Commit 81e86938 by Leonard Guetta

tpoto

parent 18fede95
 ... ... @@ -386,7 +386,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd-funct Let $A$ a $1$\nbd-category, $X$ a free $\oo$\nbd-category and $f : X \to A$ be an $\oo$\nbd-functor. The functor \begin{align*} A &\to \oo\Cat \\ a_0 &\mapsto A/a_0 a_0 &\mapsto X/a_0 \end{align*} is a cofibrant object for the projective model structure on $\oo\Cat(A)$ induced by the folk model structure on $\oo\Cat$ (\ref{paragr:projmod}). ... ...
 ... ... @@ -908,12 +908,22 @@ The previous proposition admits the following corollary, which will be of great This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generating cofibrations is not made explicit in the statement of the theorem, it is contained in proof.) \end{proof} \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. 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{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{proposition} \begin{proof} Since every $\oo$\nbd-category is fibrant for the folk model structure on $\oo\Cat$ (see \cite[Proposition 9]{lafont2010folk}), it suffices in vertue of Ken Brown's Lemma \cite{} to show that $\tau^{i}_{\leq n}$ sends folk trivial fibrations to weak equivalences of the folk model structure on $n\Cat$. % 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 $... ... No preview for this file type  ... ... @@ -179,13 +179,13 @@ A \emph{morphism of \oo-magmas} f : X \to Y is a morphism of underlying \oo The sequence of adjunctions \[ \kappa_n \dashv \tau^s_{\leq n} \dashv \iota_n \dashv \tau^{i}_{\leq n} \tau^{i}_{\leq n} \dashv \iota_n \dashv \tau^s_{\leq n} \dashv \kappa_n$ is maximal in that $\kappa_n$ doesn't have a left adjoint and $\tau^{i}_{\leq n}$ doesn't have right adjoint. is maximal in that $\kappa_n$ doesn't have a right adjoint and $\tau^{i}_{\leq n}$ doesn't have left adjoint. The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}. The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units on lower dimensional cells. The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units. \end{paragr} \begin{paragr} For $n \geq 0$, we define the $n$-skeleton functor $\sk_n : \oo\Cat \to \oo\Cat$ as ... ... @@ -2317,3 +2317,8 @@ Putting all the pieces together, we finally have the awaited proof. Hence, we can apply the second part of Proposition \ref{prop:conduchenbasis} and $C$ is free. \end{enumerate} \end{paragr} %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End:
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!