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generalized universal bundle

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higher category theory

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Extra properties and structure

1-categorical presentations

Contents

Idea

Universal bundles – or mapping cocylinders – are intermediate steps in the computation of homotopy fibers, dual to the that way mapping cone are intermediate steps in the computation of homotopy cofibers.

It is familiar from topology that one can form the path fibration PXXP X \to X of a topological space. This can be understood as an example of a general construction where one computes homotopy pullbacks of the point – or, if things are not groupoidal, comma objects.

Since universal bundles are examples of this construction, we here speak of generalized universal bundles. Another appropriate term might be generalized path fibrations.

One generalizaton of “generalized universal bundles” is that the objects in question need not be groupoidal, i.e. they behave like directed spaces. In this case the homotopy pullbacks familiar from topology are replaced by comma object constructions. This is useful in various applications. For instance the constructions category of elements and Grothendieck construction can be understood as such directed homotopy pullbacks of the point.

See also

and in particular

Definition

Let CC be a closed monoidal category with interval object II. Then for any pointed object ptpt BBpt \stackrel{pt_B}{\to}B in CC the generalized universal BB-bundle is (if it exists) the morphism

p:E ptBB p : \mathbf{E}_{pt} B \to B

which is the total composite vertical morphism of the pullback diagram

E ptB pt pt B [I,B] d 1 B d 0 B. \array{ \mathbf{E}_{\mathrm{pt}}B &\to& pt \\ \downarrow && \downarrow^{pt_B} \\ [I,B] &\stackrel{d_1}{\to}& B \\ \downarrow^{d_0} \\ B } \,.

So the object E pt:=[I,B]× Bpt\mathbf{E}_{pt} := [I,B]\times_{B} pt is defined to be the pullback of the diagram [I,B]d 1Bpt Bpt [I,B] \stackrel{d_1}{\to} B \stackrel{pt_B}{\leftarrow} pt and the morphism E ptBB\mathbf{E}_{pt}B \to B is the composite of the left vertical morphism in the above diagram which comes from the definition of pullback and d 0d_0.

Then a (generalized) ”BB-bundle” on some object XX is a morphism PXP \to X which is the pullback of the generalized universal BB-bundle E pt\mathbf{E}_{pt} along a “classifying morphism” g:XBg : X \to B

P E pt X g B \array{ P &\to& \mathbf{E}_{pt} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& B }

This can be understood as a “(directed) homotopy pullback” of the point:

If one defines, as one does, a (possiby directed) homotopy between two morphisms f,g:ABf,g : A \to B to be a morphism η:A[I,B]\eta : A \to [I,B] such that d 0 *η=fd_0^* \eta = f and d 1 *η=gd_1^* \eta = g, then PP is the “lax pullback” (really comma object) of the point along gg

P * pt B X g B. \array{ P &\to& * \\ \downarrow &\Downarrow& \downarrow^{pt_B} \\ X &\stackrel{g}{\to}& B } \,.

The fiber of the generalized universal bundle is the loop monoid Ω ptB\Omega_{pt} B:

Ω ptB P * pt B * X g B. \array{ \Omega_{pt} B &\to& P &\to& * \\ \downarrow &\Downarrow& \downarrow &\Downarrow& \downarrow^{pt_B} \\ * &\to& X &\stackrel{g}{\to}& B } \,.

the sequence

Ω ptBiE ptBpB \Omega_{pt}B \stackrel{i}{\to} \mathbf{E}_{pt} B \stackrel{p}{\to} B

is exact in that ii is the kernel of pp in the sense of kernels of morphisms of pointed objects (see there).

Examples

Groupoid incarnations of universal principal bundles

In (higher) categorical contexts, take the interval object to the the interval category I:={ab}I := \{a \to b\}. Then

Ordinary GG-principal bundles

For C=C = Cat, B:=BGB := \mathbf{B}G a one-object groupoid corresponding to a group GG with the unique point, E ptBG=EG=G//G\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G = G//G is the action groupoid of GG acting on itself. The sequence of groupoids is

GG//GBG. G \to G//G \to \mathbf{B}G \,.

This is the universal GG-bundle in its groupoid incarnation. It is a theorem by Segal from the 1960s that indeed this maps, under geometric realization to the familiar universal GG-bundle in TopTop. Moreover, it can be seen that every GG-principal bundle PXP \to X in the ordinary sense is the pullback of EG\mathbf{E} G in the following sense:

the GG-bundle PXP \to X is classified by a nonabelian GG-valued 1-cocycle (the transition function of any of its local trivializations), which is an anafunctor

X^ g BG π X. \array{ \hat X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^\pi \\ X } \,.

(For instance X^\hat X could be the Čech groupoid of a cover of XX.)

The universal groupoid bundle EGBG\mathbf{E}G \to \mathbf{B}G may now be pulled back along this anafunctor to yield the groupoid bundle g *EGXg^* \mathbf{E}G \to X given by the total left vertical morphism in

g *EG EG X^ g BG π X. \array{ g^* \mathbf{E}G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \hat X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^\pi \\ X } \,.

This bundle of groupoids is weakly equivalent to the GG-principal bundle we started with in that there is a morphism of bundles of groupoids (with PP regarded as a bundle of discrete groupoids)

g *EG P X. \array{ g^* \mathbf{E}G &&\stackrel{\simeq}{\to}&& P \\ & \searrow && \swarrow \\ &&X } \,.

In fact that horizontal morphism is an acyclic fibration in the folk model structure, i.e. a k-surjective functor for all kk.

This is recalled in the following reference.

GG-principal 2-bundles

For C=2CatC = 2Cat, strict 2-categories , B:=BGB := \mathbf{B}G a strict one-object 2-groupoid corresponding to a strict 2-group GG with the unique point, E ptBG=EG\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G was described under the name INN(G)INN(G) in

  • Urs Schreiber, David Roberts, The inner automorphism 3-group of a strict 2-group, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244 (arXiv)

This was shown to be action bigroupoid of GG acting on itself in

  • Igor Bakovič, Bigroupoid 2-torsors PhD thesis, Munich (2008) (pdf).

One can show that every GG-principal 2-bundle as described in

  • Toby Bartels, 2-Bundles (arXiv)

  • Christoph Wockel,

  • Igor Bakovič, Bigroupoid 2-torsors PhD thesis, Munich (2008) (pdf).

indeed is recovered as the pullback of EGBG\mathbf{E} G \to \mathbf{B}G along the corresponding cocycle, along the lines described above.

The way this works is indicated briefly in the last section of Roberts-Schreiber above. A more detailed description for the moment is in the notes

Universal nn-category bundles: nn-subobject classifiers

One can take BB to be something very different from the familiar classifying groupoids. Taking it to be nCatn Cat yields the subobject classifiers of higher toposes:

It was David Roberts in the blog comment

who first pointed out that these (higher) subobject classifiers are just generalized universal bundles in the above sense.

These cases for n=0n= 0 and n=1n=1 have been considered in the context of universal category bundles in

The discussion there becomes more manifestly one of bundles if one regards all morphisms CSetC \to Set appearing there as being the right legs of anafunctors.

There is a well-understood version of this for n=(,1)n = (\infty,1), i.e. for (∞,1)-categories. This is described at universal fibration of (∞,1)-categories.

Action groupoids as generalized bundles

A morphism ρ:BF\rho : B \to F to a pointed object FF (needs not be a basepoint preserving morphism!) can be regarded as a representation of BB on the point of FF. The pullback of the universal FF-bundle along this morphism

ρ *E ptFB \rho^* \mathbf{E}_{pt} F \to B

can be addressed as the FF-bundle ρ\rho-associated to the universal BB-bundle E ptB\mathbf{E}_{pt}B.

If BB is a groupoid, then ρ *E ptF\rho^* \mathbf{E}_{pt} F is the action groupoid of BB acting on the point of FF.

Further pulling this back along a cocycle g:X^Bg : \hat X \to B of a BB-principal bundle yields the ρ\rho-accociated bundle of that.

For instance for B=BGB = \mathbf{B}G and F=VectF = Vect with ρ:BGVect\rho : \mathbf{B}G \to Vect a representation of the group GG on a vector space VV, the ρ\rho-associated Vect\mathrm{Vect}-bundle on BG\mathbf{B}G is

VV//GBG. V \to V//G \to \mathbf{B}G \,.

Pulling that further back along the cocycle g:X^BGg : \hat X \to \mathbf{B}G classifying a GG-principal bundle PXP \to X, one obtains the familiar vector bundle P× GVXP \times_G V \to X which is ρ\rho-associated to PP, along the lines described above:

g *ρ *E ptF P× GV X. \array{ g^* \rho^*\mathbf{E}_{pt}F &&\stackrel{\simeq}{\to}&& P\times_G V \\ & \searrow && \swarrow \\ &&X } \,.

Revised on August 13, 2013 10:22:27 by Tim Porter (95.147.236.73)