# Exponential Law in Algebraic topology.

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?

• See Hatcher's Proposition A.14 Commented Aug 31, 2019 at 1:55
• where is this proposition @LordSharktheUnknown, I am sorry for this trivial question but I mean at which chapter is it? Commented Aug 31, 2019 at 2:24
• A for Appendix. Commented Aug 31, 2019 at 2:25
• Thanks! I am sorry for bothering you ..... but my professor also had spoken about " the direct product and the existence and uniqueness of a universal mapping for it " is this title in Hatcher also @LordSharktheUnknown Commented Aug 31, 2019 at 2:29

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 $$Z^{X×Y}=(Z^Y)^X$$

• Commented Aug 31, 2019 at 9:24

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.