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Leonard Guetta
PhD-presentation
Commits
6253f148
Commit
6253f148
authored
Jan 19, 2021
by
Leonard Guetta
Browse files
slowly but surely
parent
e07cf015
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pres.tex
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6253f148
\documentclass
{
beamer
}
\documentclass
[handout]
{
beamer
}
%\usepackage[utf8]{inputenc}
\usepackage
{
mystyle
}
...
...
@@ -166,15 +166,15 @@
\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
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, the
$
k
$
\nbd
{}
th singular homology group of an
$
\oo
$
\nbd
{}
category
$
C
$
is
In pra
c
tice, the
$
k
$
\nbd
{}
th singular homology group of an
$
\oo
$
\nbd
{}
category
$
C
$
is
the
$
k
$
\nbd
{}
th homology group of
$
\kappa
(
N
_{
\oo
}
(
C
))
$
\[
\begin
{
aligned
}
...
...
@@ -233,7 +233,122 @@
equivalence of categories.
\end{frame}
\end{document}
\begin{frame}
\frametitle
{
Equivalence of
$
\oo
$
\nbd
{}
categories and the folk model
structure
}
For every
$
n
\in
\mathbb
{
N
}$
,
\begin{itemize}
[label=-]
\item
<2-> let
$
\sD
_
n
$
be the ``
$
n
$
\nbd
{}
globe''
$
\oo
$
\nbd
{}
category:
\begin{columns}
\column
{
0.5
\textwidth
}
\pause\pause
\[
\sD
_
0
=
\bullet
,
\]
\pause
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\sD
_
1
=
\bullet
\to
\bullet
,
\end
{
tikzcd
}
\]
\column
{
0.5
\textwidth
}
\pause
\[
\sD
_
2
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet\ar
[
r,bend left
=
50
,""
{
name
=
A,below
}
]
\ar
[
r,bend
right
=
50
,""
{
name
=
B,above
}
]
\&
\bullet
,
\ar
[
from
=
A,to
=
B,Rightarrow
]
\end
{
tikzcd
}
\]
\pause
\[
\sD
_
3
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet
\ar
[
r,bend left
=
50
,""
{
name
=
U,below,near
start
}
,""
{
name
=
V,below,near end
}
]
\ar
[
r,bend
right
=
50
,""
{
name
=
D,near start
}
,""
{
name
=
E,near end
}
]
\&\bullet
,
\ar
[
Rightarrow, from
=
U,to
=
D, bend right,""
{
name
=
L,above
}
]
\ar
[
Rightarrow, from
=
V,to
=
E, bend left,""
{
name
=
R,above
}
]
\arrow
[
phantom,"
\Rrightarrow
",from
=
L,to
=
R
]
\end
{
tikzcd
}
\]
\end{columns}
\begin{center}
etc.
\end{center}
\item
<7-> let
$
\sS
_
n
$
be the ``
$
n
$
\nbd
{}
sphere''
$
\oo
$
\nbd
{}
category:
\begin{columns}
\column
{
0.5
\textwidth
}
\pause\pause
\[
\sS
_
0
=
\emptyset
,
\]
\pause
\[
\sS
_
1
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet
\&
\bullet
\end
{
tikzcd
}
\]
\column
{
0.5
\textwidth
}
\pause
\[
\sS
_
2
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet\ar
[
r,bend left
=
50
]
\ar
[
r,bend right
=
50
]
\&
\bullet
\end
{
tikzcd
}
\]
\pause
\[
\sS
_
3
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet
\ar
[
r,bend left
=
50
,""
{
name
=
U,below,near
start
}
,""
{
name
=
V,below,near end
}
]
\ar
[
r,bend
right
=
50
,""
{
name
=
D,near start
}
,""
{
name
=
E,near end
}
]
\&\bullet
.
\ar
[
Rightarrow, from
=
U,to
=
D, bend right,""
{
name
=
L,above
}
]
\ar
[
Rightarrow, from
=
V,to
=
E, bend left,""
{
name
=
R,above
}
]
\end
{
tikzcd
}
\]
\end{columns}
\begin{center}
etc.
\end{center}
\item
<12-> let
$
i
_
n :
\sS
_
n
\to
\sD
_
n
$
be the ``boundary'' inclusion.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle
{
Equivalence of
$
\oo
$
\nbd
{}
categories and the folk model
structure
}
\begin{alertblock}
{
Theorem (Lafont,Métayer,Worytkiewicz - 2009)
}
There exists a model structure on
$
\oo\Cat
$
such that:
\begin{itemize}
[label=
$
\bullet
$
]
\item
the weak equivalences are the equivalences of
$
\oo
$
\nbd
{}
categories,
\item
the set
$
\{
i
_
n :
\sS
_
n
\to
\sD
_
n
\vert
n
\in
\mathbb
{
N
}
\}
$
is
a set of generating cofibrations.
\end{itemize}
\end{alertblock}
\pause
It is known as the
\alert
{
folk model structure
}
on
$
\oo\Cat
$
.
\pause
\begin{alertblock}
{
Theorem (Métayer - 2008)
}
The cofibrant objects of the folk model structure are exactly the
$
\oo
$
\nbd
{}
categories that are free on a polygraph.
\end{alertblock}
\end{frame}
\begin{frame}
\frametitle
{
Polygraphs
}
Terminological convention:
\begin{center}
free
$
\oo
$
\nbd
{}
category =
$
\oo
$
\nbd
{}
category
free on a polygraph.
\end{center}
\end{frame}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
...
...
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