hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
An ordinary locally small category $C$ has for any ordered pair of objects $x,y$ a hom-set $C(x,y)$—an object in the category $Set$.
For $C$ more generally an enriched category over a closed monoidal category $V$, there is – by definition – for all $x,y$ an object $C(x,y) \in obj V$ that plays the role of the “collection of morphisms” from $x$ to $y$
hom-object
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |