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geometry of physics -- principal bundles

This entry contains one chapter of the material at geometry of physics.

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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Contents

Principal bundles

Model Layer

Smooth principal bundles via Smooth groupoids

For GG a Lie group, we discuss GG-principal bundles over a smooth manifold XX as a natural construction in the context of smooth groupoids.

Cech 1-cocycles

Recall the discussion of Cech cohomology in degree 1 from geometry of physics -- smooth homotopy typesPre-smooth groupoids

Definition

Let GG be a Lie group. Write (BG) LieGrpdPreSmoothGrpd(\mathbf{B}G)_\bullet \in LieGrpd \hookrightarrow PreSmoothGrpd for the Lie groupoid

(BG) =(G) (\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow} \bullet)

with composition induced from the product in GG.

A useful schematic picture this groupoid is

(BG) ={ g 1 = g 2 g 2g 1 } (\mathbf{B}G)_\bullet = \left\{ \array{ && \bullet \\ & {}^{\mathllap{g_1}}\nearrow &=& \searrow^{\mathrlap{g_2}} \\ \bullet &&\stackrel{g_2 \cdot g_1 }{\longrightarrow}&& \bullet } \right\}

where the g iGg_i \in G are elements in the group, and the bottom morphism is labeled by forming the product in the group. (The order of the factors here is a convention whose choice, once and for all, does not matter up to equivalence.)

Remark

The nerve of (BG) (\mathbf{B}G)_\bullet, def. 1, is a simplicial object of the form

N(BG) k=G × k N(\mathbf{B}G)_k = G^{\times_k}

with face maps of the form

N(BG) =(G×G×GG×GG) N(\mathbf{B}G)_\bullet = \left( \cdots G \times G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} G \stackrel{\longrightarrow}{\longrightarrow} \bullet \right)

where the outer face maps forget the corresponding outer copy in the Cartesian product of groups, and the inner face maps are given by group multiplication in two consecutive copies.

Definition

For XX a smooth manifold, we may regard it as a Lie groupoid with only identity morphisms. Its schematic depiction is simply

X={xidx}. X = \{x \stackrel{id}{\longrightarrow} x \} \,.
Definition

Let {U iX} iI\{U_i \to X\}_{i \in I} be an open cover of a smooth manifold XX. The corresponding Cech groupoid C({U i}) C(\{U_i\})_\bullet is

C({U i}) =( i,jU i×XU jp 2p 1 iU i) C(\{U_i\})_\bullet = \left( \coprod_{i,j} U_i \underset{X}{\times} U_j \stackrel{\overset{p_1}{\longrightarrow}}{\underset{p_2}{\longrightarrow}} \coprod_i U_i \right)

with the uniquely defined composition. The schematic depiction is

C({U i}) ={ (x,j) = (x,i) (x,k)}. C(\{U_i\})_\bullet = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) && \longrightarrow&& (x,k) } \right\} \,.

This indicates that the objects of this groupoid are pairs (x,i)(x,i) consisting of a point xXx \in X and a patch U iXU_i \subset X that contains xx, and a morphism is a triple (x,i,j)(x,i,j) consisting of a point and two patches, that both contain the point, in that xU iU jx \in U_i \cap U_j. The triangle in the above cartoon symbolizes the evident way in which these morphisms compose. All this inherits a smooth structure from the fact that the U iU_i are smooth manifolds and the inclusions U iXU_i \to X are smooth functions. hence also C(U)C(U) becomes a Lie groupoid.

Remark

Given an open cover {U iX}\{U_i \to X\} there is a canonical morphism from its Cech groupoid to the manifold XX given by

C({U i}) X:(x,i)x. C(\{U_i\})_\bullet \to X \;\; :\;\; (x,i) \mapsto x \,.
Example

A morphism

g:C({U i}) (BG) g : C(\{U_i\})_\bullet \longrightarrow (\mathbf{B}G)_\bullet

is given in components precisely by a collection of functions

{g ij:U ijG} i,jI \{g_{i j} : U_{i j} \to G \}_{i,j \in I}

such that on each U i×XU kU jU_i \underset{X}{\times} U_k \cap U_j the equality g jkg ij=g ikg_{j k} g_{i j} = g_{i k} of smooth functions holds:

( (x,j) (x,i) (x,k))( g ij(x) g jk(x) g ik(x) ). \left( \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&\to&& (x,k) } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow && \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\stackrel{g_{i k}(x)}{\to}&& \bullet } \right) \,.

This is precisely a cocycle in Cech cohomology on XX relative {U i}\{U_i\} with coefficients in GG.

The universal smooth GG-principal bundle

For GG a Lie group (or any topological group), traditional literature highlights the universal principal bundle EGBGE G \to B G over the classifying space of GG, and the fact that under pullback of topological spaces this yields all isomorphism classes of smooth GG-principal bundles. But an analogous construction exists in smooth groupoids which is both simpler as well as more powerful: it modulates the full groupoid of smooth GG-principal bundles. We now discuss this smooth incarnation (EG) (\mathbf{E}G)_\bullet of EGE G.

Definition

For GG a Lie group, write EG\mathbf{E}G for the action groupoid of GG acting on itself from the right, hence for the Lie groupoid

(EG) (G×Gp 1()()G) (\mathbf{E}G)_\bullet \coloneqq \left( G \times G \stackrel{\overset{(-)\cdot (-)}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} G \right)

whose manifold of objects is GG, whose manifold of morphisms is G×GG \times G, whose source-map is projection on the first factor, whose target map is multiplication in the group, whose identity-map is g(g,e)g\mapsto (g,e) and whose composition operation is

(g 1g,h)(g 1,g)(g 1,gh). (g_1 g, h) \circ (g_1,g) \coloneqq (g_1, g h) \,.
Remark

The groupoid (EG) (\mathbf{E}G)_\bullet of def. 4 has at most one morphism for every ordered pair of objects, hence the morphisms are uniquely identified by giving their source and target.

Schematically:

(EG) ={ g 2 g 1 1g 2 = g 2 1g 3 g 1 g 1 1g 3 g 3} (\mathbf{E}G)_\bullet = \left\{ \array{ && g_2 \\ & {}^{\mathllap{g_1^{-1} g_2}}\nearrow &=& \searrow^{\mathrlap{g_2^{-1}g_3 }} \\ g_1 &&\stackrel{ g_1^{-1}g_3}{\longrightarrow}&& g_3 } \right\}

or simply

(EG) ={ g 2 = g 1 g 3}. (\mathbf{E}G)_\bullet = \left\{ \array{ && g_2 \\ & {}^{\mathllap{}}\nearrow &=& \searrow^{\mathrlap{}} \\ g_1 &&\stackrel{ }{\longrightarrow}&& g_3 } \right\} \,.

This means that it is isomorphic, as a pre-smooth groupoid, to the pair groupoid of GG.

While therefore (EG) (\mathbf{E}G)_\bullet is a rather simplistic object, it is nevertheless worthwhile to make its following properties explicit.

Proposition

There is an evident morphism of smooth groupoids

p:(EG) (BG) p\colon (\mathbf{E}G)_\bullet \to (\mathbf{B}G)_\bullet

given by

(g 1,g)g (g_1, g) \mapsto g

hence

(g 1g 2)(g 1 1g 2) (g_1 \to g_2) \mapsto (\bullet \stackrel{g_1^{-1}g_2 }{\to} \bullet)

There is an evident GG-action

(EG) ×GG (\mathbf{E}G)_\bullet \times G \longrightarrow G

given by

((g 1,g 2),h)(g 1h,g 2h). ((g_1,g_2), h) \mapsto (g_1 h, g_2 h) \,.

The projection pp is the quotient projection of this action.

Remark

The nerve of (EG) (\mathbf{E}G)_\bullet is a simplicial object of the form

N((EG) ) k=G×G × k N((\mathbf{E}G)_\bullet)_k = G \times G^{\times_k}

with face maps of the form

N((BG) ) =(G×G×G×GG×G×GG×GG). N((\mathbf{B}G)_\bullet)_\bullet = \left( \cdots G \times G \times G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} G \times G\times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} G\times G \stackrel{\longrightarrow}{\longrightarrow} G \right) \,.

If here (EG) (\mathbf{E}G)_\bullet is though of isomorphically as (G//G) (G//G)_\bullet, then
these face maps are such that all except one outermost (say the topmost) in each degree are given by group multiplication in two consecutive copies of GG, with the remaining outermost one given by projection.

If on the other hand (EG) (\mathbf{E}G)_\bullet is thought of isomorphically as the pair groupoid of GG, via remark 3, then the kkth face map in each degree is given simply by projecting out the kkth factor (starting counting at 0).

Definition

A morphism p:E B p\colon E_\bullet \to B_\bullet of pre-smooth groupoids is called a fibration if for each nn\in \mathbb{N} the functor p( n):E( n) B( n) p(\mathbb{R}^n) \colon E(\mathbb{R}^n)_\bullet \to B(\mathbb{R}^n)_\bullet is an isofibration, hence if for each object eE( n)e \in E(\mathbb{R}^n), each morphism p(e)bp(e) \to b in B( n)B(\mathbb{R}^n) has a lift through pp to a morphism eee \to e' in EE:

e ψp 1(ϕ) e E p p(e) ϕ b B. \array{ e &\stackrel{\exists \psi \in p^{-1}(\phi)}{\to} & \exists e'&&& E \\ &&&&& \downarrow^p \\ p(e) &\stackrel{\phi}{\to} & b &&& B } \,.
Proposition

The projection p:(EG) (BG) p \colon (\mathbf{E}G)_\bullet \to (\mathbf{B}G)_\bullet of prop. 1 is a fibration of smooth groupoids, def. 6. Moreover, any point inclusion *EG\ast \longrightarrow \mathbf{E}G is over each n\mathbb{R}^n an equivalence of groupoids, hence is in particular a local weak equivalence of smooth groupoids (as defined here).

In summary, the morphisms *(EG) p(BG) \ast \to (\mathbf{E}G)_\bullet \stackrel{p}{\to} (\mathbf{B}G)_\bullet constitute a factorization of the canonical *(BG) \ast \to (\mathbf{B}G)_\bullet into a local weak equivalence followed by a fibration.

Proposition

The smooth groupoid (EG) (\mathbf{E}G)_\bullet of def. 4 has the following equivalent incarnations as pre-smooth groupoids by isomorphic Lie groupoids

  1. (EG) (G//G) (\mathbf{E}G)_\bullet \simeq (G//G)_\bullet is the action groupoid of GG acting on itself by right multiplication;

  2. (EG) ((BG) I) ×(BG) *(\mathbf{E}G)_\bullet \simeq ((\mathbf{B}G)^{I})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} \ast is the pullback of the point inclusion *(BG) \ast \to (\mathbf{B}G)_\bullet along one of the projection map d 1:((BG) I) (BG) d_1 \colon ((\mathbf{B}G)^I)_\bullet \longrightarrow (\mathbf{B}G)_\bullet out of the path space object of (BG) (\mathbf{B}G)_\bullet.

Proof

The first statement is immediate from the definitions. The second is also fairly immediate, but worth making more explicit: the Lie groupoid ((BG) I) ((\mathbf{B}G)^I)_\bullet has as objects the morphisms in (BG) (\mathbf{B}G)_\bullet, hence elements of GG, and as morphisms g 1g 2g_1 \to g_2 commuting squares between these, schematically:

((BG) I) ={ h 1 g 1 g 2 h 2 }. ((\mathbf{B}G)^I)_\bullet = \left\{ \array{ \bullet &\stackrel{h_1}{\longrightarrow}& \bullet \\ {}^{\mathllap{g_1}}\downarrow && \downarrow^{\mathrlap{g_2}} \\ \bullet &\stackrel{h_2}{\longrightarrow}& \bullet } \right\} \,.

The morphism d 1d_1 projects out the top horizontal morophisms:

d 1:( h 1 g 1 g 2 h 2 )( h 1 ). d_1 \;\colon\; \left( \array{ \bullet &\stackrel{h_1}{\longrightarrow}& \bullet \\ {}^{\mathllap{g_1}}\downarrow && \downarrow^{\mathrlap{g_2}} \\ \bullet &\stackrel{h_2}{\longrightarrow}& \bullet } \right) \;\; \mapsto \;\; \left( \array{ \bullet &\stackrel{h_1}{\longrightarrow}& \bullet } \right) \,.

The pullback then restricts this image to be constant and hence produces the groupoid whose objects are still the morphisms in BG\mathbf{B}G, hence elements of GG, but whose morphisms are no longer all commuting squares, but just all commuting triangles between these, schematically:

((BG) I) ×(BG) *={ g 1 g 2 h }. ((\mathbf{B}G)^I)_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} \ast = \left\{ \array{ && \bullet \\ & {}^{\mathllap{g_1}}\swarrow && \searrow^{\mathrlap{g_2}} \\ \bullet && \stackrel{h}{\longrightarrow} && \bullet } \right\} \,.

Such a triangle exists precisely if g 2=g 1hg_2 = g_1 h, which gives EG\mathbf{E}G as in def. 4, thought of as:

  • objects = { g }\left\{ \array{ && \bullet \\ & {}^g\swarrow \\ \bullet } \right\}

  • morphisms = { g g=gh h }.\left\{ \array{ && \bullet \\ & {}^g\swarrow && \searrow^{g' = g h} \\ \bullet &&\stackrel{h}{\longrightarrow}&& \bullet } \right\} \,.

GG-Principal bundles

The traditional construction of the GG-principal bundle associated to a Cech cocycle is the following.

Definition

Let XX be a smooth manifold, {U iX} I\{U_i \to X\}_I an open cover and (g ij) i,jI(g_{i j})_{i,j \in I} a Cech cocycle of degree 1 with values in GG. Then the bundle PXP \to X associated with this data is the quotient

P( iU i×G)/ P \coloneqq \left( \coprod_{i} U_i \times G \right)/{\sim}

of the Cartesian product of the cover (regarded as the disjoint union of its patches) with GG, by the equivalence relation which identifies two elements in the product whenever they are related by the Cech cocycle:

((x,i),g)((x,j),gg ij(x)). ((x,i),g) \sim ((x,j), g \cdot g_{i j}(x)) \,.
Proposition

Let XX be a smooth manifold, {U iX} I\{U_i \to X\}_I an open cover and (g ij) i,jI(g_{i j})_{i,j \in I} a Cech cocycle of degree 1 with values in GG. Then the associated GG-bundle PP, def. 7, is equivalent, regarded as a smooth groupoid with only identity morphisms, to the pullback of the morphism (EG) (BG) (\mathbf{E}G)_\bullet \to (\mathbf{B}G)_\bullet of def. 4 along the cocycle regarded as a homomorphism of smooth groupoids C({U i}) g(BG) C(\{U_i\})_\bullet \stackrel{g}{\longrightarrow} (\mathbf{B}G)_\bullet.

P C({U i}) ×(BG) (EG) (EG) C({U i}) g (BG) X \array{ P &\overset{\simeq}{\longleftarrow}& C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (\mathbf{E}G)_\bullet &\longrightarrow& (\mathbf{E}G)_\bullet \\ \downarrow && \downarrow && \downarrow \\ && C(\{U_i\})_\bullet &\stackrel{g}{\longrightarrow}& (\mathbf{B}G)_\bullet \\ &\searrow & \downarrow^{\mathrlap{\simeq}} \\ && X }
Proof

Pullbacks of pre-smooth groupoids are computed componentwise. Hence a morphism in C({U i}) ×(BG) (EG) C(\{U_i\})_\bullet\underset{(\mathbf{B}G)_\bullet}{\times} (\mathbf{E}G)_\bullet is a pair consisting of a morphism (x,i,j)(x,i,j) in C({U i}) C(\{U_i\})_\bullet and a morphism (g 1,h)(g_1, h) in (EG) (\mathbf{E}G)_\bullet such that hh is the value of the cocycle on (x,i,j)(x,i,j).

With (EG) (\mathbf{E}G)_\bullet thought of as in remark 3, then the pullback looks like:

  • objects =

    { g (x,i)} \left\{ \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right\}
  • morphisms =

    { g g g ij(x) (x,i) (x,j)} \left\{ \array{ && \bullet \\ & {}^{g}\swarrow && \searrow^{g'} \\ \bullet &&\stackrel{g_{i j }(x)}{\to}&& \bullet \\ (x,i) &&\stackrel{}{\to}&& (x,j) } \right\}

This means that the morphisms in the pullback are of the form

((x,i),g 1) ((x,j),g 1g ij(x)) \array{ ((x,i),g_1) &&\stackrel{\simeq}{\to}&& ((x,j), g_1 g_{i j}(x) ) }

and there is at most one for any ordered pair of objects. But this means that these morphisms represent precisely the equivalence relation of def. 7: the evident projection map from this pullback to PP (with PP regarded as a groupoid with only identity morphisms) is evidently essentially surjective and fully faithful, hence an equivalence.

Remark

By the pullback construction in prop. 4, PP inherits a GG-action from that on (EG) (\mathbf{E}G)_\bullet of def. 1:

via the pasting diagram of pullbacks

P˜×G (EG) ×G P˜ (EG) C({U i}) g (BG) X. \array{ \tilde P \times G &\to& (\mathbf{E}G)_\bullet \times G \\ \downarrow && \downarrow \\ \tilde P &\to& (\mathbf{E}G)_\bullet \\ \downarrow && \downarrow \\ C(\{U_i\})_\bullet &\stackrel{g}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

The morphism P˜×GP˜\tilde P \times G \to \tilde P exhibits the principal GG-action of GG on P˜\tilde P.

Fibration categories and the Factorization lemma

We saw above that smooth GG-principal bundles for GG a Lie group may naturally be understood in terms of pullbacks of the fibrant replacement EGBG\mathbf{E}G\to \mathbf{B}G of the point inclusion *BG\ast \to \mathbf{B}G along a Cech cocycle, regarded as a homomorphism of smooth groupoids.

This is a special case of a very general construction of homotopy pullbacks which will also apply, below, to weakly principal simplicial bundles and then generally to principal infinity-bundles. We now discuss the general axiomatization of this construction via categories of fibrant objects.

Categories of Fibrant objects
Definition

A category of fibrant objects 𝒞\mathcal{C} is

  • a category with weak equivalences, i.e equipped with a subcategory WW that contains all isomorphisms

    Core(𝒞)W𝒞, Core(\mathcal{C}) \hookrightarrow W \hookrightarrow \mathcal{C} \,,

    where fMor(W)f \in Mor(W) is called a weak equivalence;

  • equipped with a further subcategory

    Core(𝒞)F𝒞, Core(\mathcal{C}) \hookrightarrow F \hookrightarrow \mathcal{C} \,,

    where fMor(F)f \in Mor(F) is called a fibration

    Those morphisms which are both weak equivalences and fibrations are called acyclic fibrations .

This data has to satisfy the following properties:

  • 𝒞\mathcal{C} has finite products, and in particular a terminal object *{\ast};

  • the pullback of a fibration along an arbitrary morphism exists, and is again a fibration;

  • acyclic fibrations are preserved under pullback;

  • weak equivalences satisfy 2-out-of-3;

  • for every object there exists a path object

    • this means: for every object BB there exists at least one object denoted B IB^I (not necessarily but possibly the internal hom with an interval object) that fits into a diagram
    (BId×IdB×B)=(BσB Id 0×d 1B×B) (B \stackrel{Id \times Id}{\to} B \times B) = (B \stackrel{\sigma}{\to} B^I \stackrel{d_0 \times d_1}{\to} B \times B)

    where σ\sigma is a weak equivalence and d 0×d 1d_0 \times d_1 is a fibration;

  • all objects are fibrant, i.e. all morphisms B*B \to {\ast} to the terminal object are fibrations.

Proposition

The category of pre-smooth groupoids (here) becomes a category of fibrant objects, def. 8 with fibrations as in def. 6 and weak equivalences the local weak equivalences (as defined here).

Factorization lemma

Let CC be a category of fibrant objects. The factorization lemma says the following.

Lemma

Every morphism f:XYf : X \to Y in CC factors as

f:XiX^pY, f : X \underoverset{\simeq}{i}{\longrightarrow} \hat X \stackrel{p}{\longrightarrow} Y \,,

where

  1. ii is a weak equivalence (even a right inverse to an acyclic fibration);

  2. pp is a fibration.

Proof

Let Y IY^I with factorization YY I(d 0,d 1)Y×YY \stackrel{\simeq}{\to} Y^I \stackrel{(d_0,d_1)}{\longrightarrow} Y \times Y be a path space object for YY. Let X^Y I× YX\hat X \coloneqq Y^I \times_Y X be the pullback of ff along one of its legs, to get the diagram

X^ X f Y I d 1 Y d 0 Y. \array{ \hat X &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ Y^I &\stackrel{d_1}{\to}& Y \\ \downarrow^{\mathrlap{d_0}} \\ Y } \,.

Take pp to be the composite vertical morphism in the above diagram, hence

p:X^Y Id 0Y. p \;\colon\; \hat X \to Y^I \stackrel{d_0}{\to} Y \,.

To see that this is indeed a fibration, notice that, by the pasting law, the above pullback diagram can be refined to a double pullback diagram as follows

X^ X×Y p 1 X (f,Id) f Y I (d 1,d 0) Y×Y p 1 Y d 0 p 2 Y. \array{ \hat X &\stackrel{}{\to}& X \times Y &\stackrel{p_1}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{(f, Id)}} && \downarrow^\mathrlap{f} \\ Y^I &\stackrel{(d_1 , d_0) }{\to}& Y \times Y &\stackrel{p_1}{\to}& Y \\ \downarrow^{\mathrlap{d_0}} & \swarrow_{p_2} \\ Y } \,.

Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism X^X×Y\hat X \to X \times Y is a fibration. Similarly, since XX is assumed to be fibrant (as all objects in a category of fibrant objects), also the projection map X×YYX \times Y \to Y is a fibration (see here).

Since pp is thereby exhibited as the composite of two fibrations

p :X^X×Y(f,Id)Y×Yp 2Y, \begin{aligned} p &: \hat X \to X \times Y \stackrel{(f ,Id)}{\to} Y \times Y \stackrel{p_2}{\to} Y \end{aligned} \,,

(the first map being a pullback of a fibration as above, the composite of the second and the third map being the projection just menioned) it is itself a fibration.

Next, by the axioms of path space objects in a category of fibrant objects we have that d 1:Y IYd_1 \;\colon\; Y^I \to Y is an acyclic fibration. Since these are stable under pullback, also X^X\hat X \to X is an acyclic fibration.

But, by the axioms, Y IYY^I \to Y has a right inverse YY IY \to Y^I. By the pullback property this induces a right inverse of X^X\hat X \to X fitting into a pasting diagram

X X^ X f f Y Y I d 1 Y Id d 0 Y. \array{ X &\to& \hat X &\to& X \\ {}^{\mathrlap{f}}\downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ Y &\to& Y^I &\stackrel{d_1}{\to}& Y \\ & {}_{\mathllap{Id}}\searrow& \downarrow^{\mathrlap{d_0}} \\ && Y } \,.

This establishes the claim.

Homotopy and Homotopy pullback

Let 𝒞\mathcal{C} be a category of fibrant objects.

Definition

Two morphism f,g:ABf,g : A \to B in 𝒞(A,B)\mathcal{C}(A,B) are

  • right homotopic, denoted fgf \simeq g, precisely if they fit into a diagram of the form

    B f d 0 A η B I g d 1 B \array{ && B \\ & {}^f\nearrow & \uparrow^{d_0} \\ A &\stackrel{\eta}{\to}& B^I \\ & {}_g\searrow & \downarrow^{\mathrlap{d_1}} \\ && B }

    for some path space object B IB^I;

  • homotopic, denoted fgf \sim g, if they become right homotopic after pulled back to a weakly equivalent domain, i.e. precisely if they fit into a diagram of the form

A f B wW d 0 A^ η B I wW d 1 A g B \array{ && A &\stackrel{f}{\to}& B \\ &{}^{w \in W}\nearrow&&& \uparrow^{d_0} \\ \hat A && \stackrel{\eta}{\to} && B^I \\ &{}_{w\in W}\searrow & && \downarrow^{d_1} \\ && A &\stackrel{g}{\to}& B }

for some object A^\hat A and for some path space object B IB^I of II

In view of this the following definition is natural.

Definition

A homotopy fiber product or homotopy pullback of two morphisms

AuCvB A \stackrel{u}{\to} C \stackrel{v}{\leftarrow} B

in a category of fibrant objects is the object A× CC I× CBA \times_C C^I \times_C B defined as the (ordinary) limit

A× CC I× CB B v C I d 0 C d 1 A u C. \array{ A \times_C C^I \times_C B &\longrightarrow& &\longrightarrow & B \\ \downarrow &&&& \downarrow^v \\ & &C^I & \stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{d_1}} \\ A &\stackrel{u}{\to} & C } \,.
Proposition

The homotopy fiber product in def. 10 is isomorphic to the ordinary fiber product of either of the two morphisms with the fibration replacement of the other as given by the factorization lemma, def. 1.

Proof

By basic properties of limits the defining limit in def. 10 may be computed by two consecutive pullbacks.

A× CC I× CB A×C(C I×CB) C I×CB B v C I d 0 C d 1 A u C. \array{ A \times_C C^I \times_C B \simeq & A \underset{C}{\times} \left(C^I \underset{C}{\times} B\right) &\to& C^I \underset{C}{\times} B &\to & B \\ & \downarrow && \downarrow && \downarrow^{\mathrlap{v}} \\ & & &C^I & \stackrel{d_0}{\to}& C \\ & \downarrow && \downarrow^{\mathrlap{d_1}} \\ & A &\stackrel{u}{\to}& C } \,.

Here the intermediate pullback is precisely the one appearing in the proof of the factorization lemma.

Example

With 𝒞\mathcal{C} the category of fibrant objects given by pre-smooth groupoids, prop. 5, then for GG a Lie group, the factorization *EGpBG\ast \to \mathbf{E}G \stackrel{p}{\to} \mathbf{B}G of prop. 2 is the one given by the factorization lemma. Hence a pullback of p:EGBGp \colon \mathbf{E}G\to \mathbf{B}G as in prop. 4 is equivalently the homotopy pullback of *BG\ast \to \mathbf{B}G.

Weakly principal simplicial bundles

(… under construction …)

It is no coincidence that the above statement looks akin to the maybe more familiar statement which says that equivalence classes of GG-principal bundles are classified by homotopy-classes of morphisms of topological spaces

π 0Top(X,BG)π 0GBund(X), \pi_0 Top(X, \mathbf{B}G) \simeq \pi_0 G Bund(X) \,,

where BG\mathbf{B}G \in Top is the topological classifying space of GG. The category Top of topological spaces, regarded as an (∞,1)-category, is the archetypical (∞,1)-topos the way that Set is the archetypical topos. And it is equivalent to ∞Grpd, the (,1)(\infty,1)-category of bare ∞-groupoids. What we are seeing above is a first indication of how cohomology of bare \infty-groupoids is lifted to a richer (,1)(\infty,1)-topos to cohomology of \infty-groupoids with extra structure.

In fact, all of the statements that we have considered so far become conceptually simpler in the (,1)(\infty,1)-topos. We had already remarked that the anafunctor span XC(U)gBGX \stackrel{\simeq}{\leftarrow} C(U) \stackrel{g}{\to} \mathbf{B}G is really a model for what is simply a direct morphism XBGX \to \mathbf{B}G in the (,1)(\infty,1)-topos. But more is true: that pullback of EG\mathbf{E}G which we considered is just a model for the homotopy pullback of just the point

P˜×G EG×G P˜ EG C(U) g BG X inthemodelcategory P×G G P * X BG . . inthe(,1)topos. \array{ \vdots && \vdots \\ \tilde P \times G &\to& \mathbf{E}G \times G \\ \downarrow && \downarrow \\ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X \\ {} \\ {} \\ & in\;the\;model\;category & } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ \vdots && \vdots \\ P \times G &\to& G \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G \\ . \\ . \\ \\ \\ & in\;the\;(\infty,1)-topos } \,.
Universal principal bundle
Weakly principal simplicial bundles

The principal ∞-bundles that we wish to model are already the main and simplest example of the application of these three items:

Consider an object BG[C op,sSet]\mathbf{B}G \in [C^{op}, sSet] which is an \infty-groupoid with a single object, so that we may think of it as the delooping of an ∞-group GG, let ** be the point and *BG* \to \mathbf{B}G the unique inclusion map. The good replacement of this inclusion morphism is the GG-universal principal ∞-bundle EGBG\mathbf{E}G \to \mathbf{B}G given by the pullback diagram

EG * BG Δ[1] BG BG \array{ \mathbf{E}G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G^{\Delta[1]} &\to& \mathbf{B}G \\ \downarrow \\ \mathbf{B}G }

An ∞-anafunctor XX^BGX \stackrel{\simeq}{\leftarrow} \hat X \to \mathbf{B}G we call a cocycle on XX with coefficients in GG, and the (∞,1)-pullback PP of the point along this cocycle, which by the above discussion is the ordinary limit

P EG * BG I BG X^ g BG X \array{ P &\to& \mathbf{E}G &\to& * \\ \downarrow && \downarrow && \downarrow \\ && \mathbf{B}G^I &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ \hat X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }

we call the principal ∞-bundle PXP \to X classified by the cocycle.

It is now evident that our discussion of ordinary smooth principal bundles above is the special case of this for BG\mathbf{B}G the nerve of the one-object groupoid associated with the ordinary Lie group GG.

So we find the complete generalization of the situation that we already indicated there, which is summarized in the following diagram:

P˜×G EG×G P˜ EG C(U) g BG X inthemodelcategory P×G G P * X BG . . inthe(,1)topos. \array{ \vdots && \vdots \\ \tilde P \times G &\to& \mathbf{E}G \times G \\ \downarrow && \downarrow \\ \tilde P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X \\ {} \\ {} \\ & in\;the\;model\;category & } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ \vdots && \vdots \\ P \times G &\to& G \\ \downarrow && \downarrow \\ P &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G \\ . \\ . \\ \\ \\ & in\;the\;(\infty,1)-topos } \,.

Semantic Layer

Principal \infty-bundles

Syntactic Layer

References

Revised on April 16, 2015 09:52:45 by Urs Schreiber (195.113.30.252)