# nLab enriched over category

Enriched over-categories

# Enriched over-categories

## Idea

The generalization of an over-category to enriched categories.

## Definition

Let $(V,\otimes,I)$ be a monoidal category with pullbacks, $C$ a $V$-enriched category, and $x\in C$ an object. The enriched over-category (or enriched slice category) $C/x$ is the following $V$-category:

• its objects are the morphisms $f:y\to x$ in (the underlying ordinary category of) $C$, i.e. morphisms $I\to C(y,x)$ in $V$.

• its hom-object from $f:y\to x$ to $g:z\to x$ is the pullback in $V$:

The dual concept is the enriched under-category or enriched co-slice category.

## Properties

• $C/x$ can be described as the comma object of $Id:C\to C$ and $x : I \to C$ in the 2-category $V Cat$, where $I$ denotes the unit enriched category.

• All hom-objects of $C/x$ are augmented, i.e. come with a map 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 $f:x\to y$ in $C$ induces a $V$-functor $f_!:C/x \to C/y$. If $C$ has $V$-enriched pullbacks, then $f_!$ has a right $V$-adjoint $f^*$. If $C$ is a locally cartesian closed enriched category, then $f^*$ has a further right $V$-adjoint $f_*$.