By the logic of vertical categorification, an internal -group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category with (∞,1)-pullbacks. As described there, in full generality this involves not only a weakening of the usual associativity and unit laws up to homotopy, but requires specification of coherence laws of these homotopies up to higher homotopy, and so on.
A group object in an (∞,1)-category generalizes and unifies two familiar concepts:
it is the generalization of the notion of groupal Stasheff -space from Top to more general (∞,1)-sheaf (∞,1)-toposes: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy, and possibly only partially defined (see also looping and delooping for more on this) ;
Of particular relevance are such group objects that define effective quotients
these are deloopable;
these generalize the notion of regular epimorphism;
But notice the following. Since this is defined internal to an (∞,1)-category, externally these look like genuine ∞-groupoid and ∞-group objects. For instance a group object in a (2,1)-category such as Grpd is, externally, a 2-group.
Also notice that if the ambient -category is in fact an (∞,1)-topos, then every object in there may already be thought of as an “∞-groupoid with geometric structure” (see for instance the discussion at cohesive (∞,1)-topos, but this is true more generally). The relation between the internal groupoid objects then and the objects themselves is (an oid-ification) of that of looping and delooping. Notably for any internal group object (externally an ∞-group) the corresponging ordinary object is its delooping object , and every pointed connected object in the -topos arises this way from an internal group object.
A groupoid object
being effective means that it is the Čech nerve
Accordingly, groupoid objects in an -category play a central role in the theory of principal ∞-bundles.
The following definition follows in style the definition of a complete Segal space object.
such that for all partitions of that share precisely one vertex , we have that
is a (∞,1)-pullback diagram in . Here, by a partition of that share precisely one vertex , we mean two subsets and of whose union is and whose intersection is the singleton . The linear order on then restricts to the linear order on and .
The -category of groupoid objects in is the full sub-(∞,1)-category
of the (∞,1)-category of (∞,1)-functors on those objects that are groupoid objects.
If one requires the above condition only for those partitions that are order-preserving, then this yields the definition of a (pre-)category object in an (∞,1)-category.
of is a (∞,1)-pullback diagram in .
This is HTT, below prop. 184.108.40.206.
A group object is a groupoid object for which is a terminal object.
It follows (HTT, prop. 220.127.116.11) that a group object is of the form
A groupoid object in an -category
Analogously, the (∞,1)-colimit
over the simplicial diagram is the corresponding -quotient.
We state in prop. 1 below a list of equivalent conditions that characterize a simplicial object in an (∞,1)-category as a groupoid object. This uses the following basic notions, which we review here for convenience.
for the precomposition of with the canonical projection. Moreover, write
For and the following are equivalent
In one direction: the limit is the terminal object in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (Lurie, prop. 18.104.22.168)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.
For every and every , the morphism is an weak equivalence in the model structure for quasi-categories
Using remark 3 this means equivalently that the simplicial object is a groupoid precisely if the following
The -category of groupoid objects in is a reflective sub-(∞,1)-category
This is HTT, prop. 22.214.171.124. In nice cases the image of this reflective subcategory are the effective epimorphisms:
This appears below HTT, cor. 126.96.36.199.
Write for the augmented simplex category (including the object ).
An augmented simplicial object is the right Kan extension of its restriction to and
precisley if is a groupoid object in and the diagram
is a (∞,1)-pullback in .
is called the Cech nerve of if the equivalent conditions of this proposition are satisfied.
More generally, this is true for every (∞,1)-topos.
This is HTT, theorem 188.8.131.52 (4) iv).
hence the object universally filling the diagram
Since this is the beginning of the Cech nerve of , is naturally equipped with the structure of an -group object in .
from the full sub-(∞,1)-category of the under-(∞,1)-category of pointed objects on those that are also 0-connected (hence those that have an essentially unique point) with the -category of group objects in .
This is HTT, lemma 184.108.40.206 (1)
The inverse to we write
For we call its delooping.
the (∞,1)-colimit over the group object we have that is reproduced as the Cech nerve of
The object is the delooping object of the group object .
For more on this see also principal ∞-bundle.
The group objects themselves are modeled by a model structure on the category of simplicial groups.
Their delooping spaces are modeled by a model structure on the category of simplicial sets with a single vertex.
The Quillen equivalence itself is in section 6 there.
to the standard model structure on simplicial sets.
This means that a morphism in is a
precisely if it is so in .
This appears as (GoerssJardine, ch V, theorem. 2.3).
are those in the standard model structure on simplicial sets.
This appears as (GoerssJardine, ch V, prop. 6.2).
The -unit and counit are weak equivalences:
This appears as (GoerssJardine, ch. V prop. 6.3).
Groupoid objects in -categories are the topic of section 6.1.2 in
Model category presentations of groupoid objects in by groupoidal complete Segal spaces are discussed in
Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)
A standard textbook reference on the model categories presentation of -groups in is chapter V of
Discussion from the point of view of category objects in an (∞,1)-category is in