This entry contains one chapter of geometry of physics.
previous chapters: homotopy types, smooth homotopy types, geometry of physics -- stable homotopy types
next chapters, principal bundles, representations and associated bundles
The modern mathematical terminology group is short for group of symmetries (e.g. Klein 1872). Mathematicians and physicist tend to marvel at the ubiquity and profoundness that the concept of symmetry groups has turned out to exhibit since its conception in the 19th century. Indeed it is a fundamental concept in a sense whose full depth becomes clear (only) in homotopy theory: groups are equivalently the pointed connected homotopy types.
In traditional literature this fact is fully appreciated typically only in rather advanced corners of algebraic topology, and even that mostly just somewhat secretly. But while it is true that this fact has very sophisticated consequences, at its heart it is a simple fundamental fact that is visible and useful already in elementary group theory and representation theory.
We being in
with a discussion of the simplest case of ordinary discrete groups from this natural perspective of regarding them as pointed connected homotopy 1-types, their delooping groupoids. Then we gradually generalize to the study of infinity-group objects in general (infinity,1)-toposes.
We discuss how, via looping and delooping, discrete groups are equivalent to pointed connected groupoids.
Write
Grpd for the (2,1)-category of groupoids (objects are groupoids, 1-morphisms are functors between these and 2-morphisms are natural transformations between those, which are nessecarily natural isomorphisms),
Grp for the 1-category of groups (discrete groups), also regarded as a (2,1)-category;
$Grpd^{\ast/}$ for the $(2,1)$-category of pointed objects in Grpd,
$Grpd_{\geq 1} \hookrightarrow Grpd$ for the full sub-(2,1)-category on connected groupoids, those for which $\pi_0 \simeq \ast$;
$Grp^{\ast/}_{\geq 1}$ for the pointed objects in connected groupoids.
$\pi_1(X,x) \in Grp$ for the fundamental group of a pointed groupoid $(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}$ at the given basepoint.
$\mathbf{B}G \in Grpd$, given a group $G$, for the groupoid $(G\stackrel{\longrightarrow}{\longrightarrow} \ast)$, with composition given by the product in the group. There are two possible choices of conventions, we agree that
The (2,1)-category $Grp_{\geq 1}$ of connected groupoids is equivalent to its full sub-(2,1)-category on those objects of the form $\mathbf{B}G$, for $G$ a group.
Given a connected groupoid $X$, pick any basepoint $x\in X$ and consider the canonical inclusion $\mathbf{B}\pi_1(X,x) \longrightarrow X$. By construction this is fully faithful and by assumption of connectedness it is essentially surjective, hence it is an equivalence of groupoids.
The hom-groupoids between connected groupoids with fundamental groups $G$ and $H$, respectively, are equivalent to the action groupoids of the set of group homomorphisms $G \to H$ acted on by conjugation with elements of $H$:
Given two group homomorphisms $\phi_1, \phi_2 \colon G \longrightarrow H$ then an isomorphism between them in this hom-groupoid is an element $h \in H$ such that
By direct inspection of the naturality square for the natural transformations which are the morphisms in $Grpd(\mathbf{B}G, \mathbf{B}H)$:
The operation of forming $\pi_1$ is equivalently the operation of forming the homotopy fiber product of the point inclusion with itself, and hence extends to a (2,1)-functor
Restricted to connected groupoids among the pointed groupoids, the functor $\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp$ of remark 1 is an equivalence of (2,1)-categories.
It is clear that the functor is essentially surjective: for $G$ any group then $\pi_1(\mathbf{B}G,\ast) \simeq G$.
The more interesting point to notice is that $\pi_1$ is indeed a fully faithful (2,1)-functor, in that for any $(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}$ then the functor
is an equivalence of hom-groupoids. By prop. 1 it is sufficient to check this for $X = \mathbf{B}G$ and $Y = \mathbf{B}H$ with their canonical basepoints, hence to check that for any two groups $G,H$ the functor
is an equivalence.
To see this, observe that, by definition of pointed objects via the undercategory under the point, a morphism in $Grpd^{\ast/}$ between groupoids of this form $\mathbf{B}(-)$ is a diagram in $Grp$ (unpointed) of the form
where the natural isomorphism is equivalently just the choice of an element $h \in H$. Hence these morphisms are pairs $(\phi,h)$ of a group homomorphism and an element of the domain.
We claim that the (2,1)-functor $\pi_1$ takes such $(\phi,h)$ to the homomorphism $Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H$. To see this, consider via remark 1 this functor as forming loops:
This shows that on a morphism as above this acts by forming the pasting
Unwinding the whiskering of natural transformations here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.
One observes now that these extra labels $h$ are precisely the information that “trivializes” the conjugation action which in prop. 2 prevents the bare set of group homomorphism: a 2-morphism $(\phi_1, h_1) \Rightarrow (\phi_2,h_2)$ in $Grp^{\ast/}$ is a natural isomorphism of groupoids
(encoding a conjugation relation $\phi_2 = Ad_{h} \circ \phi_1$ as above) such that we have the pasting relation
But this says in components that $h_2 = h_1\cdot h$. Hence there is a at most one morphism in $Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))$ from $(\phi_1,h_1)$ to $(\phi_2,h_2)$: it exists if $\phi_2 = Ad_h \circ \phi_1$ and $h_2 = h_1\cdot h$.
But since, by the previous argument, the functor $\pi_1$ takes $(\phi_1,h_1)$ to $Ad_{h_1^{-1}} \circ \phi_1$, this means that such a morphism exists precisely if both $(\phi_1,h_1)$ and $(\phi_2,h_2)$ are taken to the same group homomorphism by $\pi_1$
This establishes that $\pi_1$ is alspo an equivalence on all hom-groupoids.
This proof also shows that $\mathbf{B}(-)$ is in fact the inverse equivalence:
There is an equivalence of (2,1)-categories between pointed connected groupoids and plain groups
given by forming loop space objects and by forming deloopings.
With the above perspective on ordinary groups, it is now essentially clear what higher groups in homotopy theory are to be. We discuss this in full generality below, but of course it serves to highlight the first higher case, that of 2-groups.
The idea is clear: where a group is equivalently a pointed connected homotopy 1-type/groupoid in that it is the loop space object of such a pointed connected type, a 2-group is equivalently a pointed connected homotopy 2-type/2-groupoid in that it is its loop space object.
In general this means that a 2-group is a groupoid that is equipped with the structure of a group for which the usual axioms (associativity, inverses) hold (only) up to coherent homotopy. One hence speaks of weak 2-groups.
But it turns out that in this low degree there is not too much space for such weakening to happen. Indeed, every 2-groupoid/homotopy 2-type has a model by a strict 2-groupoid, see also at homotopy hypothesis – for homotopy 2-types. Accordingly, every (discrete) 2-group is equivalent to one which is a groupoid equipped with strict group structure, where the axioms hold “on the nose”. The algebraic data encoded by such is known as a crossed module of groups.
At geometry of physics -- homotopy types is discussed how the Dold-Kan correspondence allows to think of chain complexes in non-negative degree as homotopy types, namely as Kan complexes underlying simplicial abelian groups which are equivalent to the chain complexes, via the normalized chain complex operation, see the section geometry of physics – homotopy types – Dold-Kan correspondence.
But by the discussion above of simplicial groups, this means that actually the Dold-Kan correspondence identifies chain complexes with a certain class of abelian ∞-groups.
(…)
While, under the Dold-Kan correspondences?, chain complexes are abelian ∞-groups, they do not exhaust the space of all objects that deserve to be called such. The general concept of abelian ∞-groups is the concept called spectrum in algebraic topology, stable homotopy types.
One of the remarkable conceptual simplifications brought about by general homotopy theory pertains to the general concept of cohomology: effectively every flavor of cohomology that has been considered turns out to be nothing but the theory of (infinity,1)-categorical hom spaces in a suitable (infinity,1)-topos.
Specifically for the case of group cohomology, this is the following simple statement.
Let $A$ be an abelian group. For $n \in \mathbb{N}$ write
for the Kan complex underlying the simplicial group which is the image of the chain complex $A[-1]$ concentrated on $A$ in degree $n$ under the Dold-Kan correspondence $DK \colon Ch_{\bullet \geq 0}(Ab)\stackrel{\simeq}{\longrightarrow} sAb \stackrel{forget}{\longrightarrow} KanCplx$
For $G$ any discrete group, not necessarily abelian, write
for the nerve of the groupoid (G\stackrel{\longrightarrow}{\longrightarrow} \ast).
See at geometry of physics -- homotopy types for detailed discussion of these two constructions.
Given a discrete group $G$, then a degree-$n$ cocycle in group cohomology of $G$ with coefficients in $A$ is equivalently a morphism
A homotopy between two such morphisms is equivalently a coboundary between two such cocycles.
The group cohomology group of $G$ with coefficients in $A$ is the equivalence classes of cocycles modulo coboundaries, hence is the connected components of the hom-groupoid:
It is instructive to spell this out in low degree.
Let $G$ be a discrete group and $A$ an abelian discrete group, regarded as being equipped with the trivial $G$-action.
Then a group 2-cocycle on $G$ with coefficients in $A$ is a function
such that for all $(g_1, g_2, g_3) \in G \times G \times G$ it satisfies the equation
(called the group 2-cocycle condition).
For $c, \tilde c$ two such cocycles, a coboundary $h \colon c \to \tilde c$ between them is a function
such that for all $(g_1,g_2) \in G \times G$ the equation
holds in $A$, where
is the group 2-coboundary encoded by $h$.
The degree-2 group cohomology is the set
of equivalence classes of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an abelian group under pointwise addition of cocycles in $A$
where
The 2-simplices in $(\mathb{B}G)_\bullet$ are
and the 3-simplices are
The 2-simplices in $(\mathb{B}^2 A)_\bullet$ are
and the 3-simplices are
Therefore a homomorphism of Kan complexes/simplicial sets $c \colon (\mathbf{B}G)_\bullet \to (\mathbf{B}^2 A)_\bullet$ is in degree 2 a function
i.e. a map $c \;\colon\; G \times G \to K$. To be a simplicial homomorphism this has to extend to 3-simplices as:
Hence this is the cocycle condition.
A similar argument gives the coboundaries.
We discuss now how in the computation of $H^2_{Grp}(G,A)$ one may concentrate on the normalized cocycles.
A group 2-cocycle $c \colon G \times G \to A$, def. 5 is called normalized if
For $c \colon G \times G \to A$ a group 2-cocycle, we have for all $g \in G$ that
The cocycle condition (1) evaluated on
says that
hence that
Similarly the 2-cocycle condition applied to
says that
hence that
Every group 2-cocycle $c \colon G \times G \to A$ is cohomologous to a normalized one, def. 1.
By lemma 1 it is sufficient to show that $c$ is cohomologous to a cocycle $\tilde c$ satisfying $\tilde c(e,e) = e$. Now given $c$, Let $h \colon G \to A$ be given by
Then $\tilde c \coloneqq c + d c$ has the desired property, with (2):
Above in corollary 1 we had seen that ordinary groups $G$ are equivalent to pointed connected homotopy 1-types, their deloopings $\ast \to \mathbf{B}G$.
This statement has an immediate generalization to any (∞,1)-topos $\mathbf{H}$: while one may define ∞-group algebraically as follows, it is most convenient to define them via looping, as below.
Given an (∞,1)-topos $\mathbf{H}$, then group object in $\mathbf{H}$ (an ∞-group) is an object $G \in \mathbf{H}$ equipped with the structure of an A-∞ algebra (i.e. a product operation which satisfies associativity up to higher coherent homotopy), such that the 0-truncation $\tau_0 G$ is an ordinary group object.
Given any object $X$ with a base point $x \colon \ast \to X$, then the loop space object $\Omega_x X$ canonically has the structure of an ∞-group, def. 2, where the product operation is given by concatenation of loops.
The looping operation of example 1 constitutes an equivalence of (∞,1)-categories
between pointed connected objects in $\mathbf{H}$ and ∞-group objects in $\mathbf{H}$.
The inverse equivalence $\mathbf{B}$ we call the delooping operation.
For $\mathbf{H} =$ ∞Grpd this is the May recognition theorem. For general $\mathbf{H}$ this is Lurie, "Higher Algebra", theorem 5.1.3.6.
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