The notion of enriched slice categories is the generalization of the notion of slice categories from plain category theory of locally small categories to enriched category theory of $\mathcal{V}$-enriched categories.
Let
$(V,\otimes,I)$ be a monoidal category with pullbacks (e.g. a BΓ©nabou cosmos),
$C$ a $V$-enriched category,
$x\in C$ an object.
Then the enriched slice category (or enriched over-category) $C/x$ is the following $V$-enriched category:
its objects are the morphisms $f \,\colon\, y\to x$ in (the underlying ordinary category of) $C$, i.e. morphisms $I \to C(y,x)$ in $V$.
its hom-object from $f \,\colon\, y\to x$ to $g \,\colon\, z\to x$ is the pullback in $V$:
The dual concept is the enriched co-slice category or enriched under-category, the enriched generalization of the notion of co-slice category.
$C/x$ can be described as the comma object of $Id \,\colon\,C\to C$ and $x \,\colon\, I \to C$ in the 2-category $V Cat$, where $I$ denotes the unit enriched category.
All hom-objects of $C/x$ are augmented, i.e. equipped with a morphism to the unit object. This seems somewhat curious unless $V$ is cartesian monoidal (or at least semicartesian) in which case $I$ is the terminal object $1$.
Any morphism $f \,\colon\, x\to y$ in $C$ induces a $V$-enriched functor $f_! \,\colon\, C/x \to C/y$. If $C$ has $V$-enriched pullbacks, then $f_!$ has a right $V$-enriched adjoint $f^*$. If $C$ is a locally cartesian closed enriched category, then $f^*$ has a further right $V$-adjoint $f_*$.
Last revised on October 11, 2021 at 06:26:30. See the history of this page for a list of all contributions to it.