# Schreiber list of notation and constructions in categories of fibrant objects

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

# Universal bundles

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

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

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

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

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

This yields a factorization of $f$ as

$(C \stackrel{f}{\to} D) \;\;=\;\; (C \stackrel{\sigma_f}{\to} \mathbf{E}_{f}D \stackrel{p_f}{\to} D) \,.$

# Loop space objects

If $pt_D : {*} \to D$ is a given point of $D$ (${*}$ is the terminal object) then we write $i_D : \Omega_{pt} D \to E_{pt}D$ for the morphjism in the pullback diagram

$\array{ \Omega_{pt} D &\stackrel{}{\to}& E_{pt}D \\ \downarrow && \downarrow^{p_{pt_D}} \\ {*} &\stackrel{pt_D}{\to}& D } \,.$

This is the loop space object of $D$ at the point $pt_D$.

# Delooping

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

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

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

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

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

Last revised on September 4, 2009 at 14:46:28. See the history of this page for a list of all contributions to it.