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Leonard Guetta
PhD-presentation
Commits
e07cf015
Commit
e07cf015
authored
Jan 18, 2021
by
Leonard Guetta
Browse files
qsdf
parent
452b3570
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pres.tex
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e07cf015
...
...
@@ -120,15 +120,16 @@
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
%\pause $\W^{\Th}$:=class of Thomason equivalences.
\pause
By definition, the
nerve functor induces
\[
\overline
{
N
_{
\oo
}}
:
\Ho
(
\oo\Cat
^{
\Th
}
)
\to
\Ho
(
\Psh
{
\Delta
}
)
.
\overline
{
N
_{
\oo
}}
:
\Ho
(
\oo\Cat
^{
\Th
}
)
\to
\Ho
(
\Psh
{
\Delta
}
)
,
\]
W
here:
w
here:
\begin{itemize}
[label=
$
\bullet
$
]
\item
$
\Ho
(
\oo\Cat
^{
\Th
}
)
$
is the localization of
$
\oo\Cat
$
with respect to
$
\W
^{
\Th
}$
,
the Thomason equivalences
,
\item
$
\Ho
(
\Psh
{
\Delta
}
)
$
is the localization of
$
\Psh
{
\Delta
}$
with
respect to weak equivalences of simplicial sets.
\end{itemize}
...
...
@@ -143,7 +144,7 @@
\end{alertblock}
\pause
In other words:
\begin{center}
Homotopy theory of
$
\oo
$
\nbd
{}
categories induced by Thomason equivalences
\\
$
\cong
$
\\
Homotopy theory of spaces
Homotopy theory of
$
\oo
$
\nbd
{}
categories induced by Thomason equivalences
\\
$
\cong
$
\\
Homotopy theory of spaces
.
\end{center}
\end{frame}
\begin{frame}
...
...
@@ -152,7 +153,7 @@
\[
\kappa
\colon
\Psh
{
\Delta
}
\to
\Ch
,
\]
W
here
$
\Ch
$
is the category of non-negatively graded chain complexes.
\pause
w
here
$
\Ch
$
is the category of non-negatively graded chain complexes.
\pause
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
...
...
@@ -173,12 +174,65 @@
\]
\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
)
$
,
In pratice, 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
)
)
$
\[
H
_
k
^{
\sing
}
(
C
)
:
=
H
_
k
(
N
_{
\oo
}
(
C
))
.
\begin
{
aligned
}
H
_
k
^{
\sing
}
(
C
)
&
:
=
H
_
k
(
\sH
^{
\sing
}
(
C
))
\\
&
=
H
_
k
(
\kappa
(
N
_{
\oo
}
(
C
)))
.
\end
{
aligned
}
\]
\end{frame}
\end{frame}
\begin{frame}
\frametitle
{
Equivalence of
$
\oo
$
\nbd
{}
categories and the folk model
structure
}
\begin{block}
{
Definition
}
Let
$
C
$
be an
$
\oo
$
\nbd
{}
category and
$
x,y
$
two
$
n
$
\nbd
{}
cells of
$
C
$
.
We say that
\alert
{$
x
\sim
_{
\oo
}
y
$}
if there exist
$
(
n
+
1
)
$
\nbd
{}
cells
$
r : x
\to
y
$
and
$
\overline
{
r
}
: y
\to
x
$
such that
\[
r
\comp
_
n
\overline
{
r
}
\sim
_{
\oo
}
1
_
y
\text
{
and
}
\overline
{
r
}
\comp
_
n
r
\sim
_{
\oo
}
1
_
x.
\]
(This definition is co-inductive.)
\end{block}
\pause
\begin{exampleblock}
{
Example 1
}
Let
$
x
$
and
$
y
$
be two objects of a 1-category. We have
$
x
\sim
_{
\oo
}
y
$
if and only if
$
x
$
and
$
y
$
are
\alert
{
isomorphic
}
.
\end{exampleblock}
\pause
\begin{exampleblock}
{
Example 2
}
Let
$
x
$
and
$
y
$
be two objects of a
$
2
$
\nbd
{}
category. We have
$
x
\sim
_{
\oo
}
y
$
if and only if
$
x
$
and
$
y
$
are
\alert
{
equivalent
}
.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle
{
Equivalence of
$
\oo
$
\nbd
{}
categories and the folk model
structure
}
\begin{block}
{
Definition
}
A morphism
$
f : C
\to
D
$
of
$
\oo\Cat
$
is an
\alert
{
equivalence of
$
\oo
$
\nbd
{}
categories
}
if:
\begin{itemize}
[label=
$
\bullet
$
]
\item
<2-> for every
$
0
$
\nbd
{}
cell
$
y
$
of
$
D
$
, there exists a
$
0
$
\nbd
{}
cell
$
x
$
of
$
C
$
such that
\[
f
(
x
)
\sim
_{
\oo
}
y,
\]
\item
<3-> for every parallel
$
n
$
\nbd
{}
cells
$
x
$
and
$
x'
$
of
$
C
$
and for
every
$
(
n
+
1
)
$
\nbd
{}
cell
$
\beta
: f
(
x
)
\to
f
(
x'
)
$
of
$
D
$
, there
exists an
$
(
n
+
1
)
$
\nbd
{}
cell
$
\alpha
: x
\to
x'
$
of
$
C
$
such that
\[
f
(
\alpha
)
\sim
_{
\oo
}
\beta
.
\]
\end{itemize}
\end{block}
\pause\pause\pause
When
$
C
$
and
$
D
$
are (1-)categories, we recover the usual notion of
equivalence of categories.
\end{frame}
\end{document}
%%% Local Variables:
%%% mode: latex
...
...
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