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Leonard Guetta
PhD-presentation
Commits
452b3570
Commit
452b3570
authored
Jan 18, 2021
by
Leonard Guetta
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Slowly but surely
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pres.tex
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452b3570
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...
@@ -4,6 +4,18 @@
\usepackage
{
mystyle
}
\usetheme
{
Madrid
}
\usecolortheme
{
beaver
}
%gets rid of bottom navigation bars
\setbeamertemplate
{
footline
}
[frame number]
{}
%gets rid of bottom navigation symbols
\setbeamertemplate
{
navigation symbols
}{}
%gets rid of footer
%will override 'frame number' instruction above
%comment out to revert to previous/default definitions
\setbeamertemplate
{
footline
}{}
\title
{
Homology of strict
$
\omega
$
-categories
}
%\subtitle{PhD defense}
\author
{
Léonard Guetta
}
...
...
@@ -19,6 +31,16 @@
% \tableofcontents
% \end{frame}
\begin{frame}
\frametitle
{
Preliminary conventions
}
In this talk:
\begin{itemize}
\item
<2->
$
\oo
$
\nbd
{}
category = strict
$
\omega
$
\nbd
{}
category
\item
<3->
$
n
$
\nbd
{}
category =
$
\oo
$
\nbd
{}
category with only unit cells above
dimension
$
n
$
\item
<4-> the functor
$
n
\Cat
\to
\oo\Cat
$
is an inclusion
\end{itemize}
\end{frame}
%%% oo-categories as spaces
\begin{frame}
...
...
@@ -78,6 +100,84 @@
\end
{
aligned
}
\]
\end{block}
\pause
This yields the
\alert
{
nerve functor
}
for
$
\oo
$
\nbd
{}
categories
\[
\begin
{
aligned
}
N
_{
\oo
}
:
\oo\Cat
&
\to
\Psh
{
\Delta
}
\\
C
&
\mapsto
N
_{
\oo
}
(
C
)
.
\end
{
aligned
}
\]
\end{frame}
\begin{frame}
\frametitle
{$
\oo
$
\nbd
{}
categories as spaces
}
TODO : Exemple en basse dimension
\end{frame}
\begin{frame}
\frametitle
{$
\oo
$
\nbd
{}
categories as spaces
}
\begin{block}
{
Definition
}
A morphism
$
f
\colon
C
\to
D
$
of
$
\oo\Cat
$
is a
\emph
{
Thomason equivalence
}
if
$
N
_{
\oo
}
(
f
)
\colon
N
_{
\oo
}
(
C
)
\to
N
_{
\oo
}
(
D
)
$
is a weak equivalence of
simplicial sets.
\end{block}
\pause
$
\W
^{
\Th
}$
:=class of Thomason equivalences.
\pause
By definition, the
nerve functor induces
\[
\overline
{
N
_{
\oo
}}
:
\Ho
(
\oo\Cat
^{
\Th
}
)
\to
\Ho
(
\Psh
{
\Delta
}
)
.
\]
Where:
\begin{itemize}
[label=
$
\bullet
$
]
\item
$
\Ho
(
\oo\Cat
^{
\Th
}
)
$
is the localization of
$
\oo\Cat
$
with respect to
$
\W
^{
\Th
}$
,
\item
$
\Ho
(
\Psh
{
\Delta
}
)
$
is the localization of
$
\Psh
{
\Delta
}$
with
respect to weak equivalences of simplicial sets.
\end{itemize}
% Where $\Ho(-)$ stands for the localized category (or better
% the localized pre-derivator or even weak $(\oo,1)$\nbd{}category).
\end{frame}
\begin{frame}
\frametitle
{$
\oo
$
\nbd
{}
categories as spaces
}
\begin{alertblock}
{
Theorem (Gagna, 2018)
}
$
\overline
{
N
_{
\oo
}}
:
\Ho
(
\oo\Cat
^{
\Th
}
)
\to
\Ho
(
\Psh
{
\Delta
}
)
$
is an
equivalence of categories (or better an equivalence of derivators, or of weak
$
(
\infty
,
1
)
$
\nbd
{}
categories).
\end{alertblock}
\pause
In other words:
\begin{center}
Homotopy theory of
$
\oo
$
\nbd
{}
categories induced by Thomason equivalences
\\
$
\cong
$
\\
Homotopy theory of spaces
\end{center}
\end{frame}
\begin{frame}
\frametitle
{
Singular homology of
$
\oo
$
\nbd
{}
categories
}
Recall that we have the (normalized) chain complex functor
\[
\kappa
\colon
\Psh
{
\Delta
}
\to
\Ch
,
\]
Where
$
\Ch
$
is the category of non-negatively graded chain complexes.
\pause
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
Hence,
\[
\overline
{
\kappa
}
\colon
\Ho
(
\Psh
{
\Delta
}
)
\to
\Ho
(
\Ch
)
,
\]
where
$
\Ho
(
\Ch
)
$
is the localization of
$
\Ch
$
with respect to quasi-isomorphisms.
\end{frame}
\begin{frame}
\frametitle
{
Singular homology of
$
\oo
$
\nbd
{}
categories
}
\begin{block}
{
Definition
}
The
\emph
{
singular homology functor
}
$
\sH
^{
\sing
}
\colon
\Ho
(
\oo\Cat
^{
\Th
}
)
\to
\Ho
(
\Ch
)
$
is defined as the composition
\[
\Ho
(
\oo\Cat
^{
\Th
}
)
\overset
{
\overline
{
N
_{
\oo
}}}{
\longrightarrow
}
\Ho
(
\Psh
{
\Delta
}
)
\overset
{
\overline
{
\kappa
}}{
\longrightarrow
}
\Ho
(
\Ch
)
.
\]
\end{block}
\pause
In pratice, this means that the
$
k
$
\nbd
{}
th singular homology group of an
$
\oo
$
\nbd
{}
category
$
C
$
is
the
$
k
$
\nbd
{}
th homology group of
$
N
_{
\oo
}
(
C
)
$
,
\[
H
_
k
^{
\sing
}
(
C
)
:
=
H
_
k
(
N
_{
\oo
}
(
C
))
.
\]
\end{frame}
\end{document}
%%% Local Variables:
...
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