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Leonard Guetta
PhD-presentation
Commits
2a65308a
Commit
2a65308a
authored
Jan 22, 2021
by
Leonard Guetta
Browse files
very very slowly but surely
parent
05c221bd
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pres.tex
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2a65308a
...
...
@@ -499,6 +499,7 @@
\]
which is natural in
$
C
$
. We refer to it as the
\alert
{
canonical
comparison map
}
.
\pause
\begin{block}
{
Definition
}
An
$
\oo
$
\nbd
{}
category
$
C
$
is
\alert
{
homogically coherent
}
if the
map
...
...
@@ -584,13 +585,38 @@
\begin{block}
{
Proposition
}
Let
$
C
$
be an
$
\oo
$
\nbd
{}
category. Suppose that there exists
$
d : I
\to
\oo\Cat
$
such that:
% \begin{enumerate}[label=($\roman$)]
% \item \[
% \hocolim^{\pol}_I(d)\simeq \hocolim^{\Th}_I(d) \simeq C
% \]
% \end{enumerate}
\end{block}
\end{frame}
\begin{enumerate}
[label=(
\roman*
)]
\item
<2->
$
\displaystyle\hocolim
^{
\pol
}_
I
(
d
)
\simeq
\hocolim
^{
\Th
}_
I
(
d
)
\simeq
C,
$
\item
<3-> for each
$
i
\in
\Ob
(
I
)
$
, the
$
\oo
$
\nbd
{}
category
$
d
(
i
)
$
is
homologically coherent.
\end{enumerate}
\pause\pause
Then
$
C
$
is homologically coherent.
\end{block}
% \pause
% Often, we will use:
\end{frame}
\begin{frame}
\frametitle
{
In practice
}
\begin{exampleblock}
{
Corollary
}
Let
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
A
\ar
[
r,"u"
]
\ar
[
d,"v"
]
\&
B
\ar
[
d
]
\\
C
\ar
[
r
]
\&
D
\ar
[
from
=
1
-
1
,to
=
2
-
2
,"
\ulcorner
",phantom
]
\end
{
tikzcd
}
\]
be a cocartesian square in
$
\oo\Cat
$
. If
\begin{itemize}
[label=
$
\bullet
$
]
\item
<2->
$
A
$
,
$
B
$
and
$
C
$
are homologically coherent,
\item
<3->
$
u
$
or
$
v
$
is a folk cofibration,
\item
<4-> the square is homotopy cocartesian w.r.t Thomason equivalence,
\end{itemize}
then
$
D
$
is homologically coherent.
\end{exampleblock}
\end{frame}
\end{document}
%%% Local Variables:
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