸.. 其實我只是在 refer 這個笑話而已..

http://flolac.iis.sinica.edu.tw/lambdawan/node/181

I know topos only by name, so I’m hoping to learn something. 🙂

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!