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Leonard Guetta
memoire
Commits
439a60a7
Commit
439a60a7
authored
Oct 27, 2020
by
Leonard Guetta
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edited a few typos in chapter 2. Did not yet do what said in previous commit
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homtheo.tex
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439a60a7
...
...
@@ -44,7 +44,7 @@ is poorly behaved. For example, \fi
\gamma
^
*
:
\underline
{
\Hom
}
(
\ho
(
\C
)
,
\D
)
\to
\underline
{
\Hom
}
(
\C
,
\D
)
\]
is fully faithful and its essential image consists of functors
$
F~:~
\C
~
\to
~
\D
$
that sends morphism of
$
\W
$
to isomorphisms of
$
\D
$
.
that sends
the
morphism
s
of
$
\W
$
to isomorphisms of
$
\D
$
.
We shall always consider that
$
\C
$
and
$
\ho
(
\C
)
$
have the same class of
objects and implicitly use the equality
...
...
@@ -90,7 +90,7 @@ For later reference, we put here the following definition.
is commutative. Let
$
G :
(
\C
,
\W
)
\to
(
\C
',
\W
'
)
$
be another morphism of
localizers. A
\emph
{$
2
$
\nbd
{}
morphism of localizers
}
from
$
F
$
to
$
G
$
is simply a
natural transformation
$
\alpha
: F
\Rightarrow
G
$
. The universal property of
the localization implies that there exists a
canonical
natural transformation
the localization implies that there exists a
unique
natural transformation
\[
\begin
{
tikzcd
}
\ho
(
\C
)
\ar
[
r,bend left,"
\overline
{
F
}
",""
{
name
=
A,below
}
]
\ar
[
r,bend right,"
\overline
{
G
}
"',""
{
name
=
B,above
}
]
&
\ho
(
\C
'
)
...
...
@@ -109,8 +109,8 @@ For later reference, we put here the following definition.
\end{paragr}
\begin{remark}
\label
{
remark:localizedfunctorobjects
}
Since we always consider that for every localizer
$
(
\C
,
\W
)
$
the categories
$
\C
$
and
$
\ho
(
\C
)
$
have the same objects and the localization functor is the
identity on objects, it follows that for a morphism of localizer
${
F :
(
\C
,
\W
)
and
$
\ho
(
\C
)
$
have the same
class of
objects and the localization functor is the
identity on objects, it follows that for a morphism of localizer
s
${
F :
(
\C
,
\W
)
\to
(
\C
',
\W
'
)
}$
, we tautologically have
\[
\overline
{
F
}
(
X
)=
F
(
X
)
...
...
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