# Exponential notation for function spaces

If $$X$$ and $$Y$$ are sets, the set of func­tions from $$X$$ to $$Y$$ (of­ten writ­ten $$X \to Y$$) is some­times also writ­ten $$Y^X$$. This lat­ter no­ta­tion, which we’ll call ex­po­nen­tial no­ta­tion, is re­lated to the no­ta­tion for finite pow­ers of sets (e.g., $$Y^3$$ for the set of triples of el­e­ments of $$Y$$) as well as the no­ta­tion of ex­po­nen­ti­a­tion for num­bers.

Without fur­ther ado, here are some rea­sons this is good no­ta­tion.

• A func­tion $$f : X \to Y$$ can be thought of as an ”$$X$$ wide” tu­ple of el­e­ments of $$Y$$. That is, a tu­ple of el­e­ments of $$Y$$ where the po­si­tions in the tu­ple are given by el­e­ments of $$X$$, gen­er­al­iz­ing the no­ta­tion $$Y^n$$ which de­notes the set of $$n$$ wide tu­ples of el­e­ments of $$Y$$. Note that if $$|X| = n$$, then $$Y^X \cong Y^n$$.

• This no­tion of ex­po­nen­ti­a­tion to­gether with carte­sian product as mul­ti­pli­ca­tion and dis­joint union as ad­di­tion satisfy the same re­la­tions as ex­po­nen­ti­a­tion, mul­ti­pli­ca­tion, and ad­di­tion of nat­u­ral num­bers. Namely,

• $$Z^{X \times Y} \cong (Z^X)^Y$$ (this iso­mor­phism is called cur­ry­ing)

• $$Z^{X + Y} \cong Z^X \times Z^Y$$

• $$Z^1 \cong Z$$ (where $$1$$ is a one el­e­ment set, since there is one func­tion into $$Z$$ for ev­ery el­e­ment of $$Z$$)

• $$Z^0 \cong 1$$ (where $$0$$ is the empty set, since there is one func­tion from the empty set to any set)

More gen­er­ally, $$Y^X$$ is good no­ta­tion for the ex­po­nen­tial ob­ject rep­re­sent­ing $$\text{Hom}_{\mathcal{C}}(X, Y)$$ in an ar­bi­trary carte­sian closed cat­e­gory $$\mathcal{C}$$ for the first set of rea­sons listed above.

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.

• I don’t think this is what you mean, is it?

• Thanks, I’ve cor­rected it. That was a strange typo.