Exponential Law in Algebraic topology.

Question

I had a class in algebraic topology, our main book is Allen Hatcher, our professor defined a term called "Exponential Law" as the following:

$Hom (X \times Y, Z) \cong Hom (X, Hom (Y, Z))$

$\alpha : X \times Y \rightarrow Z $

$\tilde{\alpha} : X \rightarrow Hom (Y, Z)$

$\tilde{\alpha} (x)(y) = \alpha (x, y) $

(I may have errors in copying after my professor, forgive me if I have).

My questions are:

1-Where can I find this title in Allen Hatcher or any other book (Actually I asked my professor and he/she said that I may find it in Munkres under the title of "Mapping spaces" and I assumed that he/she means Munkres of general topology and also I did not find this exponential law ), could anyone help me in this please?

2-Why it is called exponential law?

Answer 1

1.) This has already been answered in the comments, but as an alternative source Davis and Kirk talk about it when they are discussing compactly generated weak hausdorff spaces, which I prefer.

2.) If you write $\operatorname{Hom}(X,Y)$ as $Y^X$(which is standard) then the statement becomes $$Z^{X×Y}=(Z^Y)^X$$

Answer 2

Why is it called exponential law?

Let $|X|$ be the cardinality of set $X$ if $X$ is considered to be a set.

We have, in the category of sets,

$$ | Hom(X\ Y)|\ =\ |Y|^{|X|} $$

Also

$$ |X\times Y|\ =\ |X|\cdot|Y| $$

Hence

$$ |Hom(X\!\times\! Y\,\ Z)|\ =\ |Z|^{|X\times Y|}\ =\ |Z|^{|X|\cdot|Y|} \ =\ (|Z|^{|Y|})^{|X|}\ =\ |Hom(X\,\ H(Y\ Z))| $$

This is why the bijection $\ Hom(X\!\times\! Y\,\ Z)\rightarrow Hom(X\,\ Hom(Y\ Z))\ $ is called the exponential law for the category of sets; and that's why this bijection is called the exponential law for the arbitrary category for which it is true, whenever it is true.