Converts more images to diagrams

thesis/diagrams/Apl_C_Quasi_Iso_Cube.tex

@@ -0,0 +1,6 @@
+\xymatrix @=9pt {
+	A(X') \ar[rr] \ar[dd] \ar[dr] && A(\Delta[n]) \arfib'[d][dd] \arwe[dr] & \\
+	&	C^\ast(X') \ar[rr] \ar[dd] && C^\ast(\Delta[n]) \arfib[dd] \\
+	A(X) \ar'[r][rr] \arwe[dr] && A(\del\Delta[n]) \arwe[dr] & \\
+	&	C^\ast(X) \ar[rr] && C^\ast(\del\Delta[n])
+}

thesis/diagrams/Apl_C_Quasi_Iso_Pushout.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\del\Delta[n] \ar[r] \ar[d] \xypo & X \ar[d] \\
+	\Delta[n] \ar[r]                  & X'
+}

thesis/diagrams/Apl_Extension.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\DELTA \ar[d]^{\Delta[-]} \ar@<+.6ex>[drr]^-\Apl && \\
+	\sSet \ar@{-->}@<+.6ex>[rr]^-A & \xyadj & \CDGA_\k^{op} \ar@{-->}@<+.6ex>[ll]^-K
+}

thesis/diagrams/C_Extension.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\DELTA \ar[d]^{\Delta[-]} \ar@<+.6ex>[drr]^-C && \\
+	\sSet \ar@{-->}@<+.6ex>[rr]^-{C^\ast} & \xyadj & \DGA_\k^{op} \ar@{-->}@<+.6ex>[ll]
+}

thesis/diagrams/Kan_Extension.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\DELTA \ar[d]^{\Delta[-]} \ar@<+.6ex>[drr]^-F && \\
+	\sSet \ar@{-->}@<+.6ex>[rr]^-{F_!} & \xyadj & \cat{C} \ar@{-->}@<+.6ex>[ll]^-{F^\ast}
+}

thesis/diagrams/Kreck_Exact_Sequence.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\cdots \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[r] \ar[d] & H_i(E^k, F) \ar[r] \ar[d] & H_i(E^{k+1}, F) \ar[r] \ar[d] & \cdots \\
+	\cdots \ar[r] & H_{i+1}(B^{k+1}, B^k) \ar[r] & H_i(B^k, b_0) \ar[r] & H_i(B^{k+1}, b_0) \ar[r] & \cdots
+}

thesis/diagrams/Serre_Hurewicz_Square.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\pi_n(X(n)) \ar[r]^\iso \ar[d]^\iso & H_n(X(n)) \ar[d]^{\C-iso} \\
+	\pi_n(X) \ar[r] & H_n(X)
+}

thesis/diagrams/Serre_Whitehead_LES.tex

@@ -0,0 +1,4 @@
+\xymatrix{
+	\cdots \ar[r] & \pi_{i+1}(B_f, A) \ar[r] \ar[d] & \pi_i(A) \ar[r]^{f_\ast} \ar[d] & \pi_i(B) \ar[r] \ar[d] & \pi_i(B_f, A) \ar[r] \ar[d] & \cdots \\
+	\cdots \ar[r] & H_{i+1}(B_f, A) \ar[r] & H_i(A) \ar[r]^{f_\ast} & H_i(B) \ar[r] & H_i(B_f, A) \ar[r] & \cdots \\
+}

thesis/notes/A_K_Quillen_Pair.tex

@@ -30,5 +30,5 @@ Since $A$ is a left adjoint, it preserves all colimits and by functoriality it p
 
 \begin{corollary}
 	$A$ and $K$ induce an adjunction on the homotopy categories:
-	$$ \Ho{\sSet} \leftadj \opCat{\Ho{\CDGA}}. $$
+	$$ \Ho(\sSet) \leftadj \opCat{\Ho(\CDGA)}. $$
 \end{corollary}

thesis/notes/CDGA_Of_Polynomials.tex

@@ -2,8 +2,6 @@
 \section{CDGA of Polynomials}
 \label{sec:cdga-of-polynomials}
 
-\newcommand{\Apl}[0]{{A_{PL}}}
-
 We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span.
 
 \Definition{apl}{

thesis/notes/Polynomial_Forms.tex

@@ -19,11 +19,11 @@ A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimi
 
 Furthermore we have $F_! \circ \Delta[-] \iso F$. In short we have the following:
 
-\cimage[scale=0.5]{Kan_Extension}
+\cdiagram{Kan_Extension}
 
 In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get:
 
-\cimage[scale=0.5]{Apl_Extension}
+\cdiagram{Apl_Extension}
 
 
 \subsection{The cochain complex of polynomial forms}
@@ -49,7 +49,7 @@ Another way to model the $n$-simplex is by the singular cochain complex associat
 $$ C_n = C^\ast(\Delta^n; \k). $$
 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
 
-\cimage[scale=0.5]{C_Extension}
+\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.
 
@@ -95,11 +95,11 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis
 \Proof{
 	Assume we have a simplicial set $X$ such that $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphism. We can add a simplex by considering pushouts of the following form:
 
-	\cimage[scale=0.5]{Apl_C_Quasi_Iso_Pushout}
+	\cdiagram{Apl_C_Quasi_Iso_Pushout}
 
 	We can apply our two functors to it, and use the natural transformation $\oint$ to obtain the following cube:
 
-	\cimage[scale=0.5]{Apl_C_Quasi_Iso_Cube}
+	\cdiagram{Apl_C_Quasi_Iso_Cube}
 
 	Note that $A(\Delta[n]) \we C^\ast(\Delta[n])$ by \CorollaryRef{apl-c-quasi-iso}, $A(X) \we C^\ast(X)$ by assumption and $A(\del \Delta[n]) \we C^\ast(\del \Delta[n])$ by induction. Secondly note that both $A$ and $C^\ast$ send injective maps to surjective maps, so we get fibrations on the right side of the diagram. Finally note that the front square and back square are pullbacks, by adjointness of $A$ and $C^\ast$. Apply the cube lemma (\LemmaRef{cube-lemma}, \cite[Lemma 5.2.6]{hovey}) to conclude that also $A(X') \we C^\ast(X')$.
 

thesis/notes/Serre.tex

@@ -16,7 +16,6 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational
 	\end{itemize}
 }
 
-\renewcommand{\C}{\mathcal{C}}
 \Definition{serre-class}{
 	A class $\C \subset \Ab$ is a \Def{Serre class} if
 	\begin{itemize}
@@ -73,7 +72,7 @@ In the following arguments we will consider fibrations and need to compute homol
 
 	For the induction step, consider the long exact sequence in homology for the triples $(E^{k+1}, E^k, F)$ and $(B^{k+1}, B^k, b_0)$:
 
-	\cimage[scale=0.5]{Kreck_Exact_Sequence}
+	\cdiagram{Kreck_Exact_Sequence}
 
 	The morphism in the middle is a $\C$-iso by induction. We will prove that the left morphism is a $\C$-iso which implies by the five lemma that the right morphism is one as well.
 
@@ -108,7 +107,7 @@ In the following arguments we will consider fibrations and need to compute homol
 
 	Considering this claim for all $j < n$ gives a chain of $\C$-isos $H_i(X(n)) \to H_i(X(n-1)) \to \cdot \to H_i(X(2)) = H_i(X)$ for all $i \leq n$. Consider the following diagram:
 
-	\cimage[scale=0.5]{Serre_Hurewicz_Square}
+	\cdiagram{Serre_Hurewicz_Square}
 
 	where the map on the top is an isomorphism by the classical Hurewicz theorem (and $X(n)$ is $(n-1)$-connected), the map on the left is an isomorphism by the Whitehead tower and the map on the right is a $\C$-iso by the claim.
 
@@ -134,7 +133,7 @@ In the following arguments we will consider fibrations and need to compute homol
 \Proof{
 	Consider the mapping cylinder $B_f$ of $f$, i.e. factor the map $f$ as a cofibration followed by a trivial fibration $f: A \cof B_f \fib B$. The inclusion $A \subset B_f$ gives a long exact sequence of homotopy groups and homology groups:
 
-	\cimage[scale=0.5]{Serre_Whitehead_LES}
+	\cdiagram{Serre_Whitehead_LES}
 
 	We now have the equivalence of the following statements:
 	\begin{enumerate}

thesis/preamble.tex

@@ -82,6 +82,14 @@
 \newcommand{\artcof}{\ar@{^{(}->}|\simeq}
 \newcommand{\arfib}{\ar@{->>}}
 \newcommand{\artfib}{\ar@{->>}|\simeq}
+\newcommand{\arwe}{\ar|\simeq}
+
+% adjunction symbol for xymatrices
+\newcommand{\xyadj}{\raisebox{0.2\height}{\scalebox{0.5}{$\perp$}}}
+
+% pushout and pullback for xymatrices (makes empty arrow with text)
+\newcommand{\xypo}{\ar@{}[dr]|(.75){\scalebox{1.2}{$\ulcorner$}}}
+\newcommand{\xypb}{\ar@{}[dr]|(.25){\scalebox{1.2}{$\lrcorner$}}}
 
 %\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}}
 \newcommand{\leftadj}{\rightleftarrows}
@@ -104,6 +112,8 @@
 \renewcommand{\deg}[1]{{|{#1}|}}
 \newcommand{\Char}[1]{char({#1})}
 \newcommand{\RH}{\widetilde{H}}			% reduced homology
+\renewcommand{\C}{\mathcal{C}}			% Serre mod C class
+\newcommand{\Apl}[0]{{A_{PL}}}			% Apl simplicial set of polynomials
 
 % restriction of a function
 \newcommand\restr[2]{{% we make the whole thing an ordinary symbol

thesis/test_diagram.sh

@@ -4,5 +4,5 @@ set -e
 file=$1
 sed "s|__INPUT__|$file|" test_diagram.tex | xelatex -file-line-error -output-directory=build
 mv build/texput.pdf test_diagram.pdf
-exo-open test_diagram.pdf
+#exo-open test_diagram.pdf
 rm build/texput.*