hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
Given objects $x$ and $y$ in a locally small category, the hom-set $hom(x,y)$ is the collection of all morphisms from $x$ to $y$. In a closed category, the hom-set may also be called the external hom to distinguish it from the internal hom.
For a category $C$ enriched over a category $V$, the “hom-set” $C(x,y)$ is an object of $V$, the hom-object.
For $C = (C_0, C_1, s,t,e, c)$ an internal category, the generalized objects of $C$ are morphisms $x: X \to C_0$ and $y: Y \to C_0$, and the “hom-set” becomes the pullback $C(x,y)$ in
In particular, in a category with a terminal generator $*$, we may identitfy morphisms $x,y: * \to C_0$ with global objects of $C$ and form $C(x,y)$ as above.