# nLab enriched groupoid

Contents

### Context

#### Enriched category theory

enriched category theory

# Contents

## Idea

The notion of enriched groupoids is the generalization to enriched category theory of the notion of groupoids from plain category theory.

## Definition

Let $(\mathcal{V}, \times, \ast)$ be a cartesian monoidal category (serving as an base of enrichment), so that, in particular,

• the tensor unit is a terminal object $\ast$,

• for each object $v \in \mathcal{V}$ there is a diagonal morphism

$\Delta_v \,\colon\, v \to v \times v \,.$

A $\mathcal{V}$-enriched groupoid is a $\mathcal{V}$-enriched category $\mathcal{C}$ equipped for $X, Y \,\in\, Obj(\mathcal{C})$ with morphisms (in $\mathcal{V}$) of the form

$inv_{X,Y} \;\colon\; \mathcal{C}(X,Y) \longrightarrow \mathcal{C}(Y,X)$

which behave like assigning inverse morphisms in that the composition morphisms of $\mathcal{C}$

$comp_{X, Y, Z} \;\colon\; \mathcal{C}(X,Y) \otimes \mathcal{C}(Y,Z) \longrightarrow \mathcal{C}(Y,Z)$

satisfy

$comp_{Y,X,Y} \circ \big( inv_{X,Y} \times id_{\mathcal{C}(X,Y)} \big) \circ \Delta_{\mathcal{C}(X,Y)} \;\;=\;\; \mathcal{C}(X,Y) \overset{}{\longrightarrow} \ast \overset{id_Y}{\longrightarrow} \mathcal{C}(Y,Y)$

and

$comp_{X,Y,X} \circ \big( id_{\mathcal{C}(X,Y)} \times inv_{X,Y} \big) \circ \Delta_{\mathcal{C}(X,Y)} \;\;=\;\; \mathcal{C}(X,Y) \overset{}{\longrightarrow} \ast \overset{id_X}{\longrightarrow} \mathcal{C}(X,X) \,.$

## Examples

###### Example

For Set regarded with its cartesian monoidal structure, Set-enriched groupoids are ordinary groupoids.

###### Example

For Grpd regarded as a 1-category and equipped with its cartesian monoidal structure given by forming product groupoids, Grpd-enriched groupoids are strict 2-groupoids.

###### Example

The sSet-enriched groupoids are traditionally misnamed simplicial groupoids (following Dwyer & Kan (1984) and similar abuse for simplicial category), see there for more. More pedantically, $sSet$-enriched groupoids are only those sSet-internal groupoids whose set of objects is constant, cf. Exp. below.

Moreover, fundamental $\infty$-groupoids incarnated as sSet-enriched groupoids aka βDwyer-Kan simplicial groupoidsβ are (mis-)named Dwyer-Kan loop groupoids.

###### Example

Generally, a $\mathcal{V}$-enriched groupoid with a single object is equivalently the delooping groupoid of a group object internal to $\mathcal{V}$.

For instance, a one-object sSet-groupoid (Exp. ) is the delooping groupoid of a simplicial group.

###### Example

An internal groupoid in $\mathcal{V}$ is equivalently a $\mathcal{V}$-enriched groupoid if its $\mathcal{V}$-object of objects is in the image of the coproduct-preserving functor $Set \to \mathcal{V}$ (using here the assumption that $\mathcal{V}$ is a BΓ©nabou cosmos, hence cocomplete, hence with all coproducts).

## References

The notion of enriched groupoids is folklore, originating in the special case of enrichment over sSet, where it is traditionally discussed, going back to Dwyer & Kan (1980), (1984), in the guise of simplicial objects in Grpd with discrete simplicial set of objects and then referred to (inaccurately) as simplicial groupoids (see there for more).

Later

switches to the terminology of enriched groupoids over simplicial sets (but still does not give the definition of enriched groupoids).

References that make the general definition of enriched groupoids explicit:

Last revised on June 6, 2023 at 08:33:04. See the history of this page for a list of all contributions to it.