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Leonard Guetta
memoire
Commits
b5de6425
Commit
b5de6425
authored
Jan 09, 2020
by
Leonard Guetta
Browse files
First commit. The writing of my dissertation has officially begun.
parents
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3
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main.tex
0 → 100644
View file @
b5de6425
\documentclass
[12pt,a4paper]
{
report
}
\usepackage
{
mystyle
}
\title
{
Homology of
$
\omega
$
-categories
}
\author
{
Léonard Guetta
}
\begin{document}
\maketitle
\include
{
omegacat
}
\end{document}
mystyle.sty
0 → 100644
View file @
b5de6425
\ProvidesPackage
{
mystyle
}
% Layout
\usepackage
[utf8]
{
inputenc
}
% Maths packages
\usepackage
{
amsmath,amssymb,amsthm
}
\usepackage
{
tikz-cd
}
% Theorems
\theoremstyle
{
plain
}
\newtheorem
{
theorem
}{
Theorem
}
[section]
\newtheorem
{
lemma
}
[theorem]
{
Lemma
}
\newtheorem
{
scholia
}
[theorem]
{
Scholia
}
\newtheorem
{
proposition
}
[theorem]
{
Proposition
}
\newtheorem
{
fact
}
[theorem]
{
Fact
}
\newtheorem
{
corollary
}
[theorem]
{
Corollary
}
\theoremstyle
{
definition
}
\newtheorem
{
definition
}
[theorem]
{
Definition
}
\newtheorem
{
remark
}
[theorem]
{
Remark
}
\newtheorem*
{
convention
}{
Convention
}
\newtheorem*
{
notation
}{
Notation
}
\newtheorem
{
paragr
}
[theorem]
{}
\newtheorem
{
example
}
[theorem]
{
Example
}
\theoremstyle
{
plain
}
\newtheorem*
{
theorem*
}{
Theorem
}
% Unnumbered theorem
\newtheorem
{
theoremintro
}{
Theorem
}
% Theorem environment for the introduction. Numbering not following the sections
\theoremstyle
{
definition
}
\newtheorem*
{
definition*
}{
Definition
}
%Unnumbered definition
%%%%%% MACROS %%%%%%%
% oo-categories
\newcommand
{
\oo
}{
\omega
}
\newcommand
{
\Cat
}{
\mathbf
{
Cat
}}
\newcommand
{
\ooCat
}{
\mathbf
{
\oo
Cat
}}
\newcommand
{
\nCat
}{
\mathbf
{
n Cat
}}
% Source and target
\newcommand
{
\s
}
[1]
{
s
^{
(#1)
}}
\newcommand
{
\sk
}{
\s
{
k
}}
\newcommand
{
\t
}
[1]
{
t
^{
(#1)
}}
\newcommand
{
\tk
}{
\t
{
k
}}
% ad hoc
\newcommand
{
\nbar
}{
\mathbb
{
N
}
\cup\{
\oo
\}
}
omegacat.tex
0 → 100644
View file @
b5de6425
\chapter
{$
\oo
$
-Category theory
}
\section
{$
\oo
$
-graphs,
$
\oo
$
-magmas and
$
\oo
$
-categories
}
\begin{paragr}
An
\emph
{$
\oo
$
-graph
}
$
C
$
consists of a sequence
$
(
C
_
k
)
_{
k
\in
\mathbb
{
N
}}$
of sets together with maps
\[
\begin
{
tikzcd
}
C
_{
k
-
1
}
&
\ar
[
l,"s",shift left
]
\ar
[
l,"t"',shift right
]
C
_{
k
}
\end
{
tikzcd
}
\]
for every
$
k >
0
$
, satisfying the
\emph
{
globular identities
}
:
\begin{equation*}
\left\{
\begin{aligned}
s
\circ
s
&
= s
\circ
t,
\\
t
\circ
t
&
= t
\circ
s.
\end{aligned}
\right
.
\end{equation*}
Elements of
$
C
_
k
$
are called
\emph
{$
k
$
-cells
}
.
For
$
x
$
a
$
k
$
-cell with
$
k>
0
$
,
$
s
(
x
)
$
is the
\emph
{
source
}
of
$
x
$
and
$
t
(
x
)
$
is the
\emph
{
target
}
of
$
x
$
.
%Similarly, for $n \in \mathbb{N}$, an \emph{$n$-graph} $C$ is a finite sequence $(C_k)_{0 \leq k \leq n}$ of sets together with maps
% \[ \begin{tikzcd}
% C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k}
% \end{tikzcd}
% \]
% for every $0 <k < n$, satisfying the same globular identities.
\end{paragr}
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