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bsc-thesis
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### Fixed typos

master Joshua Moerman 9 years ago
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6 changed files with 29 additions and 30 deletions
1. 8
thesis/1_CategoryTheory.tex
2. 6
thesis/2_ChainComplexes.tex
3. 10
thesis/3_SimplicialAbelianGroups.tex
4. 21
thesis/4_Constructions.tex
5. 10
thesis/5_Homotopy.tex
6. 4
thesis/DoldKan.tex

#### 8 thesis/1_CategoryTheory.tex View File

 @ -19,7 +19,7 @@ We will briefly define categories and functors to fix the notation. We will not 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 $g \circ f$. We will need the category $\Set$ of sets with functions, the category $\Ab$ of abelian groups with group homomorphisms and the category $\Top$ of topological spaces and continuous maps.   \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  A \emph{functor} $F$ from a category $\cat{C}$ 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  \begin{itemize}  \item for $f: A \to B$, we have $F_1(f): F_0(A) \to F_0(B)$,  \item $F_1(\id_A) = \id_{F_0(A)}$ and @ -71,7 +71,7 @@ This now also gives a notion of isomorphisms between functors. It can be easily   For any category $\cat{C}$ we can define the $\mathbf{Hom}$-functor. It assigns to two objects in $\cat{C}$ the set of maps between them, i.e. $$\Hom{\cat{C}}{-}{-}: \cat{C}^{op} \times \cat{C} \to \Set.$$ We will show that it indeed gives a functor in the first argument, a similar proof works for the second argument. Let $f: A' \to A$ be a map in $\cat{C}$ and $g \in \Hom{\cat{C}}{A}{B}$, then $g \circ f \in \Hom{\cat{C}}{A'}{B}$. Hence the assignment $g \mapsto g \circ f$ is a map from $\Hom{\cat{C}}{A}{B}$ to $\Hom{\cat{C}}{A'}{B}$. Note that the direction of the map if reversed. Using associativity and identity it is easily checked that this assignment is functorial. We will show that it indeed defines a functor in the first argument, a similar proof works for the second argument. Let $f: A' \to A$ be a map in $\cat{C}$ and $g \in \Hom{\cat{C}}{A}{B}$, then $g \circ f \in \Hom{\cat{C}}{A'}{B}$. Hence the assignment $g \mapsto g \circ f$ is a map from $\Hom{\cat{C}}{A}{B}$ to $\Hom{\cat{C}}{A'}{B}$. Note that the direction of the map if reversed. Using associativity and identity it is easily checked that this assignment is functorial.   \subsection{Equivalence} Recall that an \emph{isomorphism} between categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C} \to \cat{D}$ and $G: \cat{D} \to \cat{C}$ such that @ -138,10 +138,10 @@ The first definition of adjunction is useful when dealing with maps, since it gi  It is left for the reader to check that this indeed gives a group homomorphism and that the functor laws hold. There is a map $\eta: S \to U\Z[S]$ given by  $$\eta(s) = e_s.$$  And given any map $f: S \to U(A)$ for any abelian group $A$, we can define  $$\overline{f}(\phi) = \sum_{x \in \text{supp}(\phi)} \phi(x) \cdot f(x).$$  $$\overline{f}(\phi) = \sum_{x \in \text{supp}(\phi)} \phi(x) \cdot e_{f(x)}.$$  It is clear that $U(\overline{f}) \circ \eta = f$. We will leave the other details (naturality of $\eta$, $\overline{f}$ being a group homomorphism, and uniqueness w.r.t.~$U(\overline{f}) \circ \eta = f$) to the reader.    By the other description of adjunctions we have $\Hom{\Ab}{\Z[S]}{T} \iso \Hom{\Set}{S}{U(T)}$, which exactly tells us that we can define a group homomorphism from $\Z[S]$ to $T$ by only specifying it on the generators $e_s, s \in S$. This fact is used throughout the thesis.  By the other description of adjunctions we have $\Hom{\Ab}{\Z[S]}{A} \iso \Hom{\Set}{S}{U(A)}$, which exactly tells us that we can define a group homomorphism from $\Z[S]$ to $A$ by only specifying it on the generators $e_s, s \in S$. This fact is used throughout this thesis. \end{example}   \subsection{The Yoneda lemma}