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Leonard Guetta
PhD-presentation
Commits
040e2e4b
Commit
040e2e4b
authored
Jan 24, 2021
by
Leonard Guetta
Browse files
Premier jet fini. J'ai ajouté le pdf au git également.
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040e2e4b
...
...
@@ -15,7 +15,7 @@
%gets rid of footer
%will override 'frame number' instruction above
%comment out to revert to previous/default definitions
\setbeamertemplate
{
footline
}{}
%
\setbeamertemplate{footline}{}
\title
{
Homology of strict
$
\omega
$
-categories
}
%\subtitle{PhD defense}
...
...
@@ -23,15 +23,23 @@
\date
{
PhD defense, 28 January 2021
}
\institute
{
IRIF - Université de Paris
}
\AtBeginSection
[]
{
\begin{frame}
[noframenumbering,plain]
\frametitle
{
Table of Contents
}
\tableofcontents
[currentsection]
\end{frame}
}
\begin{document}
\frame
{
\titlepage
}
\frame
[noframenumbering,plain]
{
\titlepage
}
% \begin{frame}
% \frametitle{Table of Contents}
% \tableofcontents
% \end{frame}
\section
{
The setting
}
\begin{frame}
\frametitle
{
Preliminary conventions
}
In this talk:
...
...
@@ -39,9 +47,12 @@
\item
<2->
$
\oo
$
\nbd
{}
category = strict
$
\omega
$
\nbd
{}
category
\item
<3->
$
n
$
\nbd
{}
category =
$
\oo
$
\nbd
{}
category with only unit cells above
dimension
$
n
$
\item
<4-> the functor
$
n
\Cat
\to
\oo\Cat
$
is an inclusion
\item
<4->
$
1
$
\nbd
{}
category = (small) category
\item
<5-> the functor
$
n
\Cat
\to
\oo\Cat
$
is an inclusion
\end{itemize}
\end{frame}
\end{frame}
%%% oo-categories as spaces
\begin{frame}
...
...
@@ -50,7 +61,7 @@
\[
\Or
\colon
\Psh
{
\Delta
}
\to
\oo\Cat
.
\]
In pictures:
\pause
In pictures:
\[
\Or
_
0
=
\bullet
,
\]
...
...
@@ -159,7 +170,7 @@
where
$
\Ch
$
is the category of non-negatively graded chain complexes.
\pause
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
\pause
Hence,
\[
\overline
{
\kappa
}
\colon
\Ho
(
\Psh
{
\Delta
}
)
\to
\Ho
(
\Ch
)
,
...
...
@@ -396,8 +407,27 @@
\end{block}
\end{frame}
\begin{frame}
\frametitle
{
Polygraphic homology practically
}
TODO
\frametitle
{
Polygraphic homology in practice
}
Let
$
C
$
be a free
$
\oo
$
\nbd
{}
category and write
$
\Sigma
_
k
$
for its set
of generating
$
k
$
\nbd
{}
cells.
\pause
The polygraphic homology of
$
C
$
is the homology of the chain
complex
\[
\mathbb
{
Z
}
\Sigma
_
0
\overset
{
\partial
}{
\longleftarrow
}
\mathbb
{
Z
}
\Sigma
_
1
\overset
{
\partial
}{
\longleftarrow
}
\mathbb
{
Z
}
\Sigma
_
2
\overset
{
\partial
}{
\longleftarrow
}
\cdots
,
\]
\pause
where for
$
x
\in
\Sigma
_
n
$
, we have
\[
\partial
(
x
)=
\text
{
``generators in the target of x''
}
-
\text
{
``generators in the source of x''
}
.
\]
\pause
\begin{exampleblock}
{
Motivation
}
For a
\emph
{
free
}
$
\oo
$
\nbd
{}
category, the polygraphic homology is
\emph
{
a
priori
}
simpler to compute than the singular homology.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle
{
Polygraphic homology vs singular homology
}
...
...
@@ -448,6 +478,7 @@
\pause
This is what I tried to answer in my PhD.
\end{frame}
\section
{
Abstract reformulation
}
\begin{frame}
\frametitle
{
Equivalence of
$
\oo
$
\nbd
{}
categories vs Thomason
equivalences
}
...
...
@@ -554,8 +585,25 @@
for
$
k
=
2
,
3
$
, for any
$
\oo
$
\nbd
{}
category
$
C
$
?
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle
{
An abstract criterion
}
Back on the triangle:
\section
{
Detecting homologically coherent
$
\oo
$
-categories I
}
\begin{frame}
\frametitle
{
Preliminaries: oplax contractile
$
\oo
$
\nbd
{}
categories
}
\begin{block}
{
Definition
}
An
$
\oo
$
\nbd
{}
category
$
C
$
is
\alert
{
oplax contractible
}
if the
canonical morphism
\[
C
\to
\sD
_
0
\]
has an inverse ``up to an oplax transformation''.
\end{block}
\pause
\begin{exampleblock}
{
Lemma
}
Every oplax contractible
$
\oo
$
\nbd
{}
category is homologically
coherent (and has the homotopy type of a point).
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle
{
An abstract criterion to detect homological coherence
}
Back to the triangle:
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\Ho
(
\oo\Cat
^{
\folk
}
)
\ar
[
d,"
\mathcal
{
J
}
"'
]
\ar
[
dr,"
\sH
^{
\pol
}
",""
{
name
=
A,below
}
]
\&
\\
...
...
@@ -581,7 +629,7 @@
Idea: exploit that sometimes it
\emph
{
is
}
an isomorphism.
\end{frame}
\begin{frame}
\frametitle
{
An abstract criterion
}
\begin{frame}
\frametitle
{
An abstract criterion
to detect homological coherence
}
\begin{block}
{
Proposition
}
Let
$
C
$
be an
$
\oo
$
\nbd
{}
category. Suppose that there exists
$
d : I
\to
\oo\Cat
$
such that:
...
...
@@ -591,11 +639,33 @@
\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.
\pause\pause
\pause
Then
$
C
$
is homologically coherent.
\end{block}
%\pause It is our main strategy to detect homologically coherent $\oo$\nbd{}categories
% \pause
% Often, we will use:
\end{frame}
\begin{frame}
\frametitle
{
Easy application: homology of globes and spheres
}
For every
$
n
\geq
0
$
,
$
\sD
_
n
$
is oplax contractible, hence
homologically coherent.
\pause
Moreover, we have
\begin{equation}
\label
{
squaresphere
}
\tag
{$
\ast
$}
\begin{tikzcd}
[ampersand replacement=
\&
]
\sS
_{
n-1
}
\ar
[r,"i_n"]
\ar
[d,"i_n"]
\&
\sD
_
n
\ar
[d]
\\
\sD
_
n
\ar
[r]
\&
\sS
_
n,
\ar
[from=1-1,to=2-2,"\ulcorner",very near end, phantom]
\end{tikzcd}
\end{equation}
(with
$
\sS
_{
-
1
}
=
\emptyset
$
).
\pause
This square is ``folk homotopy cocartesian''
because
$
i
_
n
$
is a cofibration.
\pause
\begin{exampleblock}
{
Exceptional situation:
}
The image by
$
N
_{
\oo
}$
of
\eqref
{
squaresphere
}
in
$
\Psh
{
\Delta
}$
is a
\emph
{
cocartesian
}
square
of monos, hence homotopy cocartesian.
\pause
It follows that square
\eqref
{
squaresphere
}
is ``Thomason homotopy cocartesian''.
\end{exampleblock}
\pause
By an immediate induction,
$
\sS
_
n
$
is homologically coherent
(and has the homotopy type of an
$
n
$
\nbd
{}
sphere).
\end{frame}
\begin{frame}
\frametitle
{
The case of 1-categories
}
\begin{alertblock}
{
Theorem (G. - 2019)
}
...
...
@@ -708,47 +778,30 @@
\[
P
/
{
-
}
: A
\to
\oo\Cat
\]
is
cofibrant.
\hfill
CQFD
\end{itemize}
\end{frame}
\begin{frame}
\frametitle
{
A criterion
}
A variation of the homotopy colimit criterion:
\begin{exampleblock}
{
Proposition
}
Let
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
A
\ar
[
r,"u"
]
\ar
[
d,"v"
]
\&
B
\ar
[
d
]
\\
C
\ar
[
r
]
\&
D
\ar
[
from
=
1
-
1
,to
=
2
-
2
,"
\ulcorner
",very near end,phantom
]
\end
{
tikzcd
}
\]
be a cocartesian square in
$
\oo\Cat
$
. If
\begin{enumerate}
[label=(
\roman*
)]
\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 equivalences,
\end{enumerate}
\pause\pause\pause\pause
then
$
D
$
is homologically coherent.
\end{exampleblock}
\pause
The third condition will usually be the hard one to prove.
\end{frame}
\begin{frame}
\frametitle
{
Easy application: homology of globes and spheres
}
For every
$
n
\geq
0
$
,
$
\sD
_
n
$
is oplax contractible, hence
homologically coherent.
\pause
Moreover, we have
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\sS
_{
n
-
1
}
\ar
[
r,"i
_
n"
]
\ar
[
d,"i
_
n"
]
\&
\sD
_
n
\ar
[
d
]
\\
\sD
_
n
\ar
[
r
]
\&
\sS
_
n,
\ar
[
from
=
1
-
1
,to
=
2
-
2
,"
\ulcorner
",very near end, phantom
]
\end
{
tikzcd
}
\]
(with
$
\sS
_{
-
1
}
=
\emptyset
$
).
\pause
\begin{exampleblock}
{
Perfect situation:
}
The image by
$
N
_{
\oo
}$
of the previous square is a cocartesian square
of monos, hence homotopy cocartesian.
\end{exampleblock}
\pause
By an immediate induction,
$
\sS
_
n
$
is homologically coherent
(and has the homotopy type of an
$
n
$
\nbd
{}
sphere).
\end{frame}
\end{frame}
\section
{
Detecting homologically coherent
$
\oo
$
-categories II
}
% \begin{frame}\frametitle{A criterion}
% A variation of the homotopy colimit criterion:
% \begin{exampleblock}{Proposition}
% Let
% \[
% \begin{tikzcd}[ampersand replacement=\&]
% A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\
% C \ar[r] \& D
% \ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom]
% \end{tikzcd}
% \]
% be a cocartesian square in $\oo\Cat$. If
% \begin{enumerate}[label=(\roman*)]
% \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 equivalences,
% \end{enumerate}
% \pause\pause\pause\pause then $D$ is homologically coherent.
% \end{exampleblock}
% \pause The third condition will usually be the hard one to prove.
% \end{frame}
\begin{frame}
\frametitle
{
2-categories
}
We would like to understand which 2-categories are homologically
coherent.
...
...
@@ -756,7 +809,7 @@
\item
<2-> For simplification, we focus on
\emph
{
free
}
2-categories.
\item
<3->
This boils down to the following
: given a cocartesian square
\item
<3->
Archetypal situation to understand
: given a cocartesian square
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\sS
_
1
\ar
[
d,"i
_
1
"'
]
\ar
[
r
]
\&
P
\ar
[
d
]
\\
...
...
@@ -857,7 +910,7 @@
\end{frame}
\begin{frame}
\frametitle
{
Zoology of 2-categories: Bouquet of spheres
}
\frametitle
{
Zoology of 2-categories: Bouquet
s
of spheres
}
\begin{center}
\scalebox
{
0.85
}{
\begin{tabular}
{
l || c | c
}
...
...
@@ -919,9 +972,54 @@
\pause
This
$
2
$
\nbd
{}
category has the homotopy type of the
\alert
{
torus
}
and
is homologically coherent.
\pause
Not so easy to show ! Idea: show that it is Thomason equivalent to the
monoid
$
(
\mathbb
{
N
}
\times
\mathbb
{
N
}
,
+)
$
(which is not free)
\pause
Idea of proof: show that
$
C
$
is Thomason equivalent to the
monoid
$
(
\mathbb
{
N
}
\times
\mathbb
{
N
}
,
+)
$
(which is not free) and use the
equivalence of polygraphic and singular homologies for monoids.
\pause
Not that easy to carry out properly !
\end{frame}
\begin{frame}
\frametitle
{
Bubbles
}
\begin{block}
{
Definition
}
A
\alert
{
bubble
}
in a
$
2
$
\nbd
{}
category is a non unit
$
2
$
\nbd
{}
cell
$
\alpha
$
whose source and target are units on a
$
0
$
\nbd
{}
cell.
\end{block}
\pause
In pictures:
\[
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
A
\ar
[
r,bend left
=
75
,"
1
_
A",""
{
name
=
A,below
}
]
\ar
[
r,bend
right
=
75
,"
1
_
A"',pos
=
21
/
40
,""
{
name
=
B,above
}
]
\&
A
\ar
[
from
=
A,to
=
B,"
\alpha
",Rightarrow
]
\end
{
tikzcd
}
\text
{
or
}
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
A.
\ar
[
loop,in
=
120
,out
=
60
,distance
=
1
cm,"
\alpha
"',Rightarrow
]
\end
{
tikzcd
}
\]
\pause
\begin{block}
{
Definition
}
A
$
2
$
\nbd
{}
category is
\alert
{
bubble-free
}
if it has no bubbles.
\end{block}
\end{frame}
\begin{frame}
\frametitle
{
The bubble-free conjecture
}
The archetypal example of
\emph
{
non
}
bubble-free
$
2
$
\nbd
{}
category is
the
$
2
$
\nbd
{}
category
$
B
$
from Ara and Maltsiniotis' counter-example.
\pause
In all the examples, the free
$
2
$
\nbd
{}
categories that are homologically
coherent are exactly the bubble-free ones.
\pause
\begin{exampleblock}
{
Conjecture
}
Let
$
C
$
be a free
$
2
$
\nbd
{}
category. It is homologically coherent if and
only if it is bubble-free.
\end{exampleblock}
\end{frame}
\begin{frame}
[noframenumbering,plain]
\begin{center}
Merci pour votre attention !
\end{center}
\end{frame}
\end{document}
%%% Local Variables:
...
...
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