diff --git a/thesis/1_CategoryTheory.tex b/thesis/1_CategoryTheory.tex index 932cf5a..093c12f 100644 --- a/thesis/1_CategoryTheory.tex +++ b/thesis/1_CategoryTheory.tex @@ -14,7 +14,7 @@ We will briefly define categories an functors to fix the notation. We will not p \end{itemize} \end{definition} -Instead of writing $f \in \Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many categories have functions as maps. There is a category $\Set$ of sets with functions, a category $\Ab$ of abelian groups with group homomorphisms, a category $\Top$ of topological spaces and continuous maps, and many more. +Instead of writing $f \in \Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many categories have functions as maps. For brevity we sometimes write $gf$ instead of $g \circ f$. There is a category $\Set$ of sets with functions, a category $\Ab$ of abelian groups with group homomorphisms, a category $\Top$ of topological spaces and continuous maps, and many more. \begin{definition} A \emph{functor} $F$ from a category $\cat{C}$ and to a category $\cat{D}$ consists of a function $F_0$ from the objects of $\cat{C}$ to the objects of $\cat{D}$ and a function $F_1$ from maps in $\cat{C}$ to maps in $\cat{D}$, such that diff --git a/thesis/4_Constructions.tex b/thesis/4_Constructions.tex index a139903..d1dae69 100644 --- a/thesis/4_Constructions.tex +++ b/thesis/4_Constructions.tex @@ -21,7 +21,7 @@ $$\del_n = d_0 - d_1 + \ldots + (-1)^n d_n: A_n \to A_{n-1}.$$ In this calculation we did the following. We split the inner sum in two halves \refeqn{1} and we use the simplicial equations on the second sum \refeqn{2}. Then we do a shift of indices \refeqn{3}. By interchanging the roles of $i$ and $j$ in the second sum, we have two equal sums which cancel out. So indeed this is a chain complex. \end{proof} -Thus, associated to a simplicial abelian group $A$ we obtain a chain complex $M(A)$ with $M(A)_n = A_n$ and the boundary operator as above. This construction defines a functor +Thus, associated to a simplicial abelian group $A$ we obtain a chain complex $M(A)$ with $M(A)_n = A_n$ and the boundary operator as above. Following the book \cite{goerss} we will call the chain complex $M(X)$ the \emph{Moore complex} or \emph{unnormalized chain complex} of $X$. This construction defines a functor $$ M: \sAb \to \Ch{\Ab} $$ by assigning $M(f)_n = f_n$ for a natural transformation $f: A \to B$. It follows from a nice calculation that $M(f)$ is indeed a chain map: \begin{align*} @@ -230,8 +230,7 @@ giving us $\Hom{A}{F\Z[\Delta[-]]}{a}: \DELTA^{op} \to \Ab$. Similarly $G$ itsel Many functors to $\sAb$ can be shown to have this description.\footnote{And also many functors to $\sSet$ are of this form if we leave out all additivity requirements.} In our case we could have defined our functor $K$ as $$ K'(C) = \Hom{\Ch{\Ab}}{N\Z[\Delta[-]]}{C}. $$ -We will not show that this functor $K'$ is isomorphic to our functor $K$ defined earlier, however we will indicate that it makes sense by writing out explicit calculations for $K'(C)_0$ and $K'(C)_1$. - +We will not show that this functor $K'$ is isomorphic to our functor $K$ defined earlier, however we will indicate that it makes sense by writing out explicit calculations for $K'(C)_0$ and $K'(C)_1$. First we see that $$ K'(C)_0 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[0]}{C} = \Bigg\{ \begin{tikzpicture}[baseline=-0.5ex] \matrix (m) [matrix of math nodes, row sep=1em, column sep=1em] { @@ -245,9 +244,8 @@ $$ K'(C)_0 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[0]}{C} = \Bigg\{ \foreach \i/\j in {2/2, 3/1, 4/0} \draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i); \end{tikzpicture} \Bigg\} \iso C_0 = K(C)_0, $$ -because for $f_1, f_2, \ldots$ there is now choice at all, and for $f_0: \Z \to C_0$ we only have to choose an image for $1 \in \Z$. In the next dimension we see: - -$$ K(C)_1 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[1]}{C} = \Bigg\{ +because for $f_1, f_2, \ldots$ there is now choice at all, and for $f_0: \Z \to C_0$ we only have to choose an image for $1 \in \Z$. In the next dimension we see +$$ K'(C)_1 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[1]}{C} = \Bigg\{ \begin{tikzpicture}[baseline=-0.5ex] \matrix (m) [matrix of math nodes, row sep=1em, column sep=1em] { \cdots & 0 & \Z & \Z^2 \\ @@ -260,6 +258,5 @@ $$ K(C)_1 = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[1]}{C} = \Bigg\{ \foreach \i/\j in {2/2, 3/1, 4/0} \draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i); \end{tikzpicture} \Bigg\} \iso C_1 \oplus C_0 = K(C)_1, $$ - because again we can choose $f_1$ anyway we want, which gives us $C_1$. But then we are forced to choose $f_0(x, x) = \del(f_1(x))$ for all $x \in \Z$, so we are left with choosing an element $c \in C_0$ for defining $f(1,-1) = c$. Adding this gives $C_1 \oplus C_0$. diff --git a/thesis/references.bib b/thesis/references.bib index d1a9bef..737012f 100644 --- a/thesis/references.bib +++ b/thesis/references.bib @@ -140,3 +140,21 @@ MRREVIEWER = {Fernando Muro}, MRCLASS = {18-02}, MRNUMBER = {1712872 (2001j:18001)}, } + +% For naming the unnormalized chain complex +@book {goerss, + AUTHOR = {Goerss, Paul G. and Jardine, John F.}, + TITLE = {Simplicial homotopy theory}, + SERIES = {Progress in Mathematics}, + VOLUME = {174}, + PUBLISHER = {Birkh\"auser Verlag}, + ADDRESS = {Basel}, + YEAR = {1999}, + PAGES = {xvi+510}, + ISBN = {3-7643-6064-X}, + MRCLASS = {55U10 (18G55 55-01 55Pxx)}, + MRNUMBER = {1711612 (2001d:55012)}, +MRREVIEWER = {R. M. Vogt}, + DOI = {10.1007/978-3-0348-8707-6}, + URL = {http://dx.doi.org/10.1007/978-3-0348-8707-6}, +}