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Leonard Guetta
memoire
Commits
5e7b0879
Commit
5e7b0879
authored
Dec 27, 2020
by
Leonard Guetta
Browse files
idem
parent
39423129
Changes
1
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Inline
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2cat.tex
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5e7b0879
...
...
@@ 929,7 +929,7 @@ of $2$categories.
\[
(
\binerve
(
C
))
_{
\bullet
,m
}
=
NS
(
C
)
.
\]
The result follows
then
from Lemma
\ref
{
bisimpliciallemma
}
and the fact that the
The result
then
follows from Lemma
\ref
{
bisimpliciallemma
}
and the fact that the
weak equivalences of simplicial sets are stable by coproducts and finite
products.
\end{proof}
...
...
@@ 943,7 +943,7 @@ of $2$categories.
\[
\binerve
(
C
)
_{
\bullet
,m
}
=
N
(
V
_
m
(
C
))
.
\]
The result follows
them
from Lemma
\ref
{
bisimpliciallemma
}
.
The result
then
follows from Lemma
\ref
{
bisimpliciallemma
}
.
\end{proof}
\begin{paragr}
It also follows from Lemma
\ref
{
lemma:binervthom
}
that the bisimplicial nerve
...
...
@@ 990,7 +990,7 @@ of $2$categories.
From Proposition
\ref
{
prop:streetvsbisimplicial
}
, we deduce the proposition
below which contains two useful criteria to detect Thomason homotopy
cocartesian square of
$
2
\Cat
$
.
cocartesian square
s
of
$
2
\Cat
$
.
\end{paragr}
\begin{proposition}
\label
{
prop:critverthorThomhmtpysquare
}
Let
...
...
@@ 1036,13 +1036,14 @@ of $2$categories.
\ar
[
from
=
1

1
,to
=
2

2
,phantom,"
\ulcorner
",very near end
]
\end
{
tikzcd
}
\]
of 2
\nbd
{}
categories. If
$
A
$
,
$
B
$
and
$
C
$
are free and
\good
{}
, if at least
$
u
$
of 2
\nbd
{}
categories. If
$
A
$
,
$
B
$
and
$
C
$
are free and
\good
{}
, if at least
one of
$
u
$
or
$
f
$
is a folk cofibration and if the square is Thomason homotopy
cocartesian, then
$
D
$
is
\good
{}
.
\end{paragr}
\begin{paragr}
Let
$
n,m
\geq
0
$
. We denote by
$
A
_{
(
m,n
)
}$
the free
$
2
$
category with only one
generating
$
2
$
cell whose source is a chain of length
$
m
$
and
its
target a
generating
$
2
$
cell whose source is a chain of length
$
m
$
and
whose
target
is
a
chain of length
$
n
$
:
\[
\underbrace
{
\overbrace
{
\begin
{
tikzcd
}
[
column sep
=
small, ampersand
...
...
@@ 1128,7 +1129,7 @@ of $2$categories.
nontrivial
$
1
$
\nbd
{}
cell of
$
\Delta
_
1
$
to the target of the generating
$
2
$
cell of
$
A
_{
(
1
,
1
)
}$
. It is not hard to check that
$
\tau
$
is strong
deformation retract and thus, a couniversal Thomason equivalence (Lemma
\ref
{
lemma:pushoutstrngdefrtract
}
). Hence, the morphism
$
A
_{
(
1
,
1
)
}
\to
\ref
{
lemma:pushoutstrngdefrtract
}
). Hence, the morphism
$
\Delta
_
n
\to
A
_{
(
1
,n
)
}$
is also a (couniversal) Thomason equivalence and the square is
Thomason homotopy cocartesian (Lemma
\ref
{
lemma:hmtpycocartsquarewe
}
). Now,
the morphism
$
\tau
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is also a folk cofibration and
...
...
@@ 1147,7 +1148,7 @@ of $2$categories.
\end
{
tikzcd
}
\]
where
$
\sigma
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is the
$
2
$
functor that sends the
unique non trivial
$
1
$
\nbd
{}
cell of
$
\Delta
_
1
$
the source of the generating
unique non trivial
$
1
$
\nbd
{}
cell of
$
\Delta
_
1
$
to
the source of the generating
$
2
$
\nbd
{}
cell of
$
A
_{
(
1
,
1
)
}$
, we can prove that
$
A
_{
(
m,
1
)
}$
is
\good
{}
and has
the homotopy type of a point.
...
...
@@ 1163,7 +1164,7 @@ of $2$categories.
where
$
\tau
$
is the
$
2
$
functor that sends the unique nontrivial
$
1
$
cell of
$
\Delta
_
1
$
to the target of the generating
$
2
$
cell of
$
A
_{
(
m,
1
)
}$
. This
$
2
$
functor is once again a folk cofibration, but it is
\emph
{
not
}
in general
a couniversal Thomason equivalence (it would if we had made the hypothesis that
a couniversal Thomason equivalence (it would
be
if we had made the hypothesis that
$
m
\neq
0
$
, but we did not). However, since we made the hypothesis that
$
n
\neq
0
$
, it follows from Lemma
\ref
{
lemma:istrngdefrtract
}
that
$
i :
\Delta
_
1
\to
\Delta
_
n
$
is a couniversal Thomason equivalence. Hence, the previous square
...
...
@@ 1193,7 +1194,7 @@ results at the end of the previous section.
\item
generating
$
2
$
\nbd
{}
cells:
$
\alpha
: f
\Rightarrow
1
_
A
$
,
$
\beta
: g
\Rightarrow
1
_
A
$
.
\end{itemize}
In picture, this gives
In picture
s
, this gives
\[
\begin
{
tikzcd
}
[
column sep
=
huge
]
A
\ar
[
r,bend left
=
75
,"f",""
{
name
=
A,below
}
]
\ar
[
r,bend
...
...
@@ 1204,8 +1205,8 @@ results at the end of the previous section.
\end
{
tikzcd
}
\text
{
or
}
\begin
{
tikzcd
}
A.
\ar
[
loop,in
=
50
,out
=
130
,distance
=
1
.
5
cm,"f"
'
,""
{
name
=
A,below
}
]
\ar
[
loop,in
=
50
,out
=
130
,distance
=
1
.
5
cm,"g",""
{
name
=
B,above
}
]
A.
\ar
[
loop,in
=
50
,out
=
130
,distance
=
1
.
5
cm,"f",""
{
name
=
A,below
}
]
\ar
[
loop,in
=
50
,out
=
130
,distance
=
1
.
5
cm,"g"
'
,""
{
name
=
B,above
}
]
\ar
[
from
=
A,to
=
1

1
,Rightarrow,"
\alpha
"
]
\ar
[
from
=
B,to
=
1

1
,Rightarrow,"
\beta
"
]
\end
{
tikzcd
}
...
...
@@ 1281,7 +1282,7 @@ Let us now get into more sophisticated examples.
\]
and let
$
F : P
\to
P'
$
be the unique
$
2
$
\nbd
{}
functor such that
\begin{itemize}
[label=]
\item
$
F
(
A
)=
A'
$
and
$
F
(
B
'
)=
B
$
,
\item
$
F
(
A
)=
A'
$
and
$
F
(
B
)=
B
'
$
,
\item
$
F
(
f
)=
F
(
g
)=
h
$
,
\item
$
F
(
\alpha
)=
\gamma
$
and
$
F
(
\beta
)=
1
_
h
$
.
\end{itemize}
...
...
@@ 1363,9 +1364,9 @@ Let us now get into more sophisticated examples.
\]
Let
$
H :
\sS
_
2
\to
P''
$
be the unique
$
2
$
\nbd
{}
functor such that:
\begin{itemize}
[label=]
\item
$
H
(
\overline
{
A
}
)=(
\overline
{
B
}
)=
A''
$
,
\item
$
H
(
\overline
{
A
}
)=
H
(
\overline
{
B
}
)=
A''
$
,
\item
$
H
(
i
)=
l
$
and
$
H
(
j
)=
1
_{
A''
}$
,
\item
$
H
(
\delta
)=
\lambda
$
and
$
H
(
\epsilon
)=
\
lambda
$
.
\item
$
H
(
\delta
)=
\lambda
$
and
$
H
(
\epsilon
)=
\
mu
$
.
\end{itemize}
Let us prove that
$
H
$
is a Thomason equivalence using Corollary
\ref
{
cor:criterionThomeqII
}
. In order to do so, we have to compute
$
V
_
k
(
H
)
:
...
...
@@ 1403,13 +1404,13 @@ Let us now get into more sophisticated examples.
and three arrows. More generally, it is a tedious but harmless exercise to
prove that for every
$
k>
0
$
, the category
$
V
_
k
(
P''
)
$
is the
free category on the graph that has one object
$
A''
$
and
$
2
k
+
1
$
arrows which are of
either
one of the following forms:
arrows which are of one of the following forms:
\begin{itemize}
[label=]
\item
$
(
1
_
l,
\cdots
,
1
_
l,
\lambda
,
1
^
2
_{
A''
}
,
\cdots
,
1
^
2
_{
A''
}
)
$
,
\item
$
(
1
_
l,
\cdots
,
1
_
l,
\mu
,
1
^
2
_{
A''
}
,
\cdots
,
1
^
2
_{
A''
}
)
$
,
\item
$
(
1
_
l,
\cdots
,
1
_
l
)
$
.
\end{itemize}
Once again, the functor
$
V
_
k
(
H
)
$
comes from a morphism
a
reflexive graphs and
Once again, the functor
$
V
_
k
(
H
)
$
comes from a morphism
of
reflexive graphs and
is obtained by ``killing the generator
$
(
1
_
j,
\cdots
,
1
_
j
)
$
''. Hence, it is a
Thomason equivalence and thus, so is
$
H
$
. This proves that
$
P''
$
has the
homotopy type of
$
\sS
_
2
$
.
...
...
@@ 1490,7 +1491,7 @@ Let us now get into more sophisticated examples.
Notice that we have
$
F
\circ
G
=
\mathrm
{
id
}_{
P'
}$
, which means that
$
P'
$
is a
retract of
$
P
$
. In particular,
$
\sH
^{
\sing
}
(
P
)
$
is a retract of
$
\sH
^{
\sing
}
(
P'
)
$
and since
$
P'
$
has the homotopy type of a
$
K
(
\mathbb
{
Z
}
,
2
)
$
(see
\ref
{
paragr:bubble
}
), this proves that
$
P
$
ha
ve
$
K
(
\mathbb
{
Z
}
,
2
)
$
(see
\ref
{
paragr:bubble
}
), this proves that
$
P
$
ha
s
nontrivial singular homology groups in all even dimension. But since it is a
free
$
2
$
\nbd
{}
category, all its polygraphic homology groups are trivial strictly above
dimension
$
2
$
, which means that
$
P
$
is
\emph
{
not
}
\good
{}
.
...
...
@@ 1703,7 +1704,7 @@ Now let $\sS_2$ be labelled as
\end
{
tikzcd
}
\]
Let us prove that this
$
2
$
\nbd
{}
category is
\good
{}
. Let
$
P
_
0
$
be the
sub
$
1
$
category of
$
P
$
spanned by
$
A
$
,
$
B
$
and
$
g
$
, let
$
P
_
1
$
be the
sub
$
1
$
\nbd
{}
category of
$
P
$
spanned by
$
A
$
,
$
B
$
and
$
g
$
, let
$
P
_
1
$
be the
sub
$
2
$
\nbd
{}
category of
$
P
$
spanned by
$
A
$
,
$
B
$
,
$
g
$
,
$
h
$
,
$
\gamma
$
and
$
\delta
$
and let
$
P
_
2
$
be the sub
$
2
$
\nbd
{}
category of
$
P
$
spanned by
$
A
$
,
$
B
$
,
$
f
$
,
$
g
$
,
$
\alpha
$
and
$
\beta
$
. The
$
2
$
\nbd
{}
categories
$
P
_
1
$
and
$
P
_
2
$
...
...
@@ 1831,7 +1832,7 @@ Now let $\sS_2$ be labelled as
A
\ar
[
r,"f",shift left
]
\ar
[
r,"g"',shift right
]
&
B
\ar
[
r,"h",shift left
]
\ar
[
r,"i"',shift right
]
&
C.
\end
{
tikzcd
}
\]
This implies that square
\ref
{
squarebouquetvertical
}
is cocartesian for
$
k
=
0
$
and in
This implies that square
\
eq
ref
{
squarebouquetvertical
}
is cocartesian for
$
k
=
0
$
and in
virtue of Corollary
\ref
{
cor:hmtpysquaregraph
}
it is also Thomason homotopy
cocartesian for this value of
$
k
$
. For
$
k>
0
$
, the category
$
V
_
k
(
P'
)
$
has two objects
$
A
$
and
$
B
$
and an arrow
$
A
\to
B
$
is a
$
k
$
\nbd
{}
tuple of one of the following form
...
...
@@ 1861,7 +1862,7 @@ Now let $\sS_2$ be labelled as
\item
$
(
1
_
h,
\cdots
,
1
_
h
)
$
,
\item
$
(
1
_
i,
\cdots
,
1
_
i
)
$
,
\end{itemize}
and with no other arrows. This implies that square
\ref
{
squarebouquetvertical
}
and with no other arrows. This implies that square
\
eq
ref
{
squarebouquetvertical
}
is cocartesian for every
$
k>
0
$
and in virtue of Corollary
\ref
{
cor:hmtpysquaregraph
}
it is also Thomason homotopy cocartesian for these values of
$
k
$
.
Altogether, this proves that square
\eqref
{
squarebouquetbis
}
is Thomason
...
...
@@ 1878,7 +1879,7 @@ homotopy type of the torus.
\item
generating
$
2
$
\nbd
{}
cell:
$
\alpha
: g
\comp
_
0
f
\Rightarrow
f
\comp
_
0
g
$
.
\end{itemize}
In picture, this gives:
In picture
s
, this gives:
\[
\begin
{
tikzcd
}
A
\ar
[
r,"f"
]
\ar
[
d,"g"'
]
&
A
\ar
[
d,"g"
]
\\
...
...
@@ 1907,7 +1908,7 @@ homotopy type of the torus.
\item
$
F
(
\alpha
)=
1
_{
(
1
,
1
)
}$
.
\end{itemize}
This last equation makes sense since
$
(
1
,
1
)=(
0
,
1
)+(
1
,
0
)=(
1
,
0
)+(
0
,
1
)
$
. For every
$
1
$
\nbd
{}
cell
$
w
$
of
$
P
$
(encoded as a finite words
o
n the alphabet
$
\{
f,g
\}
$
)
$
1
$
\nbd
{}
cell
$
w
$
of
$
P
$
(encoded as a finite words
i
n the alphabet
$
\{
f,g
\}
$
)
such that
$
f
$
appears
$
n
$
times and
$
g
$
appears
$
m
$
times, we have
$
F
(
w
)=(
n,m
)
$
. Let us prove that
$
F
$
is a Thomason equivalence using a dual of
\cite
[Corollaire 5.26]
{
ara2020theoreme
}
(see Remark 5.20 of op.
\
cit.). If we
...
...
@@ 2044,7 +2045,7 @@ homotopy type of the torus.
\end{definition}
\begin{paragr}
The archetypal example of a
$
2
$
\nbd
{}
category that is
\emph
{
not
}
bubblefree
is
$
B
^
2
\mathbb
{
N
}$
. Another nonbubble
$
2
$
\nbd
{}
category i
f
the one from Paragraph
\ref
{
paragr:anothercounterexample
}
. It is
is
$
B
^
2
\mathbb
{
N
}$
. Another nonbubble
$
2
$
\nbd
{}
category i
s
the one from Paragraph
\ref
{
paragr:anothercounterexample
}
. It is
remarkable that of all the free
$
2
$
\nbd
{}
categories we have seen so far, these
are the only examples that are non
\good
{}
. This motivates the following conjecture.
\end{paragr}
...
...
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