% (find-angg "LATEX/2017lucatelli-fibs.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2017lucatelli-fibs.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2017lucatelli-fibs.pdf"))
% (defun b () (interactive) (find-zsh "bibtex 2017lucatelli-fibs; makeindex 2017lucatelli-fibs"))
% (defun e () (interactive) (find-LATEX "2017lucatelli-fibs.tex"))
% (defun u () (interactive) (find-latex-upload-links "2017lucatelli-fibs"))
% (find-xpdfpage "~/LATEX/2017lucatelli-fibs.pdf")
% (find-sh0 "cp -v  ~/LATEX/2017lucatelli-fibs.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2017lucatelli-fibs.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2017lucatelli-fibs.pdf
%               file:///tmp/2017lucatelli-fibs.pdf
%           file:///tmp/pen/2017lucatelli-fibs.pdf
% http://angg.twu.net/LATEX/2017lucatelli-fibs.pdf
\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage{color}                % (find-LATEX "edrx15.sty" "colors")
\usepackage{colorweb}             % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15}               % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex            % (find-angg "LATEX/edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-dn4ex "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
\begin{document}

\catcode`\^^J=10
\directlua{dednat6dir = "dednat6/"}
\directlua{dofile(dednat6dir.."dednat6.lua")}
\directlua{texfile(tex.jobname)}
\directlua{verbose()}
\directlua{output(preamble1)}
\def\expr#1{\directlua{output(tostring(#1))}}
\def\eval#1{\directlua{#1}}
\def\pu{\directlua{pu()}}

\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
\directlua{dofile "edrxpict.lua"} % (find-LATEX "edrxpict.lua")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end



\def\bfX{\mathbf{X}}
\def\bfB{\mathbf{B}}
\def\bfA{\mathbf{A}}

Definition 2.1. Let $P:\bfX→\bfB$ be a functor. A morphism $φ:Y→X$ in
$\bfX$ over $u:=P(φ)$ is called {\sl cartesian} iff for all $v:K→J$ in
$\bfB$ and $θ:Z→X$ with $P(θ)=u∘v$ there is a unique morphism $ψ:Z→Y$
with $P(ψ)=v$ and $θ=φ∘ψ$.
%
%D diagram fib-1
%D 2Dx     100  +20  +20 +20   +15
%D 2D  100 Z
%D 2D        ->
%D 2D  +20      Y -> X   \bfX \bfA
%D 2D
%D 2D  +10 K
%D 2D        ->
%D 2D  +20      J -> I   \bfB
%D 2D
%D # ren ==>
%D
%D (( Z Y --> .plabel= b ψ
%D    Y X  -> .plabel= b φ
%D    Z X  -> .plabel= a θ
%D    
%D    K J  -> .plabel= b v
%D    J I  -> .plabel= b u
%D    K I  -> .plabel= a u∘v
%D
%D    \bfX \bfB -> .plabel= r P
%D    \bfX \bfA =
%D ))
%D enddiagram
%D
$$\pu
  \diag{fib-1}
$$

It is clear then:

Lemma: Let $P:\bfA→\bfB$ be a functor. A morphism $φ:Y→X$ of $\bfA$ is
cartesian if and only if $(\bfA/X)(-,φ) ≅ (\bfB/P(X))(P(-),P(φ))$.

(or?...)

Lemma: Let $P:\bfX→\bfB$ be a functor. A morphism $φ:Y→X$ of $\bfX$ is
cartesian if and only if $(\bfX/X)(-,φ) ≅ (\bfB/P(X))(P(-),P(φ))$.


\bsk
\bsk

Definition 2.2. $P:\bfX→\bfB$ is a {\sl fibration} or {\sl category
  fibred over $\bfB$} iff for all $u:J→I$ in $\bfB$ and $X∈P(I)$ there
is a cartesian arrow $φ:Y→X$ over $u$ called a {\sl cartesian lifting}
of $X$ along $u$.

Thereby:

Theorem: $P:\bfA→\bfB$ is a fibration if and only if for every $X$ of
$\bfA$, $P:\bfA/X→\bfB/P(X)$ is surjective on objects and has a fully
faithful right adjoint (the ``cartesian lifting'').

Proof: in fact, both cases happen if and only if for each $X$ of
$\bfA$ and $u$ of $\bfB/P(X)$, there is $\overline{u}$ such that
$(\bfA/X)(-,\overline{u}) ≅ (\bfB/P(X))(P(-),u)$ and
$P(\overline{u})=u$.




%D diagram adj
%D 2Dx     100 +40
%D 2D  100 A0  A1
%D 2D
%D 2D  +20 A2  A3
%D 2D
%D 2D  +20 A4  A5
%D 2D
%D ren A0 A1 ==> (P(Z),P(θ)) (Z,θ)
%D ren A2 A3 ==> (J,u) (Y,φ)
%D ren A4 A5 ==> \bfB/P(X) \bfA/X
%D
%D (( A0 A1 <-| 
%D    A2 A3 |->
%D    A0 A2 -> .plabel= l v
%D    A1 A3 -> .plabel= r ψ
%D    A4 A5 <- sl^ .plabel= a P
%D    A4 A5 -> sl_ .plabel= b \text{(c.l.)}
%D ))
%D enddiagram
%D
$$\pu
  \diag{adj}
$$





\end{document}

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