homotopy theory, (∞,1)-category theory, homotopy type theory
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Recall that a simplicial set is a combinatorial model for a topological space. This relation is most immediate when the simplicial set is in fact a Kan complex (an ∞-groupoid).
A simplicial group is a simplicial set with the structure of a group on it. It turns out that this necessarily means that it is also a Kan complex. Therefore a simplicial group is
an ∞-groupoid with an extra group structure on it;
a model for a topological space with a group structure.
Accordingly (as discussed at group) a simplicial group $G$ gives rise to
a one-object $\infty$-groupoid $\mathbf{B} G$ whose explicit standard realization as a simplicial set is denoted $\bar W G$
an $\infty$-groupoid $\mathbf{E} G$ whose explicit standard realization as a simplicial set (even a simplicial group, again) is denoted $W G$
such that there is a fibration
which is the universal G-bundle.
Simplicial abelian groups are models for connective modules over the Eilenberg-Mac Lane spectrum $H \mathbf{Z}$; see Dold-Kan correspondence and stable Dold-Kan correspondence.
A simplicial group, $G$, is a simplicial object in the category Grp of groups.
The category of simplicial groups is the category of functors from $\Delta^{op}$ to Grp. It will be denoted $\Simp\Grp$.
The simplicial set underlying any simplicial group (by forgetting the group structure) is a Kan complex.
This is due to (Moore, 1954)
In fact, not only are the horn fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.
Let $G$ be a simplicial group.
Here is the explicit algorithm that computes the horn fillers:
Let $(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)$ give a horn in $G_{n-1}$, so the $y_i$s are $(n-1)$ simplices that fit together as if they were all but one, the $k^{th}$ one, of the faces of an $n$-simplex. There are three cases:
if $k = 0$:
if $0\lt k \lt n$:
if $k=n$:
The filler for any horn can be chosen to be a product of degenerate elements.
The simplicial homotopy groups of a simplicial group, $G$, can be calculated as the homology groups of the Moore complex of $G$. This is, in general, a non-Abelian chain complex.
A simplicial group can be considered as a simplicial groupoid having exactly one object. If $G$ is a simplicial group, the suggested notation for the corresponding simplicially enriched groupoid would be $\mathbf{B}G$ according to notational conventions suggested elsewhere in the nLab.
There is a functor due to Dwyer and Kan, called the Dwyer-Kan loop groupoid that takes a simplicial set to a simplicial groupoid. This has a left adjoint $\overline{W}$ (see below), called the classifying space functor, and together they give an equivalence of categories between the homotopy category of simplical sets and that of simplicial groupoids. We thus have that all homotopy types are modelled by simplicial groupoids … and for connected homotopy types by simplicial groups. One important fact to note in this equivalence is that it shifts dimension by 1, so if $G(K)$ is the simplicial group corresponding to the connected simplicial set $K$ then $\pi_k(K)$ is the same as $\pi_{k-1}(G(K))$. This is important when considering algebraic models for a homotopy n-type.
Let $G$ be a simplicial group and $G_0$ its group of 0-cells, regarded as a simplicially constant simplicial group. Write $G/G_0$ for the evident quotient of simplicial groups.
The evident morphisms
form a fiber sequence in sSet.
One checks that for $X$ any simplicial set and $G$ a simplicial group acting freely on it, the quotient map
is a Kan fibration. This is for instance (Weibel, exercise 8.2.6). By the disucssion at fiber sequence it is therefore sufficient to observe that
is an ordinary pullback of simplicial sets. This is clear since the action of $G_0$ on $G$ is degreewise free (being the action of a subgroup).
Let $(G_1 \stackrel{\delta}{\to} G_0)$ be a crossed module of groups, write
for groupoid which is the corresponding strict 2-group and write $N[G_1 \to G_0]$ for the nerve being the corresponding simplicial group. Then the above says that
is a fiber sequence of groupoids.
from simplicial abelian groups to the underlying simplicial sets has a left adjoint
from simplicial sets to abelian simplicial groups, the free simplicial abelian group functor that sends the set $X_n$ of $n$-simplices to the free abelian group $(\mathbb{Z}X)_n = \mathbb{Z} X_n$ over it.
This functor $\mathbb{Z}$ has the following properties:
it preserves weak equivalences
$\mathbb{Z} X$ is a cofibrant simplicial group
(…)
Let $sSet_0 \hookrightarrow$ sSet be the category of reduced simplicial sets (simplicial sets with a single 0-cell).
For $X \in sSet_0$ define $\Omega X \in sGrpd$ by
and
Simplicial groups are models for ∞-groups. This is exhibited by the model structure on simplicial groups. See also models for group objects in ∞Grpd.
Another equivalent model is that of connected Kan complexes.
At the abstract level of (∞,1)-category theory this equivalence is induced by forming loop space objects and delooping
This (∞,1)-equivalence is modeled by a Quillen equivalence of model categories whose right adjoint Quillen functor is the operation $\overline{W}$ discussed above.
This is for instance in GoerssJardine, chapter 5.
See also group object in an (∞,1)-category – models for groups in ∞Grpd.
The category $sAb$ of simplicial abelian groups is naturally a monoidal category, with the tensor product being degreewise that of abelian groups. This is indeed a closed monoidal category. For $A, B$ The internal hom $[A,B]$ is the simplicial abelian group whose underlying simplicial set is
where $\mathbf{Z}[-] : sSet \to sAb$ is degreewise the free abelian group functor.
For $G$ a simplicial group, we describe its delooping Kan complex $\mathbf{B}G \in sSet$ and the corresponding generalized universal bundle $\mathbf{E}G \to \mathbf{B}G$ such that the ordinary pullback
in sSet models the homotopy pullback in $sSet_{Quillen}$ / (∞,1)-pullback in ∞Grpd
in the standard model structure on simplicial sets and hence produces the principal ∞-bundle $P_\bullet \to X_\bullet$ classified by $X_\bullet \to \mathbf{B}G$.
For all these constructions exist very explicit combinatorial formulas that go by the symbols
$\overline{W}G$ for the delooping $\mathbf{B}G$
$W G$ for the generalized universal bundle $\mathbf{E}G$
$\tau : X_\bullet \to G_\bullet$ (called the twisting function) for the cocycle $X_\bullet \to \mathbf{B}G$;
$X_\bullet \times_g W G$ for $P_\bullet$ (called twisted Cartesian product ).
All of these constructions are functorial and hence lift from the context of simplicial sets to that of simplicial presheaves over some site $C$. There they provide models for strict group objects, delooping and principal ∞-bundles in the corresponding (∞,1)-toposes over $C$. In particular in the projective model structure on $[C^{op}, sSet]$ the pullback of the objectwise $W G \to \overline{W}G$ is still a homotopy pullback and models the corresponding principal $\infty$-bundles.
A simplicial group $G$ is a group object internal to the category of Kan complexes. Accordingly, there should be a Kan complex $\mathbf{B}G$ which is the delooping of $G$, i.e. a Kan complex with an essentially unique object, such that the loop space object of that Kan complex reproduces $G$.
An explicit construction of $\mathbf{B}G$ from $G$ goes traditionally by the symbol $\bar W G \in KanCplx$. Another one by $d B G$.
It is immediate to deloop the simplicial group $G$ to the simplicial groupoid that in degree $k$ is the 1-groupoid with a single object and $G_k$ as its collection of morphisms.
For $\mathcal{G}$ a simplicial groupoid that on objects is a constant simplicial set, define a simplicial set $\bar W \mathcal{G}$ as follows.
$(\overline{W}\mathcal{G})_0 := ob(\mathcal{G}_0)$, the set of objects of the groupoid of 0-simplices (and hence of the groupoid at each level);
$(\overline{W}\mathcal{G})_1 = Mor(\mathcal{G}_0)$, the collection of morphisms of the groupoid $\mathcal{G}_0$:
and for $n \geq 2$,
Here $s$ and $t$ are generic symbols for the domain and codomain mappings of all the groupoids involved. The face and degeneracy mappings between $\overline{W}(\mathcal{G})_1$ and $\overline{W}(\mathcal{G})_0$ are the source and target maps and the identity maps of $\mathcal{G}_0$, respectively; whilst the face and degeneracy maps at higher levels are given as follows:
The face and degeneracy maps are given by
$d_0(h_{n-1}, \ldots, h_0) = (h_{n-2}, \ldots, h_0)$;
for $0 \lt i\lt n$, $d_i(h_{n-1}, \ldots, h_0) = (d_{i-1}h_{n-1}, d_{i-2}h_{n-2}, \ldots, d_0h_{n-i}h_{n-i-1},h_{n-i-2}, \ldots , h_0)$;
and
whilst
and,
For $G$ a simplicial group and $\mathcal{G}$ the corresponding one-object simplicial groupoid, one writes
The above construction has a straightforward internalization to contexts other than Set. For instance if $G$ is a simplicial object in topological groups or in Lie groups, then $\overline{W}G$ with
is a simplicial object in this context (topological spaces, smooth manifolds, etc.)
In particular, if $C$ is a small category and $G : C^{op} \to sSet$ is a simplicial presheaf that is objectwise a simplicial group, then we have the simplicial presheaf
For $G$ a simplicial group, write $B G$ for the bisimplicial set obtained by taking degreewise the nerve of the delooping groupoid. Write $d B G \in$ sSet for its delooping.
This is shown for instance in (JardineLuo) and in (CegarraRemedios).
If $G$ is an ordinary group, regarded as a simplicially constant simplicial group, then $\overline{W}G$ is the usual bar complex of $G$:
For $X_\bullet$ a simplicial set a morphism
in sSet corresponds precisely to what is called a twisting function, a family of maps
satisfying the relations
Simplicial groups model all ∞-groups in ∞Grpd. Accordingly all principal ∞-bundles in ∞Grpd should be modeled by simplicial principal bundles.
(principal action)
Let $G$ be a simplicial group. For $P$ a Kan complex, an action of $G$ on $E$
is called principal if it is degreewise principal, i.e. if for all $n \in \mathbb{N}$ the only elements $g \in G_n$ that have any fixed point $e \in E_n$ in that $\rho(e,g) = e$ are the neutral elements.
The canonical action
of any simplicial group on itself is principal.
(simplicial principal bundle)
For $G$ a simplicial group, a morphism $P \to X$ of Kan complexes equipped with a $G$-action on $P$ is called a $G$-simplicial principal bundle if
the action is principal;
the base is isomorphic to the quotient $E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E})$ by the action:
A simplicial $G$-principal bundle $P \to X$ is necessarly a Kan fibration.
This appears as Lemma 18.2 in MaySimpOb.
For $G$ a simplicial group, define the simplicial set $W G$ to be the decalage of $\overline{W}G$
By the discussion at homotopy pullback this means that for $X_\bullet$ any Kan complex, an ordinary pullback diagram
in sSet exhibits $P_\bullet$ as the homotopy pullback in $sSet_{Quillen}$ / (∞,1)-pullback in ∞Grpd
i.e. as the homotopy fiber of the cocycle $g$.
We call $P_\bullet := X_\bullet \times^g W G$ the simplicial $G$-principal bundle corresponding to $g$.
Let $\{\phi : X_n \to G_{(n-1)}\}$ be the twisting function corresponding to $g : X_\bullet \to \overline{W}G$ by the above discussion.
Then the simplicial set $P_\bullet := X_\bullet \times_{g} W G$ is explicitly given by the formula called the twisted Cartesian product $X_\bullet \times^\phi G_\bullet$:
its cells are
with face and degeneracy maps
$d_i (x,g) = (d_i x , d_i g)$ if $i \gt 0$
$d_0 (x,g) = (d_0 x, \phi(x) d_0 g)$
$s_i (x,g) = (s_i x, s_i g)$.
Here are some pointers on where precisely in the literature the above statements can be found.
One useful reference is
There the abbreviation PCTP ( principal twisted cartesian product ) is used for what above we called just twisted Cartesian products.
The fact that every PTCP $X \times_\phi G \to X$ defined by a twisting function $\phi$ arises as the pullback of $W G \to \overline{W}G$ along a morphjism of simplicial sets $X \to \overline{W}G$ can be found there by combining
the last sentence on p. 81 which asserts that pullbacks of PTCPs $X \times_\phi G \to X$ along morphisms of simplicial sets $f : Y \to X$ yield PTCPs corresponding to the composite of $f$ with $\phi$;
section 21 which establishes that $W G \to \bar W G$ is the PTCP for some universal twisting function $r(G)$.
lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y \to \bar W G$ with the universal twisting function $r(G)$. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.
An explicit version of the statement that twisted Cartesian products are nothing but pullbacks of a generalized universal bundle is on page 148 of
On page 239 there it is mentioned that
is a model for the loop space object fiber sequence
One place in the literature where the observation that $W G$ is the decalage of $\overline{W}G$ is mentioned fairly explicitly is page 85 of
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex
A standard reference for the case of abelian simplicial groups is chapter 5 of
Also chapter IV of
and chapter 8 of
The algorithm for finding the horn fillers in a simplicial group is given in the proof of theorem 17.1, page 67 there.
This proof that simplicial groups are Kan complexes is originally due to theorem 3.4 in
which apears in more detail as theorem 3 on p. 18-04 of
and is often attributed to
In fact, it seems that this is the origin of the very notion of Kan complex.
A proof is also on p. 14 of
Section 1.3.3 of
discusses simplicial groups in the context of nonabelian algebraic topology.