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Leonard Guetta
memoire
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a120b417
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a120b417
authored
Oct 09, 2020
by
Leonard Guetta
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a120b417
\chapter
{
Homotopy and homology type of free
$
2
$
-categories
}
\section
{
Preliminaries
: the case of free
$
1
$
-categories
}
\section
{
Preliminaries: the case of free
$
1
$
-categories
}
In this section, we review some homotopical results concerning free (
$
1
$
-)categories that will be of great help in the sequel.
\begin{paragr}
A
\emph
{
reflexive graph
}
$
G
$
consists of the data of two sets
$
G
_
0
$
and
$
G
_
1
$
together with
...
...
@@ -313,7 +313,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
Note also that the hypothesis that
$
A
\neq
B
$
was fundamental in the previous example as for
$
A
=
B
$
the square is
\emph
{
not
}
homotopy cocartesian.
\end{remark}
\section
{
Preliminaries
:
B
isimplicial sets
}
\section
{
Preliminaries:
b
isimplicial sets
}
\begin{paragr}
A
\emph
{
bisimplicial set
}
is a presheaf over the category
$
\Delta
\times
\Delta
$
,
\[
...
...
@@ -788,7 +788,7 @@ It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve in
This is an immediate consequence of Proposition
\ref
{
prop:streetvsbisimplicial
}
and Corollary
\ref
{
cor:bisimplicialsquare
}
.
\end{proof}
\section
{
Zoology of
$
2
$
-categories:
B
asic examples
}
\section
{
Zoology of
$
2
$
-categories:
b
asic examples
}
\begin{paragr}
\label
{
paragr:criterion2cat
}
Before embarking on computations of homology and homotopy types of
$
2
$
\nbd
{}
categories, let us recall the following particular case of Corollary
...
...
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a120b417
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memoire.bib
View file @
a120b417
...
...
@@ -9,7 +9,7 @@
publisher
=
{Narnia}
}
@article
{
ara2014vers
,
title
=
{Vers une structure de cat{\'e}gorie de mod{\`e}les {\`a} la {T}homason sur la cat{\'e}gorie des n
-
cat{\'e}gories strictes}
,
title
=
{Vers une structure de cat{\'e}gorie de mod{\`e}les {\`a} la {T}homason sur la cat{\'e}gorie des n
\nbd{}
cat{\'e}gories strictes}
,
author
=
{Ara, Dimitri and Maltsiniotis, Georges}
,
journal
=
{Advances in Mathematics}
,
volume
=
{259}
,
...
...
@@ -22,10 +22,10 @@
author
=
{Ara, Dimitri and Maltsiniotis, Georges}
,
journal
=
{M{\'e}moires de la Soci{\'e}t{\'e} Math{\'e}matique de France}
,
year
=
{2020}
,
note
=
{To appear
}
volume
=
{165
}
}
@article
{
ara2018theoreme
,
title
=
{Un th{\'e}or{\`e}me {A} de {Q}uillen pour les $\infty$
-
cat{\'e}gories strictes {I} : la preuve simpliciale}
,
title
=
{Un th{\'e}or{\`e}me {A} de {Q}uillen pour les $\infty$
\nbd{}
cat{\'e}gories strictes {I} : la preuve simpliciale}
,
author
=
{Ara, Dimitri and Maltsiniotis, Georges}
,
journal
=
{Advances in Mathematics}
,
volume
=
{328}
,
...
...
@@ -34,13 +34,13 @@
publisher
=
{Elsevier}
}
@article
{
ara2020comparaison
,
title
=
{Comparaison des nerfs $
n $-
cat{\'e}goriques}
,
title
=
{Comparaison des nerfs $
n$\nbd{}
cat{\'e}goriques}
,
author
=
{Ara, Dimitri and Maltsiniotis, Georges}
,
journal
=
{arXiv preprint arXiv:2010.00266}
,
year
=
{2020}
}
@article
{
ara2020theoreme
,
title
=
{Un th{\'e}or{\`e}me {A} pour les $\infty$-cat{\'e}gories strictes {II} : la preuve $\infty$-cat{\'e}gorique}
,
title
=
{Un th{\'e}or{\`e}me {A}
de {Q}uillen
pour les $\infty$-cat{\'e}gories strictes {II} : la preuve $\infty$-cat{\'e}gorique}
,
author
=
{Ara, Dimitri and Maltsiniotis, Georges}
,
volume
=
{4}
,
number
=
{1}
,
...
...
@@ -57,7 +57,7 @@ year={2020}
year
=
{2020}
}
@article
{
ara2019quillen
,
title
=
{A
Q
uillen Theorem
B
for strict $\infty$
-
categories}
,
title
=
{A
{Q}
uillen Theorem
{B}
for strict $\infty$
\nbd{}
categories}
,
author
=
{Ara, Dimitri}
,
journal
=
{Journal of the London Mathematical Society}
,
volume
=
{100}
,
...
...
@@ -77,7 +77,7 @@ year={2020}
publisher
=
{Academic Press}
}
@article
{
beke2001sheafifiableII
,
title
=
{Sheafifiable homotopy model categories, II}
,
title
=
{Sheafifiable homotopy model categories,
{
II}
}
,
author
=
{Beke, Tibor}
,
journal
=
{Journal of Pure and Applied Algebra}
,
volume
=
{164}
,
...
...
@@ -98,7 +98,7 @@ year={2020}
}
@book
{
bousfield1972homotopy
,
title
=
{Homotopy limits, completions and localizations}
,
author
=
{Bousfield, Aldridge K
night
and Kan, Daniel M
arinus
}
,
author
=
{Bousfield, Aldridge K
.
and Kan, Daniel M
.
}
,
volume
=
{304}
,
year
=
{1972}
,
publisher
=
{Springer Science \& Business Media}
...
...
@@ -117,8 +117,8 @@ issn = "0001-8708",
publisher
=
"Elsevier"
}
@article
{
bullejos2003geometry
,
title
=
{On the geometry of 2
-
categories and their classifying spaces}
,
author
=
{Bullejos, Manuel and Cegarra, Antonio M}
,
title
=
{On the geometry of 2
\nbd{}
categories and their classifying spaces}
,
author
=
{Bullejos, Manuel and Cegarra, Antonio M
.
}
,
journal
=
{K-theory}
,
volume
=
{29}
,
number
=
{3}
,
...
...
@@ -165,13 +165,13 @@ publisher = "Elsevier"
}
@phdthesis
{
crans1995combinatorial
,
title
=
{On Combinatorial Models for Higher Dimensional Homotopies}
,
author
=
{Crans, Sjoerd E
rik
}
,
author
=
{Crans, Sjoerd E
.
}
,
school
=
{Universiteit Utrecht}
,
year
=
{1995}
}
@article
{
duskin2002simplicial
,
title
=
{Simplicial matrices and the nerves of weak n
-
categories.
I
. Nerves of bicategories}
,
author
=
{Duskin, John W}
,
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}
,
number
=
{10}
,
...
...
@@ -181,7 +181,7 @@ publisher = "Elsevier"
}
@article
{
dwyer1980simplicial
,
title
=
{Simplicial localizations of categories}
,
author
=
{Dwyer, William G and Kan, Daniel M}
,
author
=
{Dwyer, William G
.
and Kan, Daniel M
.
}
,
journal
=
{J. Pure Appl. Algebra}
,
volume
=
{17}
,
number
=
{3}
,
...
...
@@ -190,7 +190,7 @@ publisher = "Elsevier"
}
@incollection
{
dwyer1995homotopy
,
title
=
{Homotopy theories and model categories}
,
author
=
{Dwyer, William G and Spalinski, Jan}
,
author
=
{Dwyer, William G
.
and Spalinski, Jan}
,
booktitle
=
{Handbook of Algebraic Topology}
,
pages
=
{73--126}
,
publisher
=
{ScienceDirect}
,
...
...
@@ -198,7 +198,7 @@ publisher = "Elsevier"
}
@article
{
eilenberg1954groups
,
title
=
{On the groups {H}($\pi$,n), {II}: Methods of computation}
,
author
=
{Eilenberg, Samuel and MacLane, Saunders}
,
author
=
{Eilenberg, Samuel and Mac
Lane, Saunders}
,
journal
=
{Annals of Mathematics}
,
pages
=
{49--139}
,
year
=
{1954}
,
...
...
@@ -206,7 +206,7 @@ publisher = "Elsevier"
}
@article
{
freyd1972categories
,
title
=
{Categories of continuous functors, {I}}
,
author
=
{Freyd, Peter J and Kelly,
G
Max}
,
author
=
{Freyd, Peter J
.
and Kelly, Max}
,
journal
=
{Journal of pure and applied algebra}
,
volume
=
{2}
,
number
=
{3}
,
...
...
@@ -252,10 +252,10 @@ publisher = "Elsevier"
publisher
=
{Mathematical Sciences Publishers}
}
@unpublished
{
groth2013book
,
title
=
{Book project on
D
erivators, volume
I
}
,
title
=
{Book project on
{D}
erivators, volume
{I}
}
,
author
=
{Groth, Moritz}
,
year
=
{2016}
,
note
=
{Available at \url{http://w
ebusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs
}}
note
=
{Available at \url{http://w
ww.math.uni-bonn.de/~mrahn/monos/intro-to-der-1.pdf
}}
}
@unpublished
{
grothendieckderivators
,
title
=
{Les D{\'e}rivateurs}
,
...
...
@@ -264,7 +264,7 @@ year={~1990},
note
=
{Available at \url{http://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs}}
}
@unpublished
{
grothendieck1991letter
,
title
=
{Lettre à
T
homason}
,
title
=
{Lettre à
{T}
homason}
,
author
=
{Grothendieck, Alexandre}
,
year
=
{1991}
,
note
=
{Available at \url{http://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs}}
...
...
@@ -314,13 +314,13 @@ year={2009}
}
@book
{
hirschhorn2009model
,
title
=
{Model categories and their localizations}
,
author
=
{Hirschhorn, Philip S}
,
author
=
{Hirschhorn, Philip S
.
}
,
year
=
{2009}
,
publisher
=
{American Mathematical Soc.}
}
@book
{
hopcroft1979introduction
,
title
=
{Introduction to automata theory, languages, and computation}
,
author
=
{Hopcroft, John E and Ullman}
,
author
=
{Hopcroft, John E
.
and Ullman
, Jeffrey D.
}
,
publisher
=
{Addison-Wesley}
,
year
=
{1979}
}
...
...
@@ -408,7 +408,7 @@ publisher={Springer}
publisher
=
{Citeseer}
}
@unpublished
{
maltsiniotis2001introduction
,
title
=
{Introduction à la théorie des dérivateurs, d'après
G
rothendieck}
,
title
=
{Introduction à la théorie des dérivateurs, d'après
{G}
rothendieck}
,
author
=
{Maltsiniotis, Georges}
,
year
=
{2001}
,
note
=
{Available at \url{http://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs}}
...
...
@@ -459,13 +459,13 @@ note={In preparation}
}
@book
{
quillen1967homotopical
,
title
=
{Homotopical algebra}
,
author
=
{Quillen, Daniel G}
,
author
=
{Quillen, Daniel G
.
}
,
year
=
{1967}
,
publisher
=
{Springer}
}
@incollection
{
quillen1973higher
,
title
=
{Higher algebraic K-theory: I}
,
author
=
{Quillen, Daniel}
,
author
=
{Quillen, Daniel
G.
}
,
booktitle
=
{Higher K-theories}
,
pages
=
{85--147}
,
year
=
{1973}
,
...
...
@@ -533,7 +533,7 @@ note={In preparation}
}
@article
{
thomason1980cat
,
title
=
{Cat as a closed model category}
,
author
=
{Thomason, Robert W}
,
author
=
{Thomason, Robert W
.
}
,
journal
=
{Cahiers de topologie et g{\'e}om{\'e}trie diff{\'e}rentielle cat{\'e}goriques}
,
volume
=
{21}
,
number
=
{3}
,
...
...
@@ -542,7 +542,7 @@ note={In preparation}
}
@article
{
thomason1979homotopy
,
title
=
{Homotopy colimits in the category of small categories}
,
author
=
{Thomason, Robert W}
,
author
=
{Thomason, Robert W
.
}
,
booktitle
=
{Mathematical Proceedings of the Cambridge Philosophical Society}
,
volume
=
{85}
,
number
=
{1}
,
...
...
@@ -552,7 +552,7 @@ note={In preparation}
}
@book
{
weibel1995introduction
,
title
=
{An Introduction to Homological Algebra}
,
author
=
{Weibel, Charles A}
,
author
=
{Weibel, Charles A
.
}
,
volume
=
{38}
,
year
=
{1995}
,
publisher
=
{Cambridge University Press}
...
...
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