hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
Given sets $A$ and $B$, the function set $A^B$ is the set of all functions from $B$ to $A$. In the foundations of mathematics, the existence of such a set may be taken to follow from the existence of power sets, from the axiom of subset collection, or as an axiom (the axiom of function sets) in its own right.
Thinking of Set as a locally small category, this is a special case of a hom-set. Thinking of $\Set$ as a cartesian closed category, this is a special case of an exponential object, which is a special case of an internal hom.