Commit 033c16d3 authored by Leonard Guetta's avatar Leonard Guetta
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......@@ -916,9 +916,9 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
\begin{paragr}
We refer to the model structure of the above proposition as the \emph{folk model structure on $n\Cat$}. By definition, the functor $\iota_n : n\Cat \to \oo\Cat$ preserves weak equivalences and fibrations when $\oo\Cat$ and $n\Cat$ are equipped with the folk model structure. In particular, the adjunction $\tau^{i}_{\leq n } \dashv \iota_n$ is a Quillen adjunction. As it happens, the functor $\tau^{i}_{\leq n}$ also preserves weak equivalences.
\end{paragr}
\begin{lemma}
\begin{proposition}\label{prop:truncationhomotopical}
The functor $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$ sends weak equivalences of the folk model structure on $\oo\Cat$ to weak equivalences of the folk model structure on $n\Cat$.
\end{lemma}
\end{proposition}
\begin{proof}
% Let $C$ be an $\oo$-category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq n}(C)$ be the unit morphism of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. Let us prove that for every pair $(x,y)$ of $k$\nbd-cells of $C$, if $k \leq n$ then $x \simeq_{\oo} y$ (see \ref{paragr:ooequivalence}) if and only if $\eta_C(x)\simeq_{\oo}\eta_C(y)$.
Let $f : C \to D$ be a weak equivalence for the folk model structure on $\oo\Cat$. By definition of weak equivalences for the the folk model structure on $n\Cat$, we have to show that
......@@ -951,23 +951,52 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
This functor is fully faithful and $\Ch^{\leq n}$ may be identified with the full subcategory of $\Ch$ spanned by chain complexes $K$ such that $K_k = 0$ for every $k >n$.
% Note that when $C$ is an $n$\nbd-category, seen as an $\oo$\nbd-category with only unit cells in dimension higher that $n$ via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, then $\tau^{i}_{\leq n}(C) = C$.
Similarly to the case of $n$\nbd-categories, the functor $\iota_n : \Ch^{\leq n} \to \Ch$ has a left adjoint $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq}$, where for a chain complex $K$, $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
Similarly to the case of $n$\nbd-categories, the functor $\iota_n : \Ch^{\leq n} \to \Ch$ has a left adjoint $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq}$, where for a chain complex $K$, $\tau^{i}_{\leq n}(K)$ is the object of $\Ch^{\leq n}$ defined as
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
\]
Likewise $n$\nbd-categories again, we can use the adjunction
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
\begin{tikzcd}
\iota_n : \Ch^{\leq n}\ar[r,shift left] & \ar[l,shift right]\Ch : \tau^{i}_{\leq n}
\end{tikzcd}
\]
Likewise $n$\nbd-categories again, we can use the adjunction $\iota_n : \Ch^{\leq n} \overset{\longrightarrow}{\longleftarrow} \Ch : \tau^{i}_{\leq n}$ to create a model structure on $\Ch^{\leq n}$.
to create a model structure on $\Ch^{\leq n}$.
\end{paragr}
\begin{proposition}
There exists a model structure on $\Ch^{\geq n}$ such that:
\begin{itemize}
\item weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$ (i.e.\ a quasi-isomorphism),
There exists a model structure on $\Ch^{\leq n}$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\item fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\end{itemize}
\end{proposition}
\begin{proof}
This a typical example of a transfer a cofibrantely generated model structure along a right adjoint as in
This a typical example of a transfer a cofibrantely generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square
\[
\begin{tikzcd}
\tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\
\tau^{i}_{\leq n}(B) \ar[r] & Y
\ar[from=1-1, to=2-2, phantom, "\ulcorner",very near end]
\end{tikzcd},
\]
the morphism $\iota_n(g)$ is a weak equivalence of $\Ch$. As explained in \cite[Proposition 7.19]{dwyer1995homotopy}, there exists a set of generating trivial cofibrations of the projective model structure on $\Ch$ consisting of the maps
\[
0 \to D_k
\]
for each $k >0$, where $D_k$ is the following chain complex concentrated in degree $k$ and $k-1$
\[
0 \leftarrow \cdots \leftarrow 0 \leftarrow \mathbb{Z} \overset{\mathrm{id}}{\leftarrow} \mathbb{Z} \leftarrow 0 \leftarrow \cdots
\]
What is left to show then is that for every $k > 0$ and every object $X$ of $\Ch^{\leq n}$, the canonical inclusion map
\[
X \to X \oplus \tau^{i}_{\leq n}(D_k)
\]
is sent by $\iota_n$ to a weak equivalence of $\Ch$. This follows immediatly from the fact that homology groups commute with direct sums.
\end{proof}
\begin{lemma}
\begin{paragr}
We refer to the model structure of the previous proposition as the \emph{projective model structure on $\Ch^{\leq n}$}.
\end{paragr}
\begin{lemma}\label{lemma:unitajdcomp}
For every chain complex $K$, the unit map
\[
K \to \iota_n\tau^{i}_{\leq n}(K)
......@@ -989,9 +1018,27 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
\]
The isomorphism being obviously induced by the unit map $K \to \iota_n\tau^{i}_{\leq n}(K)$.
\end{proof}
As a consequence of this lemma, we have the analoguous of
As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:truncationhomotopical}.
\begin{proposition}
The functor $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq n}$ sends weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.
\end{proposition}
\begin{proof}
Let $f : K \to K'$ be a weak equivalence for the projective model structure on $\Ch$ and consider the naturality square
\[
\begin{tikzcd}[column sep=huge]
K \ar[d,"\eta_K"] \ar[r,"f"] & K' \ar[d,"\eta_K'"] \\
\iota_n\tau^{i}_{\leq n}(K) \ar[r,"\iota_n\tau^{i}_{\leq n}(f)"] & \iota_n\tau^{i}_{\leq n}(K'),
\end{tikzcd}
\]
where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. Now it follows from Lemma \ref{lemma:unitajdcomp} that
\[
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K'))
\]
is an isomorphism for every $k \leq n$. Since obviously $H_k(\iota_n\tau^{i}_{\leq n}(f))$ is also an isomorphism for $k > n$, this proves the result.
\end{proof}
We now investigate the relation between truncation and linearization.
\begin{paragr}
Let $C$ be $n$\nbd-category $C$. An straigtforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that
Let $C$ be $n$\nbd-category. A straigtforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that
\[
\lambda(\iota_n(C))_k=0
\]
......
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......@@ -70,6 +70,16 @@ year={2020}
year={1998},
publisher={Academic Press}
}
@article{beke2001sheafifiableII,
title={Sheafifiable homotopy model categories, II},
author={Beke, Tibor},
journal={Journal of Pure and Applied Algebra},
volume={164},
number={3},
pages={307--324},
year={2001},
publisher={Elsevier}
}
@article{bourn1990another,
title={Another denormalization theorem for abelian chain complexes},
author={Bourn, Dominique},
......
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