nLab enriched slice category

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

• $(V,\otimes,I)$ be a monoidal category with pullbacks (e.g. a Bénabou 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 $V$:

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 $f_*$.

Related concepts

• enriched slice model structure

• slice of presheaves is presheaves on slice

References

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