nLab enriched groupoid

Idea

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

Definition

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

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

• 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

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

and

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

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 “Dwyer-Kan simplicial groupoids” 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 Bénabou cosmos, hence cocomplete, hence with all coproducts).

Related concepts

• enriched category

• internal groupoid, internal group

• groupoid object in an (∞,1)-category

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

• John F. Jardine, §9.3 in: Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

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:

• Jacopo Emmenegger, Fabio Pasquali, Giuseppe Rosolini, §3 in: Elementary fibrations of enriched groupoids, Mathematical Structures in Computer Science 31 9 (2021) 958-978 [doi:10.1017/S096012952100030X]

  (discussed as categorical semantics for homotopy type theory)