# nLab enriched slice category

Enriched slice categories

### Context

#### Enriched category theory

enriched category theory

# Enriched slice categories

## Idea

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.

## Definition

###### Definition

Let

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 $V$:

###### Remark

The dual concept is the enriched co-slice category or enriched under-category, the enriched generalization of the notion of co-slice category.

## Properties

• $C/x$ 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 $f_*$.

## References

• MathOverflow, Enriched slice categories (MO:q/259829)

Last revised on October 11, 2021 at 06:26:30. See the history of this page for a list of all contributions to it.