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 $\mathcal{C}$ has for any ordered pair of objects $x,y$ a hom-set $\mathcal{C}(x,y)$βan object in the category Set.
For $\mathcal{C}$ more generally an enriched category over a closed monoidal category $\mathcal{V}$, there is β by definition β for all $x,y$ an object $\mathcal{C}(x,y) \in obj V$ that plays the role of the βcollection of morphismsβ from $x$ to $y$.
Accordingly, if here $\mathcal{V}$ is a category of βspacesβ of sorts, then one also speaks of a hom space. For instance if $\mathcal{V}$ =vector spaces such that $\mathcal{C}$ is a linear category, then hom-spaces are vector spaces.
Or if $\mathcal{V}$=topological spaces such that $\mathcal{C}$ is topologically enriched category, then hom-spaces are topological spaces. If this last example is regarded in the context of homotopy theory, then one may consider just the homotopy type of these topological spaces, which is equivalently modeled as a simplicial set (Kan complex) and thought of as an infinity-groupoid. In all these cases people still often speak of βhom spacesβ, but see at derived hom space.
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 $(-) \otimes^{\mathbb{L}} A$ |