category object in an (∞,1)-category, groupoid object
equivalences in/of $(\infty,1)$-categories
A group object in an ordinary category $C$ with pullbacks is an internal group. More generally, there is the notion of an internal groupoid in a category $C$.
By the logic of vertical categorification, an internal $\infty$-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category $C$ 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 $A_\infty$-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) ;
it generalizes the notion of equivalence relation – or rather the internal notion of congruence – from category theory to (∞,1)-category theory.
Of particular relevance are such group objects that define effective quotients
these are deloopable;
these generalize the notion of regular epimorphism;
these serve to characterize regular (∞,1)-categories – such as ∞-stack (∞,1)-topoi – as those where every such object is an effective quotient.
A groupoid object is then accordingly the many-object version of a group object.
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 $(\infty,1)$-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 $G$ any internal group object (externally an ∞-group) the corresponging ordinary object is its delooping object $\mathbf{B}G$, and every pointed connected object in the $(\infty,1)$-topos arises this way from an internal group object.
A groupoid object
being effective means that it is the Cech nerve
of its quotient stack $C_0 \sslash C_1$ (the (∞,1)-colimit over its diagram)
Accordingly, groupoid objects in an $(\infty,1)$-category play a central role in the theory of principal ∞-bundles.
Notice that one of the four characterizing properties of an (∞,1)-topos by the higher analog of the Giraud theorem is that all groupoid objects in an (infinity,1)-topos are effective.
The following definition follows in style the definition of a complete Segal space object.
(groupoidal Segal conditions)
For $C$ an (∞,1)-category, a groupoid object in $C$ is a simplicial object in an (∞,1)-category
such that for all partitions $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we have that
is a (∞,1)-pullback diagram in $C$. Here, by a partition $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we mean two subsets $S$ and $S'$ of $\{0,1,\ldots,n\}$ whose union is $\{0,1,\ldots,n\}$ and whose intersection is the singleton $\{s\}$. The linear order on $[n]$ then restricts to the linear order on $S$ and $S'$.
The $(\infty,1)$-category of groupoid objects in $C$ is the full sub-(∞,1)-category
of the (∞,1)-category of (∞,1)-functors on those objects that are groupoid objects.
This is HTT, prop. 6.1.2.6, item 4'' with HTT, def. 6.1.2.7.
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.
A groupoid object $A : \Delta^{op} \to C$ is the Cech nerve of a morphism $A_0 \to B$ if $A$ is the restriction of an augmented simplicial object $A^+ : \Delta^{op}_a \to C$ with $A^+_0 \to A^+_{-1}$ as the morphism $A_0 \to B$, such that the sub-diagram
of $A^+$ is a (∞,1)-pullback diagram in $C$.
This is HTT, below prop. 6.1.2.11.
If $A$ is the Cech nerve of a morphism $A_0 \to A_{-1}$
then the groupoid object is deloopable in the groupoid sense.
A groupoid object $A : \Delta^{op} \to C$ is an effective quotient object if the (∞,1)-colimit diagram $A^+ : \Delta_a^{op} \to C$ exists, such that $A$ is the Cech nerve of $A^+_0 \to A^+_{-1}$, i.e. of $A_0 \to \lim_\to A_\bullet$.
A group object is a groupoid object $U : \Delta^{op} \to C$ for which $U_0 \simeq *$ is a terminal object.
It follows (HTT, prop. 7.2.2.4) that a group object is of the form
A groupoid object in an $(\infty,1)$-category
is the (∞,1)-category analog of an internal equivalence relation on $A_0$, which is just a pair of morphisms
The colimit (coequalizer) of the latter diagram is the quotient of $A_0$ by the relation $R$.
Analogously, the (∞,1)-colimit
over the simplicial diagram $A : \Delta^{op} \to C$ is the corresponding $(\infty,1)$-quotient.
If we are given a model category presentation of the (∞,1)-category $C$, then this (∞,1)-colimit is presented by a homotopy colimit over the corresponding simplicial diagraM a homotopy quotient .
We state in prop. 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 $K \in$ sSet a simplicial set, write $\Delta_{/K}$ for its category of simplices. For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ a simplicial object, write
for the precomposition of $X_\bullet$ with the canonical projection. Moreover, write
for the (∞,1)-limit over this composite (∞,1)-functor in $\mathcal{C}$ (if it exists). (Notice: square brackets for the composite functor, round brackets for its $(\infty,1)$-limit.)
For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent
the induced morphism of cone $(\infty,1)$-categories $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an equivalence of (∞,1)-categories;
the induced morphism of (∞,1)-limits $X(K) \to X(K')$ is an equivalence.
(The first perspective is used in (Lurie), the second in (Lurie2).)
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. 4.1.1.8)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.
Let $\mathcal{C}$ be an $(\infty,1)$-category incarnated explicitly as a quasi-category. Then a simplicial object in $\mathcal{C}$ is a groupoid object if the following equivalent conditions hold.
If $K \to K'$ is a morphism in sSet which is a weak homotopy equivalence and a bijection on vertices, then the induced morphism on slice-(∞,1)-categories
is an equivalence of (∞,1)-categories (a weak equivalence in the model structure for quasi-categories).
For every $n \geq 2$ and every $0 \leq i \leq n$, the morphism $\mathcal{C}_{/X[\Delta^n]} \to \mathcal{C}_{/X[\Lambda^n_i]}$ is an weak equivalence in the model structure for quasi-categories
(…)
Using remark this means equivalently that the simplicial object $X_\bullet$ is a groupoid precisely if the following
$X_\bullet$ satisfies the ordinary Segal conditions and the morphism $X(\Delta^2) \to X(\Lambda^2_0)$ is an equivalence.
(…)
The first items appear as (Lurie, prop. 6.1.2.6). The second ones appear in the proofs of (Lurie2, prop. 1.1.8, lemma 1.2.25).
The $(\infty,1)$-category of groupoid objects in $C$ is a reflective sub-(∞,1)-category
This is HTT, prop. 6.1.2.9. In nice cases the image of this reflective subcategory are the effective epimorphisms:
If $C = \mathbf{H}$ in an (∞,1)-semitopos there is a natural equivalence of (∞,1)-categories
between the $(\infty,1)$-category of groupoid objects in $\mathbf{H}$ and the full sub-(∞,1)-category of the arrow category of $\mathbf{H}$ (the (∞,1)-functor (∞,1)-category $Func(\Delta[1], \mathbf{H})$) on the effective epimorphisms.
This appears below HTT, cor. 6.2.3.5.
Write $\Delta_a$ for the augmented simplex category (including the object $[-1]$).
An augmented simplicial object $A^+ : \Delta_a^{op} \to C$ is the right Kan extension of its restriction to $[-1]$ and $[0]$
precisley if $A^+|_{\geq 0}$ is a groupoid object in $C$ and the diagram
is a (∞,1)-pullback in $C$.
$A$ is called the Cech nerve of $A_0 \to A_{-1}$ if the equivalent conditions of this proposition are satisfied.
In $C =$ ∞Grpd every groupoid object is an effective quotient, def. .
This is HTT, below remark 6.1.2.15 and HTT, cor. 6.1.3.20.
More generally, this is true for every (∞,1)-topos.
In $C$ is an (∞,1)-topos, then every groupoid object in $C$ is an effective quotient, def. .
This is HTT, theorem 6.1.0.6 (4) iv).
For $\mathcal{X}$ an (∞,1)-category with (∞,1)-pullbacks and for $x : * \to X$ a pointed object in $\mathcal{X}$, its loop space object at $x$ is the (∞,1)-pullback
hence the object universally filling the diagram
Since this is the beginning of the Cech nerve of $* \to X$, $\Omega_x X$ is naturally equipped with the structure of an $\infty$-group object in $\mathcal{X}$.
Let $\mathcal{X}$ be an (∞,1)-topos. Then the operation of forming loop space objects constitutes an equivalence of (∞,1)-categories
from the full sub-(∞,1)-category of the under-(∞,1)-category $*/\mathcal{X}$ of pointed objects on those that are also 0-connected (hence those that have an essentially unique point) with the $(\infty,1)$-category of group objects in $\mathcal{X}$.
This is HTT, lemma 7.2.2.11 (1)
The inverse to $\Omega$ we write
For $G \in Grp(\mathcal{X})$ we call $\mathbf{B}G$ its delooping.
When the ambient (∞,1)-category is an (∞,1)-topos then – by the $\infty$-Giraud axioms – all groupoid objects are effective, meaning that for
the (∞,1)-colimit over the group object $U_\bullet$ we have that $U_\bullet$ is reproduced as the Cech nerve of $* \to \mathbf{B}G$
The object $\mathbf{B}G$ is the delooping object of the group object $G$.
For more on this see also principal ∞-bundle.
There is a model category structure that presents the (∞,1)-category of group objects in ∞Grpd: the ∞-groups.
The group objects $G$ themselves are modeled by a model structure on the category $sGrp$ of simplicial groups.
Their delooping spaces $\mathbf{B}G$ are modeled by a model structure on the category $sSet_0$ of simplicial sets with a single vertex.
The operation of forming loop space objects constitutes a Quillen equivalence between these two model structures
The Quillen equivalence itself is in section 6 there.
There exists the transferred model structure on the category $sGrp$ of simplicial groups along the forgetful functor
to the standard model structure on simplicial sets.
This means that a morphism in $sGrp$ is a
weak equivalences
or fibration
precisely if it is so in $sSet_{Quillen}$.
This appears as (GoerssJardine, ch V, theorem. 2.3).
There is a model structure on reduced simplicial sets $sSet_0$ (simplicial sets with a single vertex) whose
weak equivalences
and cofibrations
are those in the standard model structure on simplicial sets.
This appears as (GoerssJardine, ch V, prop. 6.2).
The simplicial loop space functor $G$ and the delooping functor $\bar W(-)$ (discussed at simplicial group) constitute a Quillen equivalence
The $(G \dashv \bar W)$-unit and counit are weak equivalences:
This appears as (GoerssJardine, ch. V prop. 6.3).
Groupoid objects in $(\infty,1)$-categories are the topic of section 6.1.2 in
Model category presentations of groupoid objects in $\infty Grpd$ 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)
Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)
A standard textbook reference on $\infty$-groups in the classical model structure on simplicial sets is
Discussion from the point of view of category objects in an (∞,1)-category is in