\documentclass[11pt,a4paper]{scrartcl}

\usepackage[margin=2cm]{geometry}

\usepackage{amsmath,amssymb,amsfonts,centernot}
\usepackage{tikz-cd}

\newcommand{\id}{\operatorname{id}}
\newcommand{\Ob}{\operatorname{Ob}}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbZ}{\mathbb{Z}}
\newcommand{\bbQ}{\mathbb{Q}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\injto}{\hookrightarrow}
\newcommand{\Hom}{\operatorname{Hom}}

\newcommand{\cat}[1]{\mathsf{#1}}
\newcommand{\xto}[1]{\xrightarrow{#1}}


\begin{document}
Quick reference:
\begin{itemize}
	\itemsep0em
	\item Familiar categories: $\cat{Set}$ (sets and functions), $\cat{Group}$ (groups and homomorphisms), $\cat{AbGroup}$ (abelian groups and homomorphisms), $\cat{Vect}_k$ ($k$-vector spaces and $k$-linear maps), $\cat{Mat}_k$ (natural numbers and matrices), $\cat{CRing}$ (commutative rings and ring morphisms), $\cat{Top}$ (topological spaces and continuous maps), $\cat{Met}$ (metric spaces and continuous maps), $\cat{Hilb}_\bbR$ (real Hilbert spaces and bounded linear operators).
	\item An \textit{isomorphism} $f: A \to B$ admits a $g: B \to A$ such that $gf = \id_A$, $fg = \id_B$.
	\item An \textit{epimorphism} $e$ is right-cancellable: $fe = ge \implies f = g$.
	\item A \textit{monomorphism} $m$ is left-cancellable: $mf = mg \implies f = g$.
	\item $X \in \Ob(\cC)$ is \textit{terminal} if for any $Y \in \Ob(\cC)$, there exists a unique morphisms $Y \to X$.
	\item $X \in \Ob(\cC)$ is \textit{initial} if for any $Y \in \Ob(\cC)$, there exists a unique morphisms $X \to Y$.
	\item $(P, p_A: P \to A, p_B: P \to B)$ is a \textit{categorical product} for $A$ and $B$ if for every pair of morphisms $(f_A: X \to A, f_B: X \to B)$, there exists a unique morphisms $\varphi: X \to P$ satisfying $p_A \circ \varphi = f_A$ and $p_B \circ \varphi = f_B$. We use the notation $\varphi = (f_A, f_B)_P$.
	\item $(C, i_A: A \to C, i_B: B \to C)$ is a \textit{categorical coproduct} for $A$ and $B$ if for every pair of morphisms $(g_A: A \to Y, g_B: B \to Y)$, there exists a unique morphisms $\psi: C \to Y$ such that $\psi \circ i_A = g_A$ and $\psi \circ i_B = g_B$. We use the notation $\psi = (g_A, g_B)_C$.
		\[ \begin{tikzcd}
		& & A \\
		X \ar[urr, bend left = 15, "f_A"] \ar[drr, bend right = 15, "f_B"'] \ar[r, dashed, "\varphi"] & P \ar[ur, "p_A"'] \ar[dr, "p_B"]\\
		& & B
		\end{tikzcd}
		\hspace{100pt}
		\begin{tikzcd}
		A \ar[drr, bend left=15, "g_A"] \ar[dr, "i_A"'] \\
		& C \ar[r, dashed, "\psi"] & Y \\
		B \ar[urr, bend right=15, "g_B"'] \ar[ur, "i_B"]
		\end{tikzcd} \]
	\item Given a covariant functor $F: \cA \to \cB$ and objects $A, B \in \Ob(\cA)$, define
		\[ \Phi_{A, B}: \cA(A, B) \to \cB(FA, FB), \quad (A \xto{f} B) \mapsto \left(FA \xto{Ff} FB \right)\]
		$F$ is called \textit{full} if $\Phi_{A,B}$ is surjective for all $A, B$. $F$ is \textit{faithful} if $\Phi_{A, B}$ is \textit{injective} for all $A, B$.
	\item A functor $F: \cA \to \cB$ is an \textit{isomorphism of categories} if there is $S: \cB \to \cA$ such that $FS = \id_\cA$ and $SF = \id_\cB$.
	\item Given functors $F, S: \cA \to \cB$, a \textit{natural transformation} $\eta: F \Rightarrow S$ is a collection of maps $(FA \xto{\eta_A} SA) \in \cB$ for each object $A \in \cA$, such that whenever $(A \xto{f} B) \in \cA$, then
		\[ \begin{tikzcd}
		FA \ar[d, "{\eta_A}"] \ar[r, "Ff"] & FB \ar[d, "{\eta_B}"] \\
		SA \ar[r, "Sf"] & SB
		\end{tikzcd} \quad \quad \text{(This is a diagram in $\cB$)} \]
	\item A natural transformation $\eta: S \Rightarrow T$ is a \textit{natural equivalence} of functors if each component $\eta_A$ is an isomorphism. In this case, we write $S \cong T$.
	\item An \textit{equivalence of categories} $\cA$ and $\cB$ is a pair of functors $S: \cA \to \cB$, $T: \cB \to \cA$, together with a pair of natural isomorphisms making $\id_\cA \cong TS$ and $\id_\cB \cong ST$. 
		
\end{itemize}

\clearpage
Preliminaries:

\begin{enumerate}
\itemsep0em
	\item Isomorphism defines an equivalence relation on the objects of a category.
	\item An isomorphism is automatically both a monomorphism and an epimorphism.
	\item Monomorphisms and epimorphisms in $\cat{Set}$ are injective and surjective maps, respectively.
	\item $\bbZ \injto \bbQ$ is an epimorphism in $\cat{CRing}$. (Epis need not be surjective, and monic + epic is not iso!)
	\item Terminal and initial objects (should they exist) are unique up to unique isomorphism.
	\item Determine initial and terminal objects in all of the categories above.
\end{enumerate}

Working with products and coproducts:

\begin{enumerate}
\itemsep0em
	\item Determine/guess the products and coproducts in the categories above (when they exist). Actually prove the product and coproduct in $\cat{CRing}$. What makes $\cat{Vect}_k$, $\cat{Mat}_k$ and $\cat{AbGroup}$ special? (This is the notion of a \textit{categorical biproduct}.)
	\item In a categorical product $(A \times B, p_A, p_B)$, the projections $p_A$ and $p_B$ need not be epimorphisms. (Silly example: $\cat{Set}$. Better example??)
	\item In a categorical coproduct $(A \amalg B, i_A, i_B)$, the inclusions $i_A$, $i_B$, need not be monomorphisms. (Hint: What is the coproduct in $\cat{CRing}$, commutative rings?)
%	\item Work in $\cat{Set}$. Let $A = \{0, 1\}$ and $B = \{0, 1, 2\}$. Find the unique isomorphism $\varphi: A \times B \to P$ relating the usual cartesian product $(A \times B, p_A, p_B)$ to the product $(Q = \{0, 1, 2, 3, 4, 5\}, q_A(n) = n \,\mathrm{mod}\,2, q_B(n) = \lfloor n/3 \rfloor)$.
	\item Show that a terminal object satisfies the universal property for an empty product. Show that if $X$ is terminal, then $A \times X \cong A$ in a natural way. Write the corresponding statement for initials.
	\item * Let $\{X_i\}_{i \in I}$ be a family of objects in  a category, for some (possibly infinite) index set $I$. Define the product $\Pi_{i \in I} X_i$ and state its universal property. Show that even if all finite products exist, arbitrary products may not exist.
	\item ** Let $\cat{Field}$ be the full subcategory of $\cat{CRing}$ consisting of fields. Show that not all pairs of fields $(K, F)$ admit a product. (Bonus points: show that even in the full subcategory of characteristic-zero fields, a product need not exist).
\end{enumerate}

Functors and natural transformations:

\begin{enumerate}
\itemsep0em
	\item Show that if a category $\cA$ has a single object, and every morphism is an isomorphism, then $\cA$ is the same thing as a group. Show that a functor $F: \cA \to \cB$ between two such categories is the same thing as a group homomorphism.
	\item Define $F: \cat{Group} \to \cat{AbGroup}$ as the functor taking a group $G$ to its quotient $G/[G, G]$, where $[G, G]$ is the (normal) subgroup generated by all commutators. Show that this is a functor (why does it land in the right category? Where does it take morphisms?)
	\item Define $F: \cat{Group} \to \cat{AbGroup}$ on objects by $FG = Z(G)$, the centre of $G$. Why does $F$ not extend in a useful way to a functor?
	\item Show that $\cat{Vect}_k$ and $\cat{Mat}_k$ are not isomorphic categories. Give an explicit equivalence of categories between $\cat{Vect}_k$ and $\cat{Mat}_k$. (Moral: an isomorphism of categories is more like a homeomorphism, wheras equivalence is more like a homotopy equivalence).
	\item A \textit{pointed space} is a topological space $X$ along with a distinguished point $x \in X$. Define the category $\cat{Top}_*$ of pointed spaces, and write down the product, coproduct, initial and final objects. Let $\pi_1: \cat{Top}_* \to \cat{Group}$ be the fundamental group functor. Does it preserve any of the above?
%	\item Show that the functor $T: \cat{Vect}_k \to \cat{Alg}_k$ taking a vector space to its tensor algebra is free, i.e. it is a left adjoint to the forgetful functor.
	\item * Let $(-)^*: \cat{Vect}_k \to \cat{Vect}_k$ be the duality functor. Write down explicitly the transformation $\eta: \id_{\cat{Vect}_k} \Rightarrow (-)^{**}$, and check that everything works.
	\item * Let $\cat{Inn}_\bbR$ be the category consisting of finite-dimensional real inner product spaces $(V, \langle -, - \rangle_V)$, and morphisms $f: V \to W$ are those $\bbR$-linear isomorphisms preserving the inner product: $\langle f(u), f(v) \rangle_W = \langle u, v \rangle_V$ for all $u, v \in V$. Show that in this category there \textit{is} a natural transformation $\id_{\cat{Inn}_\bbR} \Rightarrow (-)^*$. (Hint: first write down a ``natural'' isomorphism $V \xto{\sim} V^*$).
\end{enumerate}



	
\end{document}