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Leonard Guetta
memoire
Commits
01539760
Commit
01539760
authored
Jan 21, 2021
by
Leonard Guetta
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01539760
...
...
@@ 25,7 +25,7 @@ In this section, we review some homotopical results on free
are the morphisms
$
f : G
\to
G'
$
that are injective on objects and on arrows,
i.e. such that
$
f
_
0
: G
_
0
\to
G
_
0
'
$
and
$
f
_
1
: G
_
1
\to
G'
_
1
$
are injective.
There is a ``underlying reflexive graph'' functor
There is a
n
``underlying reflexive graph'' functor
\[
U :
\Cat
\to
\Rgrph
,
\]
...
...
@@ 239,7 +239,7 @@ From the previous proposition, we deduce the following very useful corollary.
L
(
C
)
\ar
[
r,"L
(
\gamma
)
"
]
&
L
(
D
)
\end
{
tikzcd
}
\]
is
a
Thomason homotopy cocartesian.
is Thomason homotopy cocartesian.
\end{corollary}
\begin{proof}
Since the nerve
$
N
$
induces an equivalence of opprederivators
...
...
@@ 272,11 +272,11 @@ From the previous proposition, we deduce the following very useful corollary.
By working a little more, we obtain the more general result stated
in the proposition below. Let us say that a morphism of reflexive
graphs
$
\alpha
: A
\to
B
$
is
\emph
{
quasiinjective on arrows
}
when
for all
$
f
$
and
$
g
$
arrows
of
$
A
$
, if
for all
arrows
$
f
$
and
$
g
$
of
$
A
$
, if
\[
\alpha
(
f
)=
\alpha
(
g
)
,
\]
then either
$
f
=
g
$
or
$
f
$
and
$
g
$
are both units. In other words,
$
\alpha
$
then either
$
f
=
g
$
,
or
$
f
$
and
$
g
$
are both units. In other words,
$
\alpha
$
never sends a nonunit arrow to a unit arrow and
$
\alpha
$
never identifies two
nonunit arrows. It follows that if
$
\alpha
$
is quasiinjective on arrows and
injective on objects, then it is also injective on arrows and hence, a
...
...
@@ 304,7 +304,7 @@ From the previous proposition, we deduce the following very useful corollary.
L
(
C
)
\ar
[
r,"L
(
\gamma
)
"
]
&
L
(
D
)
\end
{
tikzcd
}
\]
is Thomason homotopy cocartesian
square
.
is Thomason homotopy cocartesian.
\end{proposition}
\begin{proof}
The case where
$
\alpha
$
or
$
\beta
$
is both injective on objects and
...
...
@@ 415,7 +415,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref
{
cor:hmtpysquaregraph
}
.
\end{example}
\begin{remark}
Since
$
i
_
1
:
\sS
_
0
\to
\sD
_
1
$
is a folk cofibration
% , since a Thomason homotopy
Since
$
i
_
1
:
\sS
_
0
\to
\sD
_
1
$
is a folk cofibration
% , since a Thomason homotopy
% cocartesian square in $\Cat$ is also so in $\oo\Cat$
and since every free category is obtained by recursively adding generators
starting from a set of objects (seen as a
$
0
$
category), the previous example
...
...
@@ 456,7 +456,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ar
[
from
=
1

1
,to
=
2

2
,phantom,"
\ulcorner
",very near end
]
\end
{
tikzcd
}
\]
Then, this
above
square is Thomason homotopy cocartesian. Indeed, it
Then, this square is Thomason homotopy cocartesian. Indeed, it
obviously is the image of a cocartesian square in
$
\Rgrph
$
by the
functor
$
L
$
and since the source and target of
$
f
$
are different,
the top map comes from a monomorphism of
$
\Rgrph
$
. Hence, we can
...
...
@@ 744,7 +744,7 @@ In practice, we will use the following corollary.
\ref
{
prop:bisimplicialcocontinuous
}
.
\end{proof}
\section
{
Bisimplicial nerve for 2categories
}
\label
{
section:bisimplicialnerve
}
We shall now describe a ``nerve'' for
$
2
$

categories with values in bisimplicial
We shall now describe a ``nerve'' for
$
2
$
\nbd
{}
categories with values in bisimplicial
sets and recall a few results that shows that this nerve is, in some sense,
equivalent to the nerve defined in
\ref
{
paragr:nerve
}
.
\begin{notation}
...
...
@@ 757,23 +757,23 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
\]
is the set of
$
k
$
simplices of the nerve of
$
C
$
.
\item
[]
Similarly, we write
$
N :
2
\Cat
\to
\Psh
{
\Delta
}$
instead of
$
N
_
2
$
for
the nerve of
$
2
$

categories. This makes sense since the nerve for categories
is the restriction of the nerve for
$
2
$

categories.
\item
[]
For
$
2
$

categories, we refer to the
$
\comp
_
0
$
composition of
$
2
$

cells as the
\emph
{
horizontal composition
}
and the
$
\comp
_
1
$
composition
of
$
2
$

cells as the
\emph
{
vertical composition
}
.
\item
[]
For a
$
2
$

category
$
C
$
and
$
x
$
and
$
y
$
objects of
$
C
$
, we denote by
the nerve of
$
2
$
\nbd
{}
categories. This makes sense since the nerve for categories
is the restriction of the nerve for
$
2
$
\nbd
{}
categories.
\item
[]
For
$
2
$
\nbd
{}
categories, we refer to the
$
\comp
_
0
$
composition of
$
2
$
\nbd
{}
cells as the
\emph
{
horizontal composition
}
and the
$
\comp
_
1
$
composition
of
$
2
$
\nbd
{}
cells as the
\emph
{
vertical composition
}
.
\item
[]
For a
$
2
$
\nbd
{}
category
$
C
$
and
$
x
$
and
$
y
$
objects of
$
C
$
, we denote by
\[
C
(
x,y
)
\]
the category whose objects are the
$
1
$
cells of
$
C
$
with
$
x
$
as source and
$
y
$
as target, and whose arrows are the
$
2
$

cells of
$
C
$
with
$
x
$
as
$
y
$
as target, and whose arrows are the
$
2
$
\nbd
{}
cells of
$
C
$
with
$
x
$
as
$
0
$
source and
$
y
$
as
$
0
$
target. Composition is induced by vertical
composition in
$
C
$
.
\end{itemize}
\end{notation}
\begin{paragr}
Every
$
2
$

category
$
C
$
defines a simplicial object in
$
\Cat
$
,
Every
$
2
$
\nbd
{}
category
$
C
$
defines a simplicial object in
$
\Cat
$
,
\[
S
(
C
)
:
\Delta
^{
\op
}
\to
\Cat
,
\]
where, for each
$
n
\geq
0
$
,
\[
S
_
n
(
C
)
:
=
\coprod
_{
(
x
_
0
,
\cdots
,x
_
n
)
\in
\Ob
(
C
)
^{
\times
(
n
+
1
)
}}
C
(
x
_
0
,x
_
1
)
...
...
@@ 795,7 +795,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
category.
\end{remark}
\begin{definition}
The
\emph
{
bisimplicial nerve
}
of a
$
2
$

category
$
C
$
is the bisimplicial set
The
\emph
{
bisimplicial nerve
}
of a
$
2
$
\nbd
{}
category
$
C
$
is the bisimplicial set
$
\binerve
(
C
)
$
defined as
\[
\binerve
(
C
)
_{
n,m
}
:
=
N
(
S
_
n
(
C
))
_
m,
...
...
@@ 804,7 +804,7 @@ equivalent to the nerve defined in \ref{paragr:nerve}.
\end{definition}
\begin{paragr}
\label
{
paragr:formulabisimplicialnerve
}
In other words, the bisimplicial nerve of
$
C
$
is obtained by ``uncurrying''
the functor
$
NS
(
C
)
:
\Delta
^{
op
}
\to
\Psh
{
\Delta
}$
.
the functor
$
NS
(
C
)
:
\Delta
^{
\
op
}
\to
\Psh
{
\Delta
}$
.
Since the nerve
$
N
$
commutes with products and sums, we obtain the formula
\begin{equation}
\label
{
fomulabinerve
}
...
...
@@ 847,7 +847,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\[
\alpha
=(
\alpha
_
1
,
\alpha
_
2
,
\cdots
,
\alpha
_
k
)
\]
of vertically composable
$
2
$

cells of
$
C
$
, i.e.
\
such that for
of vertically composable
$
2
$
\nbd
{}
cells of
$
C
$
, i.e.
\
such that for
every
$
1
\leq
i
\leq
k

1
$
, we have
\[
\src
(
\alpha
_
i
)=
\trgt
(
\alpha
_{
i
+
1
}
)
.
...
...
@@ 879,7 +879,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
degeneracy operators are induced by the units for the vertical composition.
\end{paragr}
\begin{lemma}
\label
{
lemma:binervehorizontal
}
Let
$
C
$
be a
$
2
$

category. For every
$
n
\geq
0
$
, we have
Let
$
C
$
be a
$
2
$
\nbd
{}
category. For every
$
n
\geq
0
$
, we have
\[
N
(
V
_
m
(
C
))
_
n
=(
\binerve
(
C
))
_{
n,m
}
.
\]
...
...
@@ 897,7 +897,7 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
theory of bisimplicial sets.
\end{paragr}
\begin{lemma}
\label
{
lemma:binervthom
}
A
$
2
$

functor
$
F : C
\to
D
$
is a Thomason equivalence if and only if
A
$
2
$
\nbd
{}
functor
$
F : C
\to
D
$
is a Thomason equivalence if and only if
$
\binerve
(
F
)
$
is a diagonal weak equivalence of bisimplicial sets.
\end{lemma}
\begin{proof}
...
...
@@ 910,9 +910,9 @@ an equivalent definition of the bisimplicial nerve which uses the other directio
\cite
[Théorème 3.13]
{
ara2020comparaison
}
.
\end{proof}
From this lemma, we deduce two useful criteria to detect Thomason equivalences
of
$
2
$

categories.
of
$
2
$
\nbd
{}
categories.
\begin{corollary}
\label
{
cor:criterionThomeqI
}
Let
$
F : C
\to
D
$
be a
$
2
$

functor. If
Let
$
F : C
\to
D
$
be a
$
2
$
\nbd
{}
functor. If
\begin{enumerate}
[label=
\alph*
)]
\item
$
F
_
0
: C
_
0
\to
D
_
0
$
is a bijection,
\end{enumerate}
...
...
@@ 924,10 +924,10 @@ of $2$categories.
\]
induced by
$
F
$
is a Thomason equivalence of
$
1
$
categories,
\end{enumerate}
then
$
F
$
is a Thomason equivalence of
$
2
$

categories.
then
$
F
$
is a Thomason equivalence of
$
2
$
\nbd
{}
categories.
\end{corollary}
\begin{proof}
By definition, for every
$
2
$

category
$
C
$
and every
$
m
\geq
0
$
, we have
By definition, for every
$
2
$
\nbd
{}
category
$
C
$
and every
$
m
\geq
0
$
, we have
\[
(
\binerve
(
C
))
_{
\bullet
,m
}
=
NS
(
C
)
.
\]
...
...
@@ 936,9 +936,9 @@ of $2$categories.
products.
\end{proof}
\begin{corollary}
\label
{
cor:criterionThomeqII
}
Let
$
F : C
\to
D
$
be a
$
2
$

functor. If for every
$
k
\geq
0
$
,
Let
$
F : C
\to
D
$
be a
$
2
$
\nbd
{}
functor. If for every
$
k
\geq
0
$
,
\[
V
_
k
(
F
)
: V
_
k
(
C
)
\to
V
_
k
(
D
)
\]
is a Thomason equivalence of
$
1
$
categories,
then
$
F
$
is a Thomason equivalence of
$
2
$

categories.
then
$
F
$
is a Thomason equivalence of
$
2
$
\nbd
{}
categories.
\end{corollary}
\begin{proof}
From Lemma
\ref
{
lemma:binervehorizontal
}
, we now that for every
$
m
\geq
0
$
,
...
...
@@ 1071,8 +1071,8 @@ of $2$categories.
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 whose target is a
Let
$
n,m
\geq
0
$
. We denote by
$
A
_{
(
m,n
)
}$
the free
$
2
$
\nbd
{}
category with only one
generating
$
2
$
\nbd
{}
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
...
...
@@ 1092,7 +1092,7 @@ of $2$categories.
for
}
0
\leq
i
\leq
m

1
\\
g
_{
j
+
1
}
: B
_
j
\to
B
_{
j
+
1
}
&
\text
{
for
}
0
\leq
j
\leq
n

1
%\\g_1 : A_0 \to B_1 & \\g_{n} : B_{n1} \to A_m &
\end
{
cases
}$
\item
generating
$
2
$

cell:
$
\alpha
: f
_{
m
}
\circ
\cdots
\circ
f
_
1
\Rightarrow
\item
generating
$
2
$
\nbd
{}
cell:
$
\alpha
: f
_{
m
}
\circ
\cdots
\circ
f
_
1
\Rightarrow
g
_
n
\circ
\cdots
\circ
g
_
1
$
.
\end{itemize}
Notice that
$
A
_{
(
1
,
1
)
}$
is nothing but
$
\sD
_
2
$
. We are going to prove that if
...
...
@@ 1113,10 +1113,10 @@ of $2$categories.
\ar
[
from
=
A,to
=
1

1
,Rightarrow,"
\alpha
"
]
\end
{
tikzcd
}
\]
and has many non trivial
$
2
$

cells, such as
$
f
\comp
_
0
\alpha
\comp
_
0
f
$
.
and has many non trivial
$
2
$
\nbd
{}
cells, such as
$
f
\comp
_
0
\alpha
\comp
_
0
f
$
.
Note that when
$
m
=
0
$
\emph
{
and
}
$
n
=
0
$
, then the
$
2
$

category
$
A
_{
(
0
,
0
)
}$
is
nothing but the
$
2
$

category
$
B
^
2
\mathbb
{
N
}$
and we have already seen that it
Note that when
$
m
=
0
$
\emph
{
and
}
$
n
=
0
$
, then the
$
2
$
\nbd
{}
category
$
A
_{
(
0
,
0
)
}$
is
nothing but the
$
2
$
\nbd
{}
category
$
B
^
2
\mathbb
{
N
}$
and we have already seen that it
is
\emph
{
not
}
\good
{}
(see
\ref
{
paragr:bubble
}
).
\end{paragr}
\begin{paragr}
...
...
@@ 1154,9 +1154,9 @@ of $2$categories.
end,"
\ulcorner
"
]
\end
{
tikzcd
}
\]
where
$
\tau
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is the
$
2
$

functor that sends the unique
where
$
\tau
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is the
$
2
$
\nbd
{}
functor that sends the unique
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
$
2
$
\nbd
{}
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
$
\Delta
_
n
\to
A
_{
(
1
,n
)
}$
is also a (couniversal) Thomason equivalence and the square is
...
...
@@ 1176,7 +1176,7 @@ of $2$categories.
end,"
\ulcorner
"
]
\end
{
tikzcd
}
\]
where
$
\sigma
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is the
$
2
$

functor that sends the
where
$
\sigma
:
\Delta
_
1
\to
A
_{
(
1
,
1
)
}$
is the
$
2
$
\nbd
{}
functor that sends the
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.
...
...
@@ 1190,9 +1190,9 @@ of $2$categories.
end,"
\ulcorner
"
]
\end
{
tikzcd
}
\]
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
where
$
\tau
$
is the
$
2
$
\nbd
{}
functor that sends the unique nontrivial
$
1
$
cell of
$
\Delta
_
1
$
to the target of the generating
$
2
$
\nbd
{}
cell of
$
A
_{
(
m,
1
)
}$
. This
$
2
$
\nbd
{}
functor is once again a folk cofibration, but it is
\emph
{
not
}
in general
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
...
...
@@ 1207,7 +1207,7 @@ of $2$categories.
Combined with the result of Paragraph
\ref
{
paragr:bubble
}
, we have proved the
following proposition.
\begin{proposition}
\label
{
prop:classificationAmn
}
Let
$
m,n
\geq
0
$
and consider the
$
2
$

category
$
A
_{
(
m,n
)
}$
. If
$
m
\neq
0
$
or
Let
$
m,n
\geq
0
$
and consider the
$
2
$
\nbd
{}
category
$
A
_{
(
m,n
)
}$
. If
$
m
\neq
0
$
or
$
n
\neq
0
$
, then
$
A
_{
(
m,n
)
}$
is
\good
{}
and has the homotopy type of a point.
If
$
n
=
m
=
0
$
, then
$
A
_{
(
0
,
0
)
}$
is not
\good
{}
and has the homotopy type of a
$
K
(
\mathbb
{
Z
}
,
2
)
$
.
...
...
@@ 1546,7 +1546,7 @@ Let us now get into more sophisticated examples.
\begin{center}
\begin{tabular}
{
 l  c  c 
}
\hline
$
2
$

category
&
\good
{}
?
&
homotopy type
\\
\hline
\hline
$
2
$
\nbd
{}
category
&
\good
{}
?
&
homotopy type
\\
\hline
\hline
{
$
\begin
{
tikzcd
}
\bullet
\ar
[
r,bend
...
...
main.tex
View file @
01539760
...
...
@@ 182,11 +182,11 @@ homotopy theory, polygraphs.
\pagestyle
{
fancy
}
\fancyhf
{}
\fancyfoot
[C]
{
\thepage
}
\fancyhead
[RO]
{
INTRODUCTION
}
\fancyhead
[RO
,LE
]
{
INTRODUCTION
}
\include
{
introduction
}
\fancyhf
{}
\fancyfoot
[C]
{
\thepage
}
\fancyhead
[RO]
{
INTRODUCTION (FRANÇAIS)
}
\fancyhead
[RO
,LE
]
{
INTRODUCTION (FRANÇAIS)
}
\include
{
introduction
_
fr
}
\fancyhf
{}
\fancyfoot
[C]
{
\thepage
}
...
...
memoire.bib
View file @
01539760
...
...
@@ 57,7 +57,7 @@ year={2020}
year
=
{2020}
}
@article
{
ara2019quillen
,
title
=
{A {Q}uillen
T
heorem {B} for strict $\infty$\nbd{}categories}
,
title
=
{A {Q}uillen
{T}
heorem {B} for strict $\infty$\nbd{}categories}
,
author
=
{Ara, Dimitri}
,
journal
=
{Journal of the London Mathematical Society}
,
volume
=
{100}
,
...
...
@@ 67,7 +67,7 @@ year={2020}
publisher
=
{Wiley Online Library}
}
@article
{
batanin1998monoidal
,
title
=
{Monoidal globular categories as a natural environment for the theory of weak
n
\nbd{}categories}
,
title
=
{Monoidal globular categories as a natural environment for the theory of weak
$n$
\nbd{}categories}
,
author
=
{Batanin, Michael A.}
,
journal
=
{Advances in Mathematics}
,
volume
=
{136}
,
...
...
@@ 137,7 +137,7 @@ publisher = "Elsevier"
publisher
=
{Citeseer}
}
@article
{
cisinski2003images
,
title
=
{Images directes cohomologiques dans les cat{\'e}gories de mod
e
les}
,
title
=
{Images directes cohomologiques dans les cat{\'e}gories de mod
{\`e}
les}
,
author
=
{Cisinski, DenisCharles}
,
journal
=
{Annales Math{\'e}matiques Blaise Pascal}
,
volume
=
{10}
,
...
...
@@ 170,7 +170,7 @@ publisher = "Elsevier"
year
=
{1995}
}
@article
{
duskin2002simplicial
,
title
=
{Simplicial matrices and the nerves of weak
n
\nbd{}categories. {I}. Nerves of bicategories}
,
title
=
{Simplicial matrices and the nerves of weak
$n$
\nbd{}categories. {I}. Nerves of bicategories}
,
author
=
{Duskin, John W.}
,
journal
=
{Theory and Applications of Categories}
,
volume
=
{9}
,
...
...
@@ 197,7 +197,7 @@ publisher = "Elsevier"
year
=
{1995}
}
@article
{
eilenberg1954groups
,
title
=
{On the groups {H}(
$
\pi
$
,n), {II}: Methods of computation}
,
title
=
{On the groups
$
{H}(\pi,n)
$
, {II}: Methods of computation}
,
author
=
{Eilenberg, Samuel and Mac Lane, Saunders}
,
journal
=
{Annals of Mathematics}
,
pages
=
{49139}
,
...
...
@@ 221,7 +221,7 @@ publisher = "Elsevier"
publisher
=
{Springer}
}
@article
{
gagna2018strict
,
title
=
{Strict
n
categories and augmented directed complexes model homotopy types}
,
title
=
{Strict
$n$\nbd{}
categories and augmented directed complexes model homotopy types}
,
author
=
{Gagna, Andrea}
,
journal
=
{Advances in Mathematics}
,
volume
=
{331}
,
...
...
@@ 480,7 +480,7 @@ note={In preparation}
publisher
=
{Springer}
}
@article
{
schreiber2013differential
,
title
=
{Differential cohomology in a cohesive inf
inity
topos}
,
title
=
{Differential cohomology in a cohesive
$\
inf
ty$\nbd{}
topos}
,
author
=
{Schreiber, Urs}
,
journal
=
{arXiv preprint arXiv:1310.7930}
,
year
=
{2013}
...
...
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