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geometry of physics -- groups

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


Contents

Groups

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.

Model Layer

1-Groups

Discrete groups as pointed connected groupoids

We discuss how, via looping and delooping, discrete groups are equivalent to pointed connected groupoids.

Write

Proposition

The (2,1)-category Grp 1Grp_{\geq 1} of connected groupoids is equivalent to its full sub-(2,1)-category on those objects of the form BG\mathbf{B}G, for GG a group.

Proof

Given a connected groupoid XX, pick any basepoint xXx\in X and consider the canonical inclusion Bπ 1(X,x)X\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.

Proposition

The hom-groupoids between connected groupoids with fundamental groups GG and HH, respectively, are equivalent to the action groupoids of the set of group homomorphisms GHG \to H acted on by conjugation with elements of HH:

Grpd(BG,BH)Grp(G,H)// adH Grpd(\mathbf{B}G, \mathbf{B}H) \simeq Grp(G,H)//_{ad}H

Given two group homomorphisms ϕ 1,ϕ 2:GH\phi_1, \phi_2 \colon G \longrightarrow H then an isomorphism between them in this hom-groupoid is an element hHh \in H such that

ϕ 2=Ad hϕ 1h 1ϕ 1()h. \phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h \,.
Proof

By direct inspection of the naturality square for the natural transformations which are the morphisms in Grpd(BG,BH)Grpd(\mathbf{B}G, \mathbf{B}H):

* * h * g 1 ϕ 1(g 1) ϕ 2(g 1) * * h * g 2 ϕ 1(g 2) ϕ 2(g 2) * * h *. \array{ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} && \downarrow^{\mathrlap{\phi_2(g_1)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_2}} && && \downarrow^{\mathrlap{\phi_1(g_2)}} && \downarrow^{\mathrlap{\phi_2(g_2)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast } \,.
Remark

The operation of forming π 1\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

π 1:Grpd */Grp. \pi_1 \colon Grpd^{\ast/} \longrightarrow Grp \,.
Proposition

Restricted to connected groupoids among the pointed groupoids, the functor π 1:Grpd 1 */Grp\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp of remark 1 is an equivalence of (2,1)-categories.

Proof

It is clear that the functor is essentially surjective: for GG any group then π 1(BG,*)G\pi_1(\mathbf{B}G,\ast) \simeq G.

The more interesting point to notice is that π 1\pi_1 is indeed a fully faithful (2,1)-functor, in that for any (X,x),(Y,y)Grpd 1 */(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1} then the functor

(π 1) X,Y:Grpd */((X,y),(Y,y))Grp(π 1(X,x),π 1(Y,y)) (\pi_1)_{X,Y} \colon Grpd^{\ast/}((X,y),(Y,y)) \longrightarrow Grp(\pi_1(X,x), \pi_1(Y,y))

is an equivalence of hom-groupoids. By prop. 1 it is sufficient to check this for X=BGX = \mathbf{B}G and Y=BHY = \mathbf{B}H with their canonical basepoints, hence to check that for any two groups G,HG,H the functor

(π 1) X,Y:Grpd */((BG,*),(BH,*))Grp(G,H) (\pi_1)_{X,Y} \;\colon\; Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast)) \longrightarrow Grp(G,H)

is an equivalence.

To see this, observe that, by definition of pointed objects via the undercategory under the point, a morphism in Grpd */Grpd^{\ast/} between groupoids of this form B()\mathbf{B}(-) is a diagram in GrpGrp (unpointed) of the form

* h BG Bϕ BH \array{ && \ast \\ & \swarrow &\swArrow_{h}& \searrow \\ \mathbf{B}G && \underset{\mathbf{B}\phi}{\longrightarrow} && \mathbf{B}H }

where the natural isomorphism is equivalently just the choice of an element hHh \in H. Hence these morphisms are pairs (ϕ,h)(\phi,h) of a group homomorphism and an element of the domain.

We claim that the (2,1)-functor π 1\pi_1 takes such (ϕ,h)(\phi,h) to the homomorphism Ad h 1ϕ:GHAd_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H. To see this, consider via remark 1 this functor as forming loops:

π 1(BG,*)={ * * g * BG} gG. \pi_1(\mathbf{B}G,\ast) = \left\{ \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right\}_{g\in G} \,.

This shows that on a morphism as above this acts by forming the pasting

* * g * h BG ϕ BH= * * hϕ(g)h 1 * h BG ϕ BH. \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow \\ && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \;\;\;\; = \:\;\;\; \array{ && && \ast \\ && & \swarrow && \searrow \\ && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast \\ & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow \\ \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \,.

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 hh are precisely the information that “trivializes” the conjugation action which in prop. 2 prevents the bare set of group homomorphism: a 2-morphism (ϕ 1,h 1)(ϕ 2,h 2)(\phi_1, h_1) \Rightarrow (\phi_2,h_2) in Grp */Grp^{\ast/} is a natural isomorphism of groupoids

BG ϕ 1 BH id h id BG ϕ 2 BH \array{ \mathbf{B}G &\stackrel{\phi_1}{\longrightarrow}& \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &\underset{\phi_2}{\longrightarrow}& \mathbf{B}H }

(encoding a conjugation relation ϕ 2=Ad hϕ 1\phi_2 = Ad_{h} \circ \phi_1 as above) such that we have the pasting relation

* h 1 BG ϕ 1 BH id h id BG ϕ 2 BH= * h 2 BG ϕ 2 BH. \array{ && \ast \\ & \swarrow &\swArrow_{h_1}& \searrow \\ \mathbf{B}G && \stackrel{\phi_1}{\longrightarrow} && \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &&\underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \;\;\;\;\; = \;\;\;\;\; \array{ && \ast \\ & \swarrow &\swArrow_{h_2}& \searrow \\ \mathbf{B}G && \underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \,.

But this says in components that h 2=h 1hh_2 = h_1\cdot h. Hence there is a at most one morphism in Grpd */((BG,*),(BH,*))Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast)) from (ϕ 1,h 1)(\phi_1,h_1) to (ϕ 2,h 2)(\phi_2,h_2): it exists if ϕ 2=Ad hϕ 1\phi_2 = Ad_h \circ \phi_1 and h 2=h 1hh_2 = h_1\cdot h.

But since, by the previous argument, the functor π 1\pi_1 takes (ϕ 1,h 1)(\phi_1,h_1) to Ad h 1 1ϕ 1Ad_{h_1^{-1}} \circ \phi_1, this means that such a morphism exists precisely if both (ϕ 1,h 1)(\phi_1,h_1) and (ϕ 2,h 2)(\phi_2,h_2) are taken to the same group homomorphism by π 1\pi_1

Ad h 2 1ϕ 2=Ad h 1h 1 1ϕ 2=Ad h 1 1ϕ 1. Ad_{h_2^{-1}} \circ \phi_2 = Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2 = Ad_{h_1^{-1}} \circ \phi_1 \,.

This establishes that π 1\pi_1 is alspo an equivalence on all hom-groupoids.

This proof also shows that B()\mathbf{B}(-) is in fact the inverse equivalence:

Corollary

There is an equivalence of (2,1)-categories between pointed connected groupoids and plain groups

GrpBπ 1=Ω *Grpd 1 */ Grp \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} Grpd^{\ast/}_{\geq 1}

given by forming loop space objects and by forming deloopings.

Topological and Lie groups

2-Groups

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.

Discrete
Braided and abelian

Simplicial groups

Discrete
Abelian
Topological and Lie simplicial groups

Chain complexes and Spectra

Chain complexes and the Dold-Kan correspondence

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.

(…)

Spectra and the stable Dold-Kan correspondence

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.

Group cohomology

Discrete group cohomology

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 AA be an abelian group. For nn \in \mathbb{N} write

(B nA) KanCplx (\mathbf{B}^n A)_\bullet \in KanCplx

for the Kan complex underlying the simplicial group which is the image of the chain complex A[1]A[-1] concentrated on AA in degree nn under the Dold-Kan correspondence DK:Ch 0(Ab)sAbforgetKanCplxDK \colon Ch_{\bullet \geq 0}(Ab)\stackrel{\simeq}{\longrightarrow} sAb \stackrel{forget}{\longrightarrow} KanCplx

(B nA) DK(A[n]). (\mathbf{B}^n A)_\bullet \coloneqq DK(A[-n]) \,.

For GG any discrete group, not necessarily abelian, write

(BG) KanCplx (\mathbf{B}G)_\bullet \in KanCplx

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.

Proposition/Definition

Given a discrete group GG, then a degree-nn cocycle in group cohomology of GG with coefficients in AA is equivalently a morphism

(BG) (B nA) . (\mathbf{B}G)_\bullet \longrightarrow (\mathbf{B}^n A)_\bullet \,.

A homotopy between two such morphisms is equivalently a coboundary between two such cocycles.

The group cohomology group of GG with coefficients in AA is the equivalence classes of cocycles modulo coboundaries, hence is the connected components of the hom-groupoid:

H Grp n(G,A)π 0Hom(BG,B nA). H^n_{Grp}(G,A) \simeq \pi_0 Hom(\mathbf{B}G, \mathbf{B}^n A) \,.

It is instructive to spell this out in low degree.

Proposition

Let GG be a discrete group and AA an abelian discrete group, regarded as being equipped with the trivial GG-action.

Then a group 2-cocycle on GG with coefficients in AA is a function

c:G×GA c \colon G \times G \to A

such that for all (g 1,g 2,g 3)G×G×G(g_1, g_2, g_3) \in G \times G \times G it satisfies the equation

(1)c(g 1,g 2)c(g 1,g 2g 3)+c(g 1g 2,g 3)c(g 2,g 3)=0A c(g_1, g_2) - c(g_1, g_2 \cdot g_3) + c(g_1 \cdot g_2, g_3) - c(g_2, g_3) = 0 \;\;\;\; \in A

(called the group 2-cocycle condition).

For c,c˜c, \tilde c two such cocycles, a coboundary h:cc˜h \colon c \to \tilde c between them is a function

h:GA h \colon G \to A

such that for all (g 1,g 2)G×G(g_1,g_2) \in G \times G the equation

(2)c˜(g 1,g 2)=c(g 1,g 2)+(dh)(g 1,g 2) \tilde c(g_1,g_2) = c(g_1,g_2) + (d h)(g_1,g_2)

holds in AA, where

(dh)(g 1,g 2)h(g 1g 2)h(g 1)h(g 2) (d h)(g_1, g_2) \coloneqq h(g_1 g_2) - h(g_1) - h(g_2)

is the group 2-coboundary encoded by hh.

The degree-2 group cohomology is the set

H Grp 2(G,A)=2Cocycles(G,A)/Coboundaries(G,A) H^2_{Grp}(G,A) = 2Cocycles(G,A) / Coboundaries(G,A)

of equivalence classes of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an abelian group under pointwise addition of cocycles in AA

[c 1]+[c 2]=[c 1+c 2] [c_1] + [c_2] = [c_1 + c_2]

where

c 1+c 2:(g 1,g 2)c 1(g 1,g 2)+c 2(g 1,g 2). c_1 + c_2 \colon (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2) \,.
Proof

The 2-simplices in (mathbBG) (\mathb{B}G)_\bullet are

(BG) 2={ * g 1 g 2 * g 1g 2 *|g 1,g 2G}, (\mathbf{B}G)_2 = \left\{ \left. \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_1 g_2}{\to}&& {*} } \right| g_1, g_2 \in G \right\} \,,

and the 3-simplices are

(BG) 3={* g 2 * g 1 g 1g 2 g 3 * g 1g 2g 3 ** g 2 * mathrlapg 1 g 2g 3 g 3 * g 1g 2g 3 *|g 1,g 2,g 3G}. (\mathbf{B}G)_3 = \left\{ \left. \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{\mathrlap{g_1}} &&{}^{g_1 g_2}\nearrow&& \downarrow^{\mathrlap{g_3}} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{\,mathrlap{g_1}} &&\searrow^{g_2 g_3}&& \downarrow^{\mathrlap{g_3}} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& {*} } \right| g_1, g_2, g_3 \in G \right\} \,.

The 2-simplices in (mathbB 2A) (\mathb{B}^2 A)_\bullet are

(B 2A) 2={ * c * *|cA}, (\mathbf{B}^2 A)_2 = \left\{ \left. \array{ && {*} \\ & {}^{}\nearrow &\Downarrow^{\mathrlap{c}}& \searrow^{} \\ {*} &&\stackrel{}{\to}&& {*} } \right| c\in A \right\} \,,

and the 3-simplices are

(B 2A) 3={* * c 1 c 2 * ** * c 3 c 4 * *|c 1,c 2,c 3A;c 4=c 3c 1c 2}. (\mathbf{B}^2 A)_3 = \left\{ \left. \array{ {*} &&\stackrel{}{\to}&& {*} \\ \uparrow^{\mathrlap{}} &\swArrow_{c_1}&{}^{}\nearrow&\swArrow_{c_2}& \downarrow^{\mathrlap{}} \\ {*} &&\stackrel{}{\to}&& {*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{}{\to}&& {*} \\ \uparrow^{\mathrlap{}} &\swArrow_{c_3}&\searrow^{}&\swArrow_{c_4}& \downarrow^{\mathrlap{}} \\ {*} &&\stackrel{}{\to}&& {*} } \right| c_1, c_2, c_3 \in A; c_4 = c^3 - c_1 - c_2 \right\} \,.

Therefore a homomorphism of Kan complexes/simplicial sets c:(BG) (B 2A) c \colon (\mathbf{B}G)_\bullet \to (\mathbf{B}^2 A)_\bullet is in degree 2 a function

c 2:( * g 1 g 2 * g 2g 1 *)( * * c(g 1,g 2) * * * *) c_2 \;\; : \;\; \left( \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_2 g_1}{\to}&& {*} } \right) \;\;\; \mapsto \;\;\; \left( \array{ && {*} \\ & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&& {*} } \right)

i.e. a map c:G×GKc \;\colon\; G \times G \to K. To be a simplicial homomorphism this has to extend to 3-simplices as:

c 3 :(* g 2 * g 1 g 2g 1 g 3 * g 3g 2g 1 ** g 2 * g 1 g 3g 2 g 3 * g 3g 2g 1 *) (* * c(g 1,g 2) c(g 2,g 3) * ** * c(g 1,g 2g 3) c(g 2,g 3) * *). \begin{aligned} c_3 \;\;\; &: \;\;\; \left( \array{ {*} && \stackrel{g_2}{\longrightarrow} && {*} \\ \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3} \\ {*} && \stackrel{g_3 g_2 g_1}{\longrightarrow} && {*} } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&&{*} } \right) \\ & \mapsto \left( \array{ {*} &\stackrel{}{\longrightarrow} & &\stackrel{}{\longrightarrow}& {*} \\ \uparrow &\Downarrow^{c(g_1,g_2)} &\nearrow&\Downarrow^{c(g_2,g_3)}& \downarrow \\ {*} &\longrightarrow&&\longrightarrow&{*} } \;\;\;\; \stackrel{}{\Rightarrow} \;\;\;\; \array{ {*} &\longrightarrow&&\longrightarrow& {*} \\ \uparrow &\Downarrow^{c(g_1,g_2 g_3)} &\searrow &\Downarrow^{c(g_2, g_3)}& \downarrow \\ {*} &\longrightarrow&&\longrightarrow&{*} } \right) \end{aligned} \,.

Hence this is the cocycle condition.

A similar argument gives the coboundaries.

We discuss now how in the computation of H Grp 2(G,A)H^2_{Grp}(G,A) one may concentrate on the normalized cocycles.

Definition

A group 2-cocycle c:G×GAc \colon G \times G \to A, def. 5 is called normalized if

g 0,g 1G(g 0=eorg 1=e)(c(g 0,g 1)=e). \forall_{g_0,g_1 \in G} \;\; \left(g_0 = e \;or\; g_1 = e \right) \Rightarrow \left( c(g_0,g_1) = e \right) \,.
Lemma

For c:G×GAc \colon G \times G \to A a group 2-cocycle, we have for all gGg \in G that

c(e,g)=c(e,e)=c(g,e). c(e,g) = c(e,e) = c(g,e) \,.
Proof

The cocycle condition (1) evaluated on

(g 1,g,e)G 3 (g^{-1}, g, e) \in G^3

says that

c(g 1,g)+c(e,e)=c(g,e)+c(g 1,g) c(g^{-1}, g) + c(e, e) = c(g, e) + c(g^{-1}, g )

hence that

c(e,e)=c(g,e). c(e,e) = c(g, e) \,.

Similarly the 2-cocycle condition applied to

(e,g,g 1)G 3 (e, g, g^{-1}) \in G^3

says that

c(e,g)+c(g,g 1)=c(g,g 1)+c(e,e) c(e,g) + c(g,g^{-1}) = c(g,g^{-1}) + c(e,e)

hence that

c(e,g)=c(e,e). c(e,g) = c(e,e) \,.
Proposition

Every group 2-cocycle c:G×GAc \colon G \times G \to A is cohomologous to a normalized one, def. 1.

Proof

By lemma 1 it is sufficient to show that cc is cohomologous to a cocycle c˜\tilde c satisfying c˜(e,e)=e\tilde c(e,e) = e. Now given cc, Let h:GAh \colon G \to A be given by

h(g)c(g,g). h(g) \coloneqq c(g,g) \,.

Then c˜c+dc\tilde c \coloneqq c + d c has the desired property, with (2):

c˜(e,e) (c+dh)(e,e) =c(e,e)+c(ee,ee)c(e,e)c(e,e) =0. \begin{aligned} \tilde c(e,e) & \coloneqq (c + d h)(e,e) \\ & = c(e,e) + c(e \cdot e, e \cdot e) - c(e,e) - c(e,e) \\ & = 0 \end{aligned} \,.

Semantic Layer

Above in corollary 1 we had seen that ordinary groups GG are equivalent to pointed connected homotopy 1-types, their deloopings *BG\ast \to \mathbf{B}G.

This statement has an immediate generalization to any (∞,1)-topos H\mathbf{H}: while one may define ∞-group algebraically as follows, it is most convenient to define them via looping, as below.

Definition

Given an (∞,1)-topos H\mathbf{H}, then group object in H\mathbf{H} (an ∞-group) is an object GHG \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 τ 0G\tau_0 G is an ordinary group object.

Example

Given any object XX with a base point x:*Xx \colon \ast \to X, then the loop space object Ω xX\Omega_x X canonically has the structure of an ∞-group, def. 2, where the product operation is given by concatenation of loops.

Proposition

The looping operation of example 1 constitutes an equivalence of (∞,1)-categories

Grp(H)BΩH 1 */ Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\longleftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\longrightarrow}} \mathbf{H}_{\geq 1}^{\ast /}

between pointed connected objects in H\mathbf{H} and ∞-group objects in H\mathbf{H}.

The inverse equivalence B\mathbf{B} we call the delooping operation.

For H=\mathbf{H} = ∞Grpd this is the May recognition theorem. For general H\mathbf{H} this is Lurie, "Higher Algebra", theorem 5.1.3.6.

Syntactic Layer

Pointed connected types

Identity types of connected types

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Revised on February 1, 2016 12:31:17 by Urs Schreiber (86.187.73.110)