Commit 3e739f45 authored by Leonard Guetta's avatar Leonard Guetta
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I need to go home

parent ee9c9be6
......@@ -888,7 +888,14 @@ The previous proposition admits the following corollary, which will be of great
\begin{proof}
The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.
\end{proof}
\section{Polygraphic homology and truncation}
\section{Equivalence of homologies in low dimension}
\begin{paragr}
Recall that for every $n \geq 0$ we have taken the habit of identifying $n\Cat$ as a full subcategory of $\oo\Cat$ via the canonical fully faithful functor $\iota_n : n\Cat \to \oo\Cat$ (defined in \ref{paragr:defncat}) that sends an $n$\nbd-category $C$ to the $\oo$\nbd-category with the same $k$\nbd-cells as $C$ for $k\leq n$ and only unit cells for $k > n$. In particular, we abusively wrote
\[
C=\iota_n(C).
\]
Within this section, \emph{and only within this section}, we try not to make this abuse of notation and explicitly write $\iota_n$ whenever we should.
\end{paragr}
\begin{paragr}
Let $\Ch^{\leq n}$ be the category of chain complexes in degree comprised between $0$ and $n$. This means that an object $K$ of $\Ch^{\leq n}$ is a diagram of abelian groups of the form
\[
......@@ -900,14 +907,14 @@ The previous proposition admits the following corollary, which will be of great
\]
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 $k >n$.
Notice now that for an $n$\nbd-category $C$, seen as an $\oo$\nbd-category via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, the chain complex $\lambda(C)$ is such that
Notice now that for an $n$\nbd-category $C$, the chain complex $\lambda(\iota_n(C))$ is such that
\[
\lambda(C)_k=0
\lambda(\iota_n(C))_k=0
\]
for every $k > n$ and hence $\lambda(C)$ can be seen as an object of $\Ch^{\leq n}$. Thus, we can define a functor $\lambda_{\leq n } : n\Cat \to \Ch^{\leq n}$ as
for every $k > n$ and hence $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Thus, we can define a functor $\lambda_{\leq n } : n\Cat \to \Ch^{\leq n}$ as
\begin{align*}
\lambda_{\leq n} : n\Cat &\to \oo\Cat\\
C&\mapsto \lambda(C).
\lambda_{\leq n} : n\Cat &\to \Ch^{\leq n}\\
C&\mapsto \lambda(\iota_n(C)).
\end{align*}
%% this means that there exists a unique functor $\lambda_{\leq n} : n\Cat \to \Ch^{\leq n}$ such that the following square is commutative
%% \[
......@@ -918,11 +925,11 @@ The previous proposition admits the following corollary, which will be of great
%% \]
%% This functor being
Recall from \ref{paragr:defncat} that for every $n \geq 0$, the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
Recall from \ref{paragr:defncat} that for every $n \geq 0$, the canonical inclusion $\iota_n : n\Cat \to \oo\Cat $ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
\[
x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.
\]
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 $\tau^{i}_{\leq n}(C) = C$.
% 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, for a chain complex $K$, write $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
......@@ -933,6 +940,20 @@ The previous proposition admits the following corollary, which will be of great
\]
which is left adjoint to the canonical inclusion $\iota_n : \Ch^{\leq n} \to \Ch$.
\end{paragr}
\begin{lemma}
For every chain complex $K$, the unit map
\[
K \to \iota_n\tau^{i}_{\leq n}(K)
\]
induces isomorphisms
\[
H_k(K) \simeq H_k(\iota_n\tau^{i}_{\leq n}(K))
\]
for every $0 \leq k \leq n$.
\end{lemma}
\begin{proof}
\end{proof}
\begin{lemma}
The square
\[
......@@ -948,13 +969,13 @@ The previous proposition admits the following corollary, which will be of great
\end{proof}
With this lemma at hand we can prove the important following proposition which basically says that if an $\oo$\nbd-category $C$ is free up to dimension $n-1$, then for any $k$ such that $0 \leq k \leq n$ there is no need to find a cofibrant replacement in order to compute $H^{\pol}_k(C)$.
\begin{proposition}
Let $n \geq 0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $ 0 \leq k < n$, then the canonical map of $\ho(\Ch)$
Let $n \geq 0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $ 0 \leq k \leq n-1$, then the canonical map of $\ho(\Ch)$
\[
\alpha^{\pol}_C : \sH^{\pol}(C) \to \lambda(C)
\]
induces isomorphisms
\[
H_k^{\pol}(C) \to H_k(\lambda(C))
H_k^{\pol}(C) \simeq H_k(\lambda(C))
\]
for every $0 \leq k \leq n$.
\end{proposition}
......@@ -962,17 +983,17 @@ With this lemma at hand we can prove the important following proposition which b
\todo{À écrire}
\end{proof}
\begin{paragr}
Since every $\oo$\nbd-category $C$ admits $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) the $0$-th and first homology groups of the chain complex $\lambda(C)$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
Since every $\oo$\nbd-category $C$ admits $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) $H_0(\lambda(C))$ and $H_1(\lambda(C))$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
\end{paragr}
Somewhat related is the following proposition.
\begin{proposition}
Let $C$ be an $\oo$\nbd-category and $n \geq 0$. The canonical map
\[
\sH^{\pol}(C) \to \sH^{\pol}(\tau^{i}_{\leq n}(C))
\sH^{\pol}(C) \to \sH^{\pol}(\iota_n\tau^{i}_{\leq n}(C))
\]
induces isomorphisms
\[
H^{\pol}_k(C) \simeq H^{\pol}_k(\tau^{i}_{\leq n}(C))
H^{\pol}_k(C) \simeq H^{\pol}_k(\iota_n\tau^{i}_{\leq n}(C))
\]
for every $0 \leq k \leq n$.
\end{proposition}
......@@ -997,15 +1018,59 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
\begin{proof}
\todo{À écrire (facile)}
\end{proof}
Slightly less trivial is the following lemma.
\begin{paragr}
In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ induces, for every $\oo$\nbd-category $C$ and every $n \geq 0$, a canonical morphism of $n\Cat$
\[
c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C),
\]
which is natural in $C$.
\end{paragr}
\begin{lemma}
The following triangle of functors
For every $\oo$\nbd-category $C$, the canonical morphism of $\Cat$
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\tau^{i}_{\leq 1}"] \ar[d,"N_{\oo}"] & \Cat \\
\Psh{\Delta} \ar[ru,"c_1"'] &
\end{tikzcd}
c_1N_{\oo}(C) \to \tau^{i}_{\leq 1}(C)
\]
is commutative (up to an isomorphism).
is an isomorphism.
\end{lemma}
\section{Homology and Homotopy of $\oo$-categories in low dimension}
\begin{proof}
\todo{À écrire.}
\end{proof}
We can now prove the important following proposition.
\begin{proposition}
For every $\oo$\nbd-category $C$, the canonical map of $\ho(\Ch)$
\[
\alpha^{\sing}: \sH^{\sing}(C) \to \lambda(C)
\]
induces isomorphisms
\[
H^{\sing}_k(C) \simeq H_k(\lambda(C))
\]
for $k \in \{0,1\}$.
\end{proposition}
\begin{proof}
\todo{À écrire}
\end{proof}
Finally, we obtain the result we were aiming for.
\begin{proposition}
For every $\oo$\nbd-category $C$, the canonical comparison map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C)
\]
induces isomorphisms
\[
H^{\sing}_k(C) \simeq H^{\pol}_k(C)
\]
for $k \in \{0,1\}$.
\end{proposition}
%% Slightly less trivial is the following lemma.
%% \begin{lemma}
%% The following triangle of functors
%% \[
%% \begin{tikzcd}
%% \oo\Cat \ar[r,"\tau^{i}_{\leq 1}"] \ar[d,"N_{\oo}"] & \Cat \\
%% \Psh{\Delta} \ar[ru,"c_1"'] &
%% \end{tikzcd}
%% \]
%% is commutative (up to an isomorphism).
%% \end{lemma}
%%\section{Homology and Homotopy of $\oo$-categories in low dimension}
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