Schreiber list of notation and constructions in categories of fibrant

When in the context of a category of fibrant objects we use the following notation and conventions for common constructions.

Universal bundles

For f:CDf : C \to D any morphisms of fibrant objects and for D ID^I denoting a path object for DD, we write p f:E fDDp_f : \mathbf{E}_f D \to D for the composite vertical morphism in the pullback diagram

E fD C f D I d 1 D d 0 D. \array{ \mathbf{E}_f D &\to& C \\ \downarrow && \downarrow^f \\ D^I &\stackrel{d_1}{\to}& D \\ \downarrow^{d_0} \\ D } \,.

This is always a fibration and is an acyclic fibration if ff is a weak equivalence. Also the morphism

E fDC \mathbf{E}_f D \to C

is always an acyclic fibration. Moreover, this has a section σ f:CE fC\sigma_f : C \to \mathbf{E}_f C that is necessarily a weak equivalence.

This yields a factorization of ff as

(CfD)=(Cσ fE fDp fD). (C \stackrel{f}{\to} D) \;\;=\;\; (C \stackrel{\sigma_f}{\to} \mathbf{E}_{f}D \stackrel{p_f}{\to} D) \,.

Loop space objects

If pt D:*Dpt_D : {*} \to D is a given point of DD (*{*} is the terminal object) then we write i D:Ω ptDE ptDi_D : \Omega_{pt} D \to E_{pt}D for the morphjism in the pullback diagram

Ω ptD E ptD p pt D * pt D D. \array{ \Omega_{pt} D &\stackrel{}{\to}& E_{pt}D \\ \downarrow && \downarrow^{p_{pt_D}} \\ {*} &\stackrel{pt_D}{\to}& D } \,.

This is the loop space object of DD at the point pt Dpt_D.


If for a given object GG there is an object KK with a unique point pt K:*Kpt_K : {*} \to K and such that B=Ω ptKB = \Omega_{pt} K we write K=BGK = \mathbf{B}G. This is the delooping of GG.

The object E ptK\mathbf{E}_{pt} K in this case we denote by EG\mathbf{E} G as a short form for E ptBG\mathbf{E}_{pt} \mathbf{B}G. The lack of the subscript indicates the different use of the symbol E\mathbf{E} in the case that we have a unique point.

Then in this case the fibration sequence GE ptKKG \to \mathbf{E}_{pt} K \to K reads

GEGBG. G \to \mathbf{E}G \to \mathbf{B}G \,.

This is the universal GG-principal ∞-bundle \infty in the context of the given category of fibrant objects.

Created on September 1, 2009 at 16:34:48. See the history of this page for a list of all contributions to it.