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Leonard Guetta
memoire
Commits
beac890d
Commit
beac890d
authored
Oct 09, 2020
by
Leonard Guetta
Browse files
gotta go
parent
777544a7
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2
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2cat.tex
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beac890d
...
...
@@ -1168,25 +1168,48 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}.
\end{paragr}
\begin{paragr}
Let
$
P
$
be the free
$
2
$
\nbd
{}
category defined as follows:
\begin{itemize}
[label=-]
\item
generating
$
0
$
\nbd
{}
cell:
$
A
$
,
\item
generating
$
1
$
\nbd
{}
cell:
$
f : A
\to
A
$
,
\item
generating
$
2
$
\nbd
{}
cells:
$
\alpha
: f
\Rightarrow
1
_
A
$
and
$
\beta
: f
\Rightarrow
1
_
A
$
.
\end{itemize}
In picture, this gives
\begin{itemize}
[label=-]
\item
generating
$
0
$
\nbd
{}
cell
s
:
$
A
$
and
$
B
$
,
\item
generating
$
1
$
\nbd
{}
cell
s
:
$
f
,g
: A
\to
B
$
,
\item
generating
$
2
$
\nbd
{}
cells:
$
\alpha
,
\beta
,
\gamma
: f
\to
g
$
.
\end{itemize}
In picture, this gives
\[
\begin
{
tikzcd
}
[
column sep
=
huge
]
A
\ar
[
r,bend left
=
75
,"f",""
{
name
=
A,below
}
]
\ar
[
r,bend right
=
75
,"f"',""
{
name
=
B,above
}
]
\ar
[
r,"
1
_
A" pos
=
1
/
3
,""
{
name
=
C,above
}
,""
{
name
=
D,below
}
]
&
A
\ar
[
from
=
A,to
=
C,Rightarrow,"
\alpha
"
]
\ar
[
from
=
B,to
=
D,"
\beta
" pos
=
9
/
20
,Rightarrow
]
\end
{
tikzcd
}
\qquad
\quad
\text
{
or
}
\begin
{
tikzcd
}
A.
\ar
[
loop,in
=
30
,out
=
150
,distance
=
3
cm,"f",""
{
name
=
A,below
}
]
\ar
[
from
=
A,to
=
1
-
1
,bend right,Rightarrow,"
\alpha
"'
]
\ar
[
from
=
A,to
=
1
-
1
,bend left,Rightarrow,"
\beta
"
]
\begin
{
tikzcd
}
[
column sep
=
huge
]
A
\ar
[
r,bend left
=
75
,"f",""
{
name
=
A,below,pos
=
8
/
20
}
,""
{
name
=
C,below,pos
=
1
/
2
}
,""
{
name
=
E,below,pos
=
12
/
20
}
]
\ar
[
r,bend
right
=
75
,"g"',""
{
name
=
B,above,pos
=
8
/
20
}
,""
{
name
=
D,above,pos
=
1
/
2
}
,""
{
name
=
F,above,pos
=
12
/
20
}
]
&
B.
\ar
[
from
=
A,to
=
B,Rightarrow,"
\alpha
"',bend right
]
\ar
[
from
=
C,to
=
D,Rightarrow,"
\beta
"
]
\ar
[
from
=
E,to
=
F,Rightarrow,"
\gamma
",bend left
]
\end
{
tikzcd
}
\]
Now let
$
P'
$
be the sub-
$
2
$
\nbd
{}
category of
$
P
$
spanned by
$
A
$
,
$
B
$
,
$
\alpha
$
and
$
beta
$
, and let
$
P''
$
be the sub-
$
2
$
\nbd
{}
category of
$
P
$
spanned by
$
A
$
,
$
B
$
,
$
\beta
$
and
$
\gamma
$
. These
$
2
$
\nbd
{}
categories are
simply copies of
$
\sS
_
2
$
. Notice that we have a cocartesian
square
\begin{equation}
\label
{
square:bouquet
}
\begin{tikzcd}
\sD
_
1
\ar
[r,"\langle \beta \rangle"]
\ar
[d,"\langle \beta \rangle"]
&
P'
\ar
[d]
\\
P''
\ar
[r]
&
P,
\ar
[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\end{equation}
and by reasoning as in the proof of Lemma
\ref
{
lemma:squarenerve
}
, one can
show that the square induced by the nerve
\[
\begin
{
tikzcd
}
N
_{
\oo
}
(
\sD
_
1
)
\ar
[
r,"
\langle
\beta
\rangle
"
]
\ar
[
d,"
\langle
\beta
\rangle
"
]
&
N
_{
\oo
}
(
P'
)
\ar
[
d
]
\\
N
_{
\oo
}
(
P''
)
\ar
[
r
]
&
N
_{
\oo
}
(
P
)
\end
{
tikzcd
}
\]
We shall now prove that
$
P
$
is
\good
{}
using the techniques introduced in
is also cocartesian. This proves that square
\ref
{
square:bouquet
}
is
Thomason homotopy cocartesian
\todo
{
détailler?
}
and in particular that
$
P
$
has the homotopy
type of a bouqet of two
$
2
$
\nbd
{}
spheres. Since
$
\sD
_
1
$
,
$
P'
$
and
$
P''
$
are
free and
\good
{}
and since
$
\langle
\beta
\rangle
:
\sD
_
1
\to
P'
$
and
$
\langle
\beta
\rangle
:
\sD
_
1
\to
P'
$
, this proves that
$
P
$
is
\good
{}
.
\end{paragr}
\section
{
The ``Bubble-free'' conjecture
}
\begin{definition}
...
...
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