From e11f0209af87ee44b5aca5f3d1c931ae4b3e7cd9 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Mon, 5 Jan 2015 17:17:46 +0100 Subject: [PATCH] Small clean ups --- thesis/notes/A_K_Quillen_Pair.tex | 2 +- thesis/notes/Polynomial_Forms.tex | 9 --------- 2 files changed, 1 insertion(+), 10 deletions(-) diff --git a/thesis/notes/A_K_Quillen_Pair.tex b/thesis/notes/A_K_Quillen_Pair.tex index 1e23a5c..5f3b2bb 100644 --- a/thesis/notes/A_K_Quillen_Pair.tex +++ b/thesis/notes/A_K_Quillen_Pair.tex @@ -188,7 +188,7 @@ In particular if the vector space $V'$ is finitely generated, we can repeat this Now $V(n)$ is finitely generated for all $n$ by assumption. By the inductive procedure above we see that $(\Lambda V(n), d) \to A(K(\Lambda V(n), d))$ is a weak equivalence for all $n$. Hence $(\Lambda V, d) \to A(K(\Lambda V, d))$ is a weak equivalence. } -Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by \CorollaryRef{serre-whitehead} (the Serre-Whitehead theorem). To be precise: $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X))) \to A(X)$ is a weak equivalence. +Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by \CorollaryRef{serre-whitehead} (the Serre-Whitehead theorem). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X))) \to A(X)$ is a weak equivalence. \Lemma{}{ The map $X \to K(M(X))$ is a weak equivalence for simply connected rational spaces $X$ of finite type. diff --git a/thesis/notes/Polynomial_Forms.tex b/thesis/notes/Polynomial_Forms.tex index a61ff57..57e993b 100644 --- a/thesis/notes/Polynomial_Forms.tex +++ b/thesis/notes/Polynomial_Forms.tex @@ -7,7 +7,6 @@ Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define th F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y) & C \in \cat{C} \end{align*} A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial. Secondly we see readily that $F^\ast$ is functorial. By using the definition of colimit and the Yoneda lemma (Y) we can prove that $F_!$ is left adjoint to $F^\ast$: - \begin{align*} \Hom_\cat{C}(F_!(X), Y) &\iso \Hom_\cat{C}(\colim_{\Delta[n] \to X} F[n], Y) \\ @@ -19,18 +18,14 @@ A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimi \end{align*} Furthermore we have $F_! \circ \Delta[-] \iso F$. In short we have the following: - \cdiagram{Kan_Extension} - In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get: - \cdiagram{Apl_Extension} \subsection{The cochain complex of polynomial forms} In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description: - \begin{align*} A(X) &= \lim_{\Delta[n] \to X} \Apl_n @@ -38,9 +33,7 @@ In our case we take the opposite category, so the definition of $A$ is in terms &\iso \Hom_\sSet(\colim_{\Delta[n] \to X}\Delta[n], \Apl) = \Hom_\sSet(X, \Apl), \end{align*} - where the addition, multiplication and differential are defined pointwise. Conclude that we have the following contravariant functors (which form an adjoint pair): - \begin{align*} A(X) &= \Hom_\sSet(X, \Apl) & X \in \sSet \\ K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in \CDGA_\k. @@ -52,7 +45,6 @@ where the addition, multiplication and differential are defined pointwise. Concl Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's \todo{Choose: normalized or not?}: $$ C_n = C^\ast(\Delta^n; \k). $$ The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions - \cdiagram{C_Extension} where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X) \we C^\ast(X)$ for every $X \in \sSet$. Furthermore this map preserves multiplication on the homology algebras. @@ -148,4 +140,3 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis So by the five lemma we can conclude that the middle morphism is an isomorphism as well, proving the isomorphism $H^n(A(X)) \tot{\iso} H^n(C^\ast(X))$ for all $n$. This proves the statement for all $X$. } -