category object in an (∞,1)-category, groupoid object
homotopy theory, (∞,1)-category theory, homotopy type theory
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A notion of internal ∞-groupoid is a vertical categorification of internal groupoid. As described at that entry on vertical categorification, there is some flexibility possible in exactly what one may mean by this, depending for instance on which of several definitions of ordinary groupoids one starts with, and how one deals with the higher coherences that are introduced upon categorification.
One very general notion of internal $\infty$-groupoid is given by taking some standard definition of $\infty$-groupoid and writing it down internally to an (∞,1)-category. This is described below in the section
For various applications somewhat stricter or at least more rigidified models for this may be useful. Notably one may wish to speak of $\infty$-groupoids internal to an ordinary category. In terms of Kan complexes as a model for ordinary ∞-groupoids, this is discussed below in the section
Notice from the discussion there that these two aspects may and often do interplay: simplicial objects in sheaf toposes may be used to present (∞,1)-categories inside of which one may be interested in $\infty$-groupoid objects in the first sense. For instance a Lie $\infty$-group object is usefully thought of as a group object internal to the (∞,1)-topos presented by a model of simplicial sets in the category of sheaves on Diff.
Finally, a simplified version of ∞-groupoid is often sufficient and useful: that given by strict ∞-groupoids, equivalently crossed complexes. These may be straightforwardly internalized in any ordinary category with pullbacks. This is discussed in
Under the internal omega-nerve operation this will embed into the definition of Kan complexes in an ordinary category mentioned before. Even if one is interested just in strict $\infty$-groupoids, this embedding is useful in order to understand what the right notion of morphisms between these objects is, and what the extra descent conditions are that a proper internal formulation turns out to require on top of the conditions known from the external formulation (all this is discussed below). For that purpose Dominic Verity’s theorem described at
is very useful.
The special case of internal 1-groupoids in an (∞,1)-category, which are still only associative and unital up to higher homotopy, but which do not include “higher cells” as additional data, is discussed in detail at:
The general case remains to be explored.
A standard model for general ∞-groupoids is given by simplicial sets that are Kan fibrant: Kan complexes. It is straightforward to internalize the horn-filler condition that characterizes Kan complexes from the category SSet of simplicial objects in Set to one of simplicial objects in any category $C$ with a bit of suitable structure.
But for the simplicial objects in question to be usefully identified as ∞-groupoids internal to $C$, the horn filler condition is – while necessary – typically not sufficient. At least one has to be a bit careful about what it is one wants to model.
A very well understood special case that serves to give some feeling for the general situation is that where $C$ is a Grothendieck topos. In that case simplicial objects in $C$ are simplicial sheaves. There is a well developed theory of model structures on simplicial sheaves that are known to be models for ∞-stack (∞,1)-toposes.
In terms of this the true $\infty$-groupoids internal to $C$ are those simplicial sheaves that are fibrant objects in the given model category. These fibrant simplicial sheaves in particular satisfy the internal Kan filler conditon, but they are characterized by a much stronger condition: in addition they satisfy a descent condition.
Another issue highlighted by this example is that the right notion of morphisms of internal Kan complexes are not just the internal morphisms of simplicial objects: for the case of $C$ a Grothendieck topos and using the model structure on simplicial sheaves as above, the right notion of morphism between two internal Kan complexes $X$ and $A$ is a morphism of simplicial objects $\hat X \to A$ out of a cofibrant replacement of $X$, as discussed at derived hom space. So the right notion of morphisms of internal $\infty$-groupoids are $\infty$-anafunctors.
A little bit of theory for this exists for the slightly more general case that $C$ is an arbitrary elementary topos. See model structure on simplicial objects in a topos?.
For more general $C$ not much is known.
The Kan complex definition of ∞-groupoid may be internalized to more general categories. Namely:
In particular, if $C$ is a left exact category equipped with a Grothendieck pretopology, then the limits in (2) exist and one can choose single arrow coverings as distinguished ‘surjections’. So we define a Kan object in such a $C$ to be a simplicial object in $C$ satisfying the Kan conditions (3). Then we may define an $\infty$-groupoid in $C$ to be simply a Kan object in $C$. (Recall that one way to define an ordinary $\infty$-groupoid is as a Kan complex, that is a Kan object in Set.)
Another, almost equivalent, way to describe the Kan conditions expressed in this way is by saying that “the Kan conditions are satisfied” is true in the internal logic of the site $C$, or equivalently the internal logic of its Grothendieck topos of sheaves (after applying the Yoneda embedding to obtain a simplicial object in this topos). Of course, if $C$ is itself a Grothendieck topos, then it is equivalent to the topos of sheaves on itself for its canonical topology. (This is only “almost equivalent” if the Grothendieck pretopology is in fact a topology.)
If the category $C$ is a Grothendieck topos, i.e. a category of sheaves (of sets) $C = Sh(S)= Sh(S,Set)$ on some small site, then simplicial objects in $C$ are the same as simplicial sheaves on $S$
There are various (Quillen equivalent) model category structures on the categories of simplicial sheaves or presheaves on a small site; see model structure on simplicial presheaves and model structure on simplicial sheaves for more. An $\infty$-groupoid in $C$ may be taken to be a fibrant object with respect to one of these model category structures.
The two definitions are not equivalent even when $C$ is a Grothendieck topos; the second is strictly stronger. Consider, for instance, a Grothendieck topos $C = Sh(S)$ and an internal groupoid in $C$, i.e. a sheaf of groupoids on $S$, but which is not a stack. Then the nerve of this internal groupoid will satisfy the Kan conditions in the sense of the first definition (by repeating the usual proof that the nerve of a groupoid is a Kan complex in the internal logic), but it will not be fibrant as a simplicial sheaf (since it is not a stack).
On the other hand, it is true that every fibrant simplicial sheaf satisfies the Kan conditions in the internal logic, by the following argument:
The fibrant objects in any local model structure on simplicial sheaves in particular have the property that they are sheaves with values in Kan complexes.
Here a reminder on why this is so. The local model structures on simplicial sheaves are left Bousfield localizations of the injective or projective global model structure on functors. So their fibrant objects are in particular fibrant in $[S^{op},SSet]_{proj/inj}$. The fibrant objects in $[S^{op}, SSet]_{proj}$ are (by definition of projective fibrations) precisely the objectwise Kan complexes. The fibrant objects in $[S^{op},SSet]_{inj}$ are less, but still contained in the collection of objectwise Kan complexes. So also the fibrant objects in $Sh(S,SSet)_{inj,proj}^{loc}$ are in particular Kan complex valued sheaves (that in addition satisfy a descent condition).
The weighted limits over simplicial sheaves, or analogously the powering of $Sh(S,SSet)$ over SSet works objectwise, so that for $X \in Sh(S,SSet)$ we have
and of course
Therefore fibrant $X \in Sh(S,SSet)$ have the property that for all $n, k$ the canonical morphism $X_n \to X^{\Lambda_k[n]}$ is a surjection when pulled back to any representable $U \to X$. In particular, the morphism is a stalkwise epimorphism, hence an epimorphism of sheaves.
Note that the local model structure on simplicial sheaves also contains information about the cofibrant objects, which (it can be argued) are necessary to get the right notion of morphism between “internal $\infty$-groupoids.” Every Kan complex in $Set$ is cofibrant, but the same cannot be expected to be true everywhere, and in general a (weak) “internal $\infty$-functor” should be expected to be a map out of a cofibrant replacement.
Here are some examples of internal $\infty$-groupoids according to the first definition (that is, internal Kan complexes).
A classical example consists of the topological $\infty$-groupoids. The nerve construction makes a topological $\infty$-groupoid from a topological groupoid. This is actually a characterization of topological groupoids among topological categories.
In particular, if $G$ is a topological group, then $N(\mathbf{B}G)$ is a topological $\infty$-groupoid. This is relevant to the construction of the classifying spaces for continuous principal bundles.
Another classical example consists of the ∞-Lie groupoids.
Models for ∞-stacks/(∞,1)-presheaves in higher geometry by local Kan complexes of objects in a given site (for instance locally Kan simplicial manifolds for higher differential geometry) are discussed in