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Leonard Guetta
PhD-presentation
Commits
9f16f3b1
Commit
9f16f3b1
authored
Jan 25, 2021
by
Leonard Guetta
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\documentclass
{
beamer
}
\documentclass
[handout]
{
beamer
}
%\usepackage[utf8]{inputenc}
\usepackage
{
mystyle
}
...
...
@@ -154,10 +154,10 @@
\]
where:
\begin{itemize}
[label=
$
\bullet
$
]
\item
$
\Ho
(
\oo\Cat
^{
\Th
}
)
$
is the localization of
$
\oo\Cat
$
w
ith respect to
\item
$
\Ho
(
\oo\Cat
^{
\Th
}
)
$
is the localization of
$
\oo\Cat
$
w
.r.t
the Thomason equivalences,
\item
$
\Ho
(
\Psh
{
\Delta
}
)
$
is the localization of
$
\Psh
{
\Delta
}$
w
ith
respect to
weak equivalences of simplicial sets.
\item
$
\Ho
(
\Psh
{
\Delta
}
)
$
is the localization of
$
\Psh
{
\Delta
}$
w
.r.t
the
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).
...
...
@@ -523,7 +523,7 @@
\pause
Hence, both
$
\sH
^{
\pol
}$
and
$
\sH
^{
\sing
}$
are obtained as left derived
functors of
$
\lambda
$
but not w.r.t the same class of weak equivalences.
\begin{exampleblock}
{
Corollary
}
\begin{exampleblock}
{
Corollary
(abstract non-sense)
}
There is a canonical natural transformation
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
...
...
@@ -544,7 +544,7 @@
comparison map
}
.
\pause
\begin{block}
{
Definition
}
An
$
\oo
$
\nbd
{}
category
$
C
$
is
\alert
{
homogically coherent
}
if the
An
$
\oo
$
\nbd
{}
category
$
C
$
is
\alert
{
homo
lo
gically coherent
}
if the
map
\[
\pi
_
C :
\sH
^{
\sing
}
(
C
)
\to
\sH
^{
\folk
}
(
C
)
...
...
@@ -552,7 +552,7 @@
is an isomorphism.
\end{block}
\pause
Goal: Understand which
$
\oo
$
\nbd
{}
categories are homogically coherent.
Goal: Understand which
$
\oo
$
\nbd
{}
categories are homo
lo
gically coherent.
\end{frame}
\begin{frame}
\frametitle
{
Polygraphic homology is not homotopical
}
...
...
@@ -572,7 +572,7 @@
\pause
\begin{exampleblock}
{
New slogan
}
The polygraphic homology is a
way of computing the singular homology of homogically coherent
way of computing the singular homology of homo
lo
gically coherent
$
\oo
$
\nbd
{}
categories.
\end{exampleblock}
\end{frame}
...
...
@@ -597,7 +597,7 @@
for
$
k
=
2
,
3
$
, for any
$
\oo
$
\nbd
{}
category
$
C
$
?
\end{exampleblock}
\end{frame}
\section
{
Detecting homologically coherent
$
\oo
$
-categories
I
}
\section
{
Detecting homologically coherent
$
\oo
$
-categories
}
\begin{frame}
\frametitle
{
Preliminaries: oplax contractile
$
\oo
$
\nbd
{}
categories
}
\begin{block}
{
Definition
}
...
...
@@ -633,7 +633,7 @@
In other words, for a diagram
$
d : I
\to
\oo\Cat
$
, the
canonical map
\[
\hocolim
_{
I
}^{
\
folk
}
(
d
)
\to
\hocolim
_{
I
}^{
\
Th
}
(
d
)
\hocolim
_{
I
}^{
\
Th
}
(
d
)
\to
\hocolim
_{
I
}^{
\
folk
}
(
d
)
\]
is not an isomorphism in general.
\pause
...
...
@@ -710,7 +710,7 @@
A
}^{
\folk
}
A
/
a
\simeq
\colim
_{
a
\in
A
}
A
/
a
\simeq
A.
\]
\pause
Let
$
f : P
\longrightarrow
A
$
be a folk cofibrant
re
solution
of
$
A
$
.
\pause
(Note that
$
P
$
is free but need not be a
re
placement
of
$
A
$
.
\pause
(Note that
$
P
$
is free but need not be a
$
1
$
\nbd
{}
category).
% \pause How do we prove that ? Let us take a detour.
% \pause In order to do
...
...
@@ -791,7 +791,7 @@
cofibrant.
\hfill
CQFD
\end{itemize}
\end{frame}
\section
{
Detecting homologically coherent
$
\oo
$
-categories
II
}
\section
{
The case of
$
2
$
-categories
}
% \begin{frame}\frametitle{A criterion}
% A variation of the homotopy colimit criterion:
% \begin{exampleblock}{Proposition}
...
...
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