### restructure com alg

parent 12aa8582
 \section*{About} \section{Overview} The lecture content is mostly guided by Antoine Chamber-Loir's Book (Mostly) Commutative Algebra. We will not follow it one-to-one, but chapter numbers will be provided on the go. There will be some extra topics covered that are not in the book. Definitions and theorems enclosed with parenthesis are not part of the lecture and (mostly) come from the exercise sheets. \section{Overview} \subsection{Introduction/Motivation} \subsubsection*{What is it?} ... ... @@ -183,113 +181,3 @@ using this notion, then $f: S \to T$ is an $R$-algebra morphism if and only if \end{enumerate} \end{xmp} \section{The language of Categories and Functors [ACL A.3]} When looking at common features mathematical strucutres'' such as Groups, Fields, Top. Spaces, Hilbert spaces, etc., one finds that it is very hard to define them using the common method of specifying properties of subsets of sets, or sets of subsets of sets etc. The Eilenberg-MacLane notion of a category is as follows: \begin{dfn}[Category] A category $\textsf{C}$ consists of the following data: \begin{itemize} \item A collection'' of \textbf{objects} $X$ of $\textsf{C}$. \item For any objects $X,Y$ of $\textsf{C}$, we define a \emph{set} $\Hom_{\textsf{C}}(X,Y)$ of \textbf{morphisms} from $X$ to $Y$ in $\textsf{C}$. Instead of writing $f \in \Hom(X,Y)$, we will usually write $X \stackrel{f}{\to}Y$ or $f: X \to Y$. \item For every object $X$ of $\textsf{C}$, there exists an \textbf{identity-morphism} $\id_X: X \to X$. \item Morphisms can be composed. For objects $X,Y,Z$, there exists a map \begin{align*} \Hom_{\textsf{C}}(X,Y) \times \Hom_{\textsf{C}}(Y,Z) &\to \Hom_{\textsf{C}}(X,Z)\\ (g,f) &\mapsto f \circ g \end{align*} \begin{center} \begin{tikzcd}[ ] X \arrow[]{r}{g} \arrow[bend left]{rr}{f \circ g}& Y \arrow[]{r}{f}& Z \end{tikzcd} \end{center} \end{itemize} This data must respect the following axioms. \begin{itemize} \item Identity laws: $\id_Y \circ f = f \quad g \circ \id_x = g$ \item Composition is associative: $f \circ (g \circ h) = (f \circ g) \circ h$ \end{itemize} \end{dfn} \begin{rem}[] We are intentionally being obtuse when talking about collection'' of objects. The reason is that we want to talk about the collection'' of \emph{all} sets. Using the normal set theory we know from the first semester, we quickly run into problems as there is no such thing as a set of all sets. There are a number of solutions to this though. \begin{itemize} \item There are set theories that make use of \emph{classes} to enlarge the notion of set. \item Restrict what sets we are allowed to talk about. For example using \emph{Grothendieck universes}. \item Ignore the problem. \end{itemize} We will chose the third option. If you still feel unfomfortable, one can say that there is some logical formuale $\Phi_{\textsf{C}}(X)$ in one variable, which evaluates to \texttt{true}, if $X$ is an object in the category $\textsf{C}$. \end{rem} \begin{xmp}[] When we say that something is a category, we usually specify what the objects are and what the morphisms (and composition) look like. Feel free to skip those you are not familiar with. \begin{itemize} \item The category of sets $\Set$, where the objects are sets and morphisms are functions. \item The category $\Grp$, where the objects are groups and morphisms are group homomorphisms. \item $\Top$ the category of topological spaces, where the objects are topological spaces (with their topology) and where the morphisms are \emph{continuous} functions. \item $\Top_{\ast}$, with Topological spaces with a basepoint $(X,\tau,x_0)$ as objects and where the morphisms are continous, base-point preserving functions. \item $\Ring$ with Rings as objects and ring homomorphisms as morphisms. \item Given a fixed ring $R$, the category $R-\textsf{Mod}$ of $R$-modules and linear maps. \item \textsf{Graph} has graphs as objects and graph homomorphisms as morphisms. \item For a fixed field $K$, the category $\textsf{Vec}_K$ has $K$-vector spaces as objects with linear maps as morphisms. \item One rather weird category is the category of fields with field homomomorphisms. \end{itemize} The examples above are categories were objects were \emph{sets with structure} and the morphisms were \emph{structure preserving} maps. But the objects need not be sets, and morphisms need not be functions! \begin{itemize} \item The trivial category, which consists of a single object $\ast$ and its identity morphism $\ast \stackrel{\id}{\to} \ast$. \item A group (or monoid) $(G,\cdot,e)$ can be viewed as a category with a single object $\ast$, where the morphisms $g: \ast \to \ast$ are the elements $g \in G$ and where composition of morphisms is defined as the multiplication in $G$: $g \circ h := g \cdot h$. \item A poset $(P,\leq)$ is a category where the objects are the elements of $P$ and there exist unique morphisms $x \to y$ if and only if $x \leq y$. \item \textsf{Hask}, the idealized cateogory of Haskell types and functions. \item Given a category $\textsf{C}$, we can talk about its \textbf{arrow category}, where the objects are morphisms $(X \stackrel{f}{\to} Y)$ in $\textsf{C}$ and a morphism between objects $(X \stackrel{f}{\to} Y), (X' \stackrel{g}{\to}Y')$ are pairs of $\textsf{C}$-morphisms $\alpha: X \to X'$ and $\beta: Y \to Y'$ such that the following diagram commutes: \begin{center} \begin{tikzcd}[ ] %\arrow[bend right,swap]{dr}{F} X \arrow[]{r}{f} \arrow[]{d}{\alpha}& Y \arrow[]{d}{\beta}\\ X' \arrow[]{r}{g}& Y' \end{tikzcd} \end{center} i.e. such that $g \circ \alpha = \beta \circ f$. \end{itemize} \end{xmp} In general, always specify what category you are working in. The real numbers $\R$ are a Set, Topological space, Group, Field, Poset etc. \begin{rem}[] We say that a triangle \begin{tikzcd}[column sep=0.8em] & \cdot \arrow[]{dr}{h}\\ \cdot \arrow[]{ur}{g} \arrow[]{rr}{f} & & \cdot \end{tikzcd} \textbf{commutes}, if $g \circ h = f$. A diagram is commutative, if every triangle of that diagram commutes. This includes non-obvious triangles, such as those formed by morphisms obtained through composition or the identity morphisms. \end{rem}
 \section{The language of Categories and Functors [ACL A.3]} When looking at common features mathematical strucutres'' such as Groups, Fields, Top. Spaces, Hilbert spaces, etc., one finds that it is very hard to define them using the common method of specifying properties of elements or subsets of sets, or sets of subsets of sets etc. The language of category theory lets us move away from the element-set relationship and focus more on the relationship between sets. As we will see later, many constructions from Algebra, Set Theory, Topology etc. can be unified and generalized. \subsection{Basic Definitions} \input{./files/02/02a-basic-definitions} \subsection{Functors and Natural transformations} \input{./files/02/02b-functors-natural-transformations} \subsection{Isomorphisms} \input{./files/02/02c-isomorphisms} \subsection{The Yoneda Lemma} \input{./files/02/02d-yoneda}
 The Eilenberg-MacLane notion of a category is as follows: \begin{dfn}[Category] A category $\textsf{C}$ consists of the following data: \begin{itemize} \item A collection'' of \textbf{objects} $X$ of $\textsf{C}$. \item For any objects $X,Y$ of $\textsf{C}$, we define a \emph{set} $\Hom_{\textsf{C}}(X,Y)$ of \textbf{morphisms} from $X$ to $Y$ in $\textsf{C}$. Instead of writing $f \in \Hom(X,Y)$, we will usually write $X \stackrel{f}{\to}Y$ or $f: X \to Y$. \item For every object $X$ of $\textsf{C}$, there exists an \textbf{identity-morphism} $\id_X: X \to X$. \item Morphisms can be composed. For objects $X,Y,Z$, there exists a map \begin{align*} \Hom_{\textsf{C}}(X,Y) \times \Hom_{\textsf{C}}(Y,Z) &\to \Hom_{\textsf{C}}(X,Z)\\ (g,f) &\mapsto f \circ g \end{align*} \begin{center} \begin{tikzcd}[ ] X \arrow[]{r}{g} \arrow[bend left]{rr}{f \circ g}& Y \arrow[]{r}{f}& Z \end{tikzcd} \end{center} \end{itemize} This data must respect the following axioms. \begin{itemize} \item Identity laws: $\id_Y \circ f = f \quad g \circ \id_x = g$ \item Composition is associative: $f \circ (g \circ h) = (f \circ g) \circ h$ \end{itemize} \end{dfn} \begin{rem}[] We are intentionally being obtuse when talking about collection'' of objects. The reason is that we want to talk about the collection'' of \emph{all} sets. Using the normal set theory we know from the first semester, we quickly run into problems as there is no such thing as a set of all sets. There are a number of solutions to this though. \begin{itemize} \item There are set theories that make use of \emph{classes} to enlarge the notion of set. \item Restrict what sets we are allowed to talk about. For example using \emph{Grothendieck universes}. \item Ignore the problem. \end{itemize} We will chose the third option. If you still feel unfomfortable, one can say that there is some logical formuale $\Phi_{\textsf{C}}(X)$ in one variable, which evaluates to \texttt{true}, if $X$ is an object in the category $\textsf{C}$. \end{rem} \begin{xmp}[] When we say that something is a category, we usually specify what the objects are and what the morphisms (and composition) look like. Feel free to skip those you are not familiar with. \begin{itemize} \item The category of sets $\Set$, where the objects are sets and morphisms are functions. \item The category $\Grp$, where the objects are groups and morphisms are group homomorphisms. \item $\Top$ the category of topological spaces, where the objects are topological spaces (with their topology) and where the morphisms are \emph{continuous} functions. \item $\Top_{\ast}$, with Topological spaces with a basepoint $(X,\tau,x_0)$ as objects and where the morphisms are continous, base-point preserving functions. \item $\Ring$ with Rings as objects and ring homomorphisms as morphisms. \item Given a fixed ring $R$, the category $R-\textsf{Mod}$ of $R$-modules and linear maps. \item \textsf{Graph} has graphs as objects and graph homomorphisms as morphisms. \item For a fixed field $K$, the category $\textsf{Vec}_K$ has $K$-vector spaces as objects with linear maps as morphisms. \item One rather weird category is the category of fields with field homomomorphisms. \end{itemize} The examples above are categories were objects were \emph{sets with structure} and the morphisms were \emph{structure preserving} maps. But the objects need not be sets, and morphisms need not be functions! \begin{itemize} \item The trivial category, which consists of a single object $\ast$ and its identity morphism $\ast \stackrel{\id}{\to} \ast$. \item A group (or monoid) $(G,\cdot,e)$ can be viewed as a category with a single object $\ast$, where the morphisms $g: \ast \to \ast$ are the elements $g \in G$ and where composition of morphisms is defined as the multiplication in $G$: $g \circ h := g \cdot h$. \item A poset $(P,\leq)$ is a category where the objects are the elements of $P$ and there exist unique morphisms $x \to y$ if and only if $x \leq y$. \item \textsf{Hask}, the idealized cateogory of Haskell types and functions. \item Given a category $\textsf{C}$, we can talk about its \textbf{arrow category}, where the objects are morphisms $(X \stackrel{f}{\to} Y)$ in $\textsf{C}$ and a morphism between objects $(X \stackrel{f}{\to} Y), (X' \stackrel{g}{\to}Y')$ are pairs of $\textsf{C}$-morphisms $\alpha: X \to X'$ and $\beta: Y \to Y'$ such that the following diagram commutes: \begin{center} \begin{tikzcd}[ ] %\arrow[bend right,swap]{dr}{F} X \arrow[]{r}{f} \arrow[]{d}{\alpha}& Y \arrow[]{d}{\beta}\\ X' \arrow[]{r}{g}& Y' \end{tikzcd} \end{center} i.e. such that $g \circ \alpha = \beta \circ f$. \end{itemize} \end{xmp} In general, always specify what category you are working in. The real numbers $\R$ are a Set, Topological space, Group, Field, Poset etc. \begin{rem}[] We say that a triangle \begin{tikzcd}[column sep=0.8em] & \cdot \arrow[]{dr}{h}\\ \cdot \arrow[]{ur}{g} \arrow[]{rr}{f} & & \cdot \end{tikzcd} \textbf{commutes}, if $g \circ h = f$. A diagram is commutative, if every triangle of that diagram commutes. This includes non-obvious triangles, such as those formed by morphisms obtained through composition or the identity morphisms. \end{rem}
 \begin{dfn}[Functor] Let $\textsf{C},\textsf{D}$ be categories. A \textbf{functor} $F$ from $\textsf{C}$ to $\textsf{D}$, denoted $F: \textsf{C} \to \textsf{D}$ is a rule assigning \begin{enumerate} \item an object $F(X)$ of $\textsf{D}$ for any objects $X$ of $\textsf{C}$ \item A morphism $F(f): F(X) \to F(Y)$ to any morphism $f: X \to Y$. \end{enumerate} such that $F(\id_X) = \id_{F(X)}$ and for any pair of composable morphisms: $(X \stackrel{f}{\to} Y), (Y \stackrel{g}{\to}Z)$ in $\textsf{C}$, the following diagram commutes (in $\textsf{D}$) \begin{center} \begin{tikzcd}[] %\arrow[bend right,swap]{dr}{F} F(X) \arrow[]{r}{F(f)} \arrow[bend right]{rr}{F(g \circ f)} \arrow[bend left]{rr}{F(g) \circ F(f)}& F (Y) \arrow[]{r}{F(g)} & F (Z) \end{tikzcd} \end{center} \end{dfn} From now on, we will omit parenthesis for functor application unless necessary so instead of $F(X)$ or $F(f)$, we write $FX$ or $Ff$. \begin{xmp}[] Functors are plentiful in Algebra and Topology. \begin{enumerate} \item If we have a group $(G,\cdot,e)$, we can forget the group structure and only look at the underlying set $G$. Group homomorphisms turn into functions. We call this the \textbf{forgetful functor}. The same construction works for topological spaces, where we forget the topology of a topological space. \item We can turn every field into a group, by removing the $0$. Field homomorphisms then become grougroup. \item Given any ring $R$, we can obtain the group of units $R^{\times}$. This is the \textbf{units functor} from the category of rings to the category of groups. \item As you should know from Topology last semester, the \textbf{fundamental group} $\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. \end{enumerate} \end{xmp} Consider rule $\textsf{F}: \textsf{Vec}_k \to \textsf{Vec}_k$ which maps vector spaces to their dual space and linear maps to their dual maps. So for vector spaces $V,W$ and a linear map $f \in \Hom(V,W)$ we define \begin{align*} F V = V^{\ast}, \quad \text{and} \quad Ff =: f^{\ast}: W^{\ast} \to V^{\ast}, \quad f^{\ast} \beta = \beta \circ f \in V^{\ast} \end{align*} You might notice that the direction of $Ff$ is opposite to that of $f$, so it doesn't technically fit the definition we gave earlier. \begin{dfn}[] Let $\textsf{C}$ be a category. The \textbf{opposite category} $\textsf{C}^{\text{op}}$ is the category with the same objects as $\textsf{C}$ and $\Hom_{\textsf{C}^{\text{op}}}(X,Y) = \Hom_{\textsf{C}}(Y,X)$. \begin{center} \begin{tikzcd}[] & \textsf{C} & & & & \textsf{C}^{\text{op}}\\ \\ X \arrow[]{r}{f} \arrow[bend left]{rr}{g \circ f} & Y \arrow[]{r}{g} & Z & & X & Y \arrow[]{l}{f^{\text{op}}} & Z \arrow[]{l}{g^{\text{op}}} \arrow[bend right,swap]{ll}{f^{\text{op}} \circ g^{\text{op}} = (g \circ f)^{op}} \end{tikzcd} \end{center} A functor $F: \textsf{C} \to \textsf{D}^{\text{op}}$ is called a \textbf{contravariant} functor from $\textsf{C}$ to $\textsf{D}$. \end{dfn} We will stil call it a functor and (if necessary) specify wheter the functor is \textbf{covariant} or contravariant. \begin{xmp}[Hom-functor] Since $\Hom_{\textsf{C}}(X,Y)$ is a set, we can define two set-valued functors. Fix an object $X$ \begin{enumerate} \item The \textbf{covariant Hom-functor} is \begin{align*} h^{X}: \textsf{C} \to \Set, \quad h^{X}(A) = \Hom_{\textsf{C}}(X,A), \quad h^{X}(f) = (f \circ -) = (g \mapsto f \circ g) \end{align*} \begin{center} \begin{tikzcd}[ ] X \arrow[]{r}{g} & A \arrow[]{r}{f} & B \end{tikzcd} \end{center} \item The \textbf{contravariant Hom-functor} is \begin{align*} h_{X}: \textsf{C} \to \Set, \quad h_X(A) = \Hom_{\textsf{C}}(A,X), \quad h_X(f) = (- \circ f) = (g \mapsto g \circ f) \end{align*} \begin{center} \begin{tikzcd}[ ] A \arrow[]{r}{f} & B \arrow[]{r}{g} & X \end{tikzcd} \end{center} \end{enumerate} \textbf{Warning for Physicists:} The positionig of the index is opposite to that for tensors. For example, the covariant derivative $\del_{\mu}$ has a downstairs index, whereas the covariant Hom-functor $h^{X}$ has it upstairs. A mnemonic is that $h^{X}$ has morphisms that \emph{fall} from the upstairs $X$ and $h_X$ has morphisms that fall downstairs into $X$. And if we look at $h^{X}(f)$ from above to below, the $f$ comes \emph{after} $X$, so $h^{X}(f)$ is \emph{post-composition} with $f$ and $h_X(f)$ is \emph{pre-composition} with $f$. \end{xmp}
 The language of category theory lets us unify definitions from various contexts. One such definition is that of an isomorphism. \begin{dfn}[] Let $\textsf{C}$. A morphism $X \stackrel{f}{\to}Y$ in $\textsf{C}$ is said to be an \textbf{isomorphism}, if there exists a morphism $(Y \stackrel{g}{\to}X)$ such that $f \circ g = \id_Y$ and $g \circ f = \id_X$, or equivalently, if the following diagram commutes: \begin{center} \begin{tikzcd}[ ] X \arrow[bend left]{r}{f} \arrow[loop left]{l}{\id_X} & Y \arrow[bend left]{l}{g} \arrow[loop right]{r}{\id_Y} \end{tikzcd} \end{center} If that is the case, we say that $X$ and $Y$ are isomorphic and write $X \iso Y$ or $f: X \stackrel{\sim}{\to}Y$. \end{dfn} In the category $\Set$, these are the bijective maps. In $\Top$, they are the homeomorphisms. In $\Grp$ they are the group-isomorphisms etc. \begin{lem}[] If $f: X \to Y$ is an isomorphism, then the map $g$ with the properties above map is unique. \end{lem} \begin{proof} Suppose $g$ and $h$ satisfy the relations above. Then \begin{align*} g = g \circ \id_Y = g \circ (f \circ h) = (g \circ f) \circ h = \id_X \circ h = h \end{align*} \end{proof} The map $g$ is called the \textbf{inverse} of $f$ and sometimes denoted $f^{-1}$. The inverse is also an isomorphism and $(f^{-1})^{-1} = f$. \begin{lem}[] \label{lem:functor-of-iso} Let $F: \textsf{C} \to \textsf{D}$ be a functor and $X \stackrel{f}{\to} Y$ an isomorphism. Then $Ff$ is an isomorphism and $(Ff)^{-1} = F(f^{-1})$. \end{lem} \begin{proof} We check the identity relations. \begin{align*} Ff \circ F(f^{-1}) = F(f \circ f^{-1}) = F(\id_Y) = \id_{FY} \end{align*} similarly, one checks that $F(f^{-1}) \circ = \id_{FX}$. \end{proof} We will write $\text{Isom}_{\textsf{C}}(X,Y)$ for the set of isomorphisms from $X$ to $Y$ and $\Aut_{\textsf{C}}(X) = \text{Isom}_{\textsf{C}}(X,X)$. \begin{lem}[] $\Aut_{\textsf{C}}(X)$ is a group with composition. \end{lem}
 \subsection{Functors and Natural transformations} \begin{dfn}[Functor] Let $\textsf{C},\textsf{D}$ be categories. A \textbf{functor} $F$ from $\textsf{C}$ to $\textsf{D}$, denoted $F: \textsf{C} \to \textsf{D}$ is a rule assigning \begin{enumerate} \item an object $F(X)$ of $\textsf{D}$ for any objects $X$ of $\textsf{C}$ \item A morphism $F(f): F(X) \to F(Y)$ to any morphism $f: X \to Y$. \end{enumerate} such that $F(\id_X) = \id_{F(X)}$ and for any pair of composable morphisms: $(X \stackrel{f}{\to} Y), (Y \stackrel{g}{\to}Z)$ in $\textsf{C}$, the following diagram commutes (in $\textsf{D}$) \begin{center} \begin{tikzcd}[] %\arrow[bend right,swap]{dr}{F} F(X) \arrow[]{r}{F(f)} \arrow[bend right]{rr}{F(g \circ f)} \arrow[bend left]{rr}{F(g) \circ F(f)}& F (Y) \arrow[]{r}{F(g)} & F (Z) \end{tikzcd} \end{center} \end{dfn} From now on, we will omit parenthesis for functor application unless necessary so instead of $F(X)$ or $F(f)$, we write $FX$ or $Ff$. \begin{xmp}[] Functors are plentiful in Algebra and Topology. \begin{enumerate} \item If we have a group $(G,\cdot,e)$, we can forget the group structure and only look at the underlying set $G$. Group homomorphisms turn into functions. We call this the \textbf{forgetful functor}. The same construction works for topological spaces, where we forget the topology of a topological space. \item We can turn every field into a group, by removing the $0$. Field homomorphisms then become grougroup. \item Given any ring $R$, we can obtain the group of units $R^{\times}$. This is the \textbf{units functor} from the category of rings to the category of groups. \item As you should know from Topology last semester, the \textbf{fundamental group} $\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. \end{enumerate} \end{xmp} Consider rule $\textsf{F}: \textsf{Vec}_k \to \textsf{Vec}_k$ which maps vector spaces to their dual space and linear maps to their dual maps. So for vector spaces $V,W$ and a linear map $f \in \Hom(V,W)$ we define \begin{align*} F V = V^{\ast}, \quad \text{and} \quad Ff =: f^{\ast}: W^{\ast} \to V^{\ast}, \quad f^{\ast} \beta = \beta \circ f \in V^{\ast} \end{align*} You might notice that the direction of $Ff$ is opposite to that of $f$, so it doesn't technically fit the definition we gave earlier. \begin{dfn}[] Let $\textsf{C}$ be a category. The \textbf{opposite category} $\textsf{C}^{\text{op}}$ is the category with the same objects as $\textsf{C}$ and $\Hom_{\textsf{C}^{\text{op}}}(X,Y) = \Hom_{\textsf{C}}(Y,X)$. \begin{center} \begin{tikzcd}[] & \textsf{C} & & & & \textsf{C}^{\text{op}}\\ \\ X \arrow[]{r}{f} \arrow[bend left]{rr}{g \circ f} & Y \arrow[]{r}{g} & Z & & X & Y \arrow[]{l}{f^{\text{op}}} & Z \arrow[]{l}{g^{\text{op}}} \arrow[bend right,swap]{ll}{f^{\text{op}} \circ g^{\text{op}} = (g \circ f)^{op}} \end{tikzcd} \end{center} A functor $F: \textsf{C} \to \textsf{D}^{\text{op}}$ is called a \textbf{contravariant} functor from $\textsf{C}$ to $\textsf{D}$. \end{dfn} We will stil call it a functor and (if necessary) specify wheter the functor is \textbf{covariant} or contravariant. \begin{xmp}[Hom-functor] Since $\Hom_{\textsf{C}}(X,Y)$ is a set, we can define two set-valued functors. Fix an object $X$ \begin{enumerate} \item The \textbf{covariant Hom-functor} is \begin{align*} h^{X}: \textsf{C} \to \Set, \quad h^{X}(A) = \Hom_{\textsf{C}}(X,A), \quad h^{X}(f) = (f \circ -) = (g \mapsto f \circ g) \end{align*} \begin{center} \begin{tikzcd}[ ] X \arrow[]{r}{g} & A \arrow[]{r}{f} & B \end{tikzcd} \end{center} \item The \textbf{contravariant Hom-functor} is \begin{align*} h_{X}: \textsf{C} \to \Set, \quad h_X(A) = \Hom_{\textsf{C}}(A,X), \quad h_X(f) = (- \circ f) = (g \mapsto g \circ f) \end{align*} \begin{center} \begin{tikzcd}[ ] A \arrow[]{r}{f} & B \arrow[]{r}{g} & X \end{tikzcd} \end{center} \end{enumerate} \textbf{Warning for Physicists:} The positionig of the index is opposite to that for tensors. For example, the covariant derivative $\del_{\mu}$ has a downstairs index, whereas the covariant Hom-functor $h^{X}$ has it upstairs. A mnemonic is that $h^{X}$ has morphisms that \emph{fall} from the upstairs $X$ and $h_X$ has morphisms that fall downstairs into $X$. And if we look at $h^{X}(f)$ from above to below, the $f$ comes \emph{after} $X$, so $h^{X}(f)$ is \emph{post-composition} with $f$ and $h_X(f)$ is \emph{pre-composition} with $f$. \end{xmp} \subsection{Isomorphisms} The language of category theory lets us unify definitions from various contexts. One such definition is that of an isomorphism. \begin{dfn}[] Let $\textsf{C}$. A morphism $X \stackrel{f}{\to}Y$ in $\textsf{C}$ is said to be an \textbf{isomorphism}, if there exists a morphism $(Y \stackrel{g}{\to}X)$ such that $f \circ g = \id_Y$ and $g \circ f = \id_X$, or equivalently, if the following diagram commutes: \begin{center} \begin{tikzcd}[ ] X \arrow[bend left]{r}{f} \arrow[loop left]{l}{\id_X} & Y \arrow[bend left]{l}{g} \arrow[loop right]{r}{\id_Y} \end{tikzcd} \end{center} If that is the case, we say that $X$ and $Y$ are isomorphic and write $X \iso Y$ or $f: X \stackrel{\sim}{\to}Y$. \end{dfn} In the category $\Set$, these are the bijective maps. In $\Top$, they are the homeomorphisms. In $\Grp$ they are the group-isomorphisms etc. \begin{lem}[] If $f: X \to Y$ is an isomorphism, then the map $g$ with the properties above map is unique. \end{lem} \begin{proof} Suppose $g$ and $h$ satisfy the relations above. Then \begin{align*} g = g \circ \id_Y = g \circ (f \circ h) = (g \circ f) \circ h = \id_X \circ h = h \end{align*} \end{proof} The map $g$ is called the \textbf{inverse} of $f$ and sometimes denoted $f^{-1}$. The inverse is also an isomorphism and $(f^{-1})^{-1} = f$. \begin{lem}[] \label{lem:functor-of-iso} Let $F: \textsf{C} \to \textsf{D}$ be a functor and $X \stackrel{f}{\to} Y$ an isomorphism. Then $Ff$ is an isomorphism and $(Ff)^{-1} = F(f^{-1})$. \end{lem} \begin{proof} We check the identity relations. \begin{align*} Ff \circ F(f^{-1}) = F(f \circ f^{-1}) = F(\id_Y) = \id_{FY} \end{align*} similarly, one checks that $F(f^{-1}) \circ = \id_{FX}$. \end{proof} We will write $\text{Isom}_{\textsf{C}}(X,Y)$ for the set of isomorphisms from $X$ to $Y$ and $\Aut_{\textsf{C}}(X) = \text{Isom}_{\textsf{C}}(X,X)$.