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Leonard Guetta
PhD-presentation
Commits
e7cb9aa8
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e7cb9aa8
authored
Jan 19, 2021
by
Leonard Guetta
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pres.tex
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e7cb9aa8
\documentclass
[handout]
{
beamer
}
\documentclass
{
beamer
}
%\usepackage[utf8]{inputenc}
\usepackage
{
mystyle
}
...
...
@@ -107,12 +107,14 @@
N
_{
\oo
}
:
\oo\Cat
&
\to
\Psh
{
\Delta
}
\\
C
&
\mapsto
N
_{
\oo
}
(
C
)
.
\end
{
aligned
}
\]
\end{frame}
\begin{frame}
\frametitle
{$
\oo
$
\nbd
{}
categories as spaces
}
TODO : Exemple en basse dimension
\]
\pause
\begin{exampleblock}
{
Example
}
When
$
C
$
is a (1-)category,
$
N
_{
\oo
}
(
C
)
$
is nothing but the usual nerve of
$
C
$
.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle
{$
\oo
$
\nbd
{}
categories as spaces
}
\begin{block}
{
Definition
}
...
...
@@ -341,12 +343,72 @@
\end{frame}
\begin{frame}
\frametitle
{
Polygraphs
}
\begin{block}
{
Definition
}
An
$
\oo
$
\nbd
{}
category is free on a polygraph if it can be obtained
recursively from the empty category by freely
attaching cells.
\end{block}
\pause
Terminological convention:
\begin{center}
free
$
\oo
$
\nbd
{}
category =
$
\oo
$
\nbd
{}
category
free on a polygraph.
\end{center}
free on a polygraph.
\end{center}
\pause
\begin{exampleblock}
{
Important fact
}
If
$
C
$
is a free
$
\oo
$
\nbd
{}
category, the set of free generators of
$
C
$
is uniquely determined from
$
C
$
.
\end{exampleblock}
TODO : Laisser le bloc ci-dessus ?
\end{frame}
\begin{frame}
\frametitle
{
Abelianization of
$
\oo
$
\nbd
{}
categories
}
Recall that by a variation of the Dold--Kan equivalence, we have:
\[
\Ab
(
\oo\Cat
)
\simeq
\Ch
,
\]
\pause
hence, a forgetful functor
\[
\Ch\simeq
\Ab
(
\oo\Cat
)
\to
\oo\Cat
,
\]
\pause
which
has a left adjoint
\[
\lambda
:
\oo\Cat
\to
\Ch
,
\]
which we refer to as the
\alert
{
abelianization functor
}
.
\end{frame}
\begin{frame}
\frametitle
{
Polygraphic homology
}
\begin{alertblock}
{
Proposition (folklore ?)
}
The functor
$
\lambda
:
\oo\Cat
\to
\Ch
$
is left Quillen w.r.t the folk
model structure on
$
\oo\Cat
$
and the projective model structure on
$
\Ch
$
.
\end{alertblock}
\pause
\begin{block}
{
Definition
}
The
\alert
{
polygraphic homology functor
}
is the left derived functor of
$
\lambda
$
:
\[
\sH
^{
\pol
}
:
=
\LL
\lambda
\colon
\Ho
(
\oo\Cat
^{
\folk
}
)
\to
\Ho
(
\Ch
)
,
\]
where
$
\Ho
(
\ooCat
^{
\folk
}
)
$
is the localization of
$
\oo\Cat
$
w.r.t the
equivalences of
$
\oo
$
\nbd
{}
categories.
\end{block}
\end{frame}
\begin{frame}
\frametitle
{
Polygraphic homology practically
}
TODO
\end{frame}
\begin{frame}
% \frametitle{}
A natural question:
\begin{center}
Let
$
C
$
be an
$
\oo
$
\nbd
{}
category. Do we have
$
\sH
^{
\pol
}
(
C
)
\simeq
\sH
^{
\sing
}
(
C
)
$
?
\end{center}
% \pause
% Answer : In general, \textbf{no} !
\end{frame}
\end{document}
%%% Local Variables:
...
...
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