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\begin{document}

\textbf{Additive and Abelian Categories}

\begin{itemize}
	\itemsep0em
	\item A \textit{zero object} $Z \in \cC$ is an object which is both initial and terminal.
	\item A \textit{subobject} of an object $A$ in a category $\cC$ is a monomorphism $u: B \to A$. If $v: C \to A$ is another monomorphism, say that $u$ and $v$ are \textit{equivalent as subobjects of $A$} if there is an isomorphism $\phi: B \to C$ such that $u = v \phi$.
	\item A \textit{pointed category} is a category admitting a zero object.
	\item In a pointed category, a \textit{biproduct} of $X, Y$ is the data  $(B, p_X, p_Y, i_X, i_Y)$ such that $(B, p_X, p_Y)$ is a categorical product of $X$ and $Y$, $(B, i_X, i_Y)$ is a categorical coproduct of $X$ and $Y$,  and this data is \textit{compatible} in the sense that $p_X \circ i_X = \id_X$, $p_Y \circ i_Y = \id_Y$, $p_X \circ i_Y = 0_{YX}$, $p_Y \circ i_X = 0_{XY}$.
	\item The category $\cA$ is \textit{additive} if it satisfies the following conditions:
	\begin{enumerate}
		\item There is a zero object in $\cA$.
		\item For any $X, Y \in \cA$, a categorical product $X \times Y \in \cA$	 exists.
		\item Each hom-set $\cA(X, Y)$ is an abelian group, and composition of morphisms is bilinear, i.e. the maps $\cA(Y, Z) \times \cA(X, Y) \to \cA(X, Z)$, $(f, g) \mapsto f \circ g$ are $\bbZ$-bilinear.
	\end{enumerate}
	\item In an additive category, if a product $(X \times Y, p_X, p_Y)$ exists, it extends uniquely to a biproduct $(X \times Y, p_X, p_Y, i_X, i_Y)$. Similarly for a coproduct.
	\item A functor $F: \cA \to \cB$ between additive categories is an \textit{additive functor} if $F(f + g) = Ff + Fg$.
	\item Let $(X \xto{f} Y) \in \cC$, a pointed category. A \textit{kernel} of $f$ is an object $K \in \cC$, along with a map $(K \xto{\ker f} X)$, such that $f \circ (\ker f) = 0$, and whenever $(W \xto{w} X)$ satisfies $f\circ w = 0$, there exists a unique $\hat{w}: W \to K$ such that
		\[ \begin{tikzcd}
		K \ar[r, "\ker f"] \ar[rr, bend left=40, "0"] & X \ar[r, "f"] & Y \\
		& W \ar[u, "w"'] \ar[ru, "0"'] \ar[lu, dashed, "\hat{w}"]
		\end{tikzcd}\]
	\item Let $(X \xto{f} Y) \in \cC$, a pointed category. A \textit{cokernel} of $f$ is an object $C \in \cC$, along with a map $(Y \xto{\coker{f}} C)$, such that $(\coker f) \circ f = 0$, and whenever $(Y \xto{u} U)$ satisfies $u \circ f = 0$, there exists a unique $(C \xto{\hat{u}} U)$ such that
		\[ \begin{tikzcd}
 			X \ar[r, "f"] \ar[rd, "0"] \ar[rr, bend left=50, "0"] & Y \ar[r, "\coker f"] \ar[d, "u"] & C \ar[ld, dashed, "\hat{u}"] \\
 			& U
 		\end{tikzcd} \]	
	\item The \textit{image} of $f: X \to Y$ is $\im f = \ker (\coker f)$, whenever it exists.
	\item The \textit{coimage} of $f: X \to Y$ is $\coim f = \coker (\ker f)$, whenever it exists.
	\item The additive category $\cA$ is called an \textit{abelian category} if all morphisms admit kernels and cokernels, and furthermore that every monomorphism arises as a kernel, and every epimorphism arises as a cokernel.
	\item $\cA$ abelian. The sequence $(A \xto{f} B \xto{g} C) \in \cA$ is \textit{exact at $B$} if $\im f \cong \ker g$ as subobjects of $B$.
\end{itemize}

\clearpage
Easy exercises about biproducts and additive categories:
\begin{enumerate}
	\item If $Z, Z' \in \cC$ are zero objects, there is a unique isomorphism $Z \xto{\sim} Z'$.
	\item Let $\cC$ be a pointed category, and $X, Y \in \cC$ objects. Define the zero map $X \xto{0_{XY}} Y$. If $\cC$ happens to be additive, show $0_{XY}$ is necessarily the identity in the abelian group $\cC(X, Y)$.
	\item If $X$ is an object in an additive category $\cA$, then $\cA(X, X)$ is naturally a unital ring.
	\item Let $\cC$ be a category with binary products and coproducts. Given an object $A \in \cC$, define the \textit{diagonal} map $\Delta_A: A \to A \times A$ and the \textit{codiagonal} map $\nabla_A: A \amalg A \to A$.
	\item Let $\cC$ be a pointed category with binary biproducts. Show that each hom-set is naturally a commutative monoid, with $\bbN$-bilinear composition. If $\cC$ is additive, does this commutative monoid structure necessarily agree with the abelian group structure?
	\item (\textit{A more efficient definition of biproducts}) Let $\cC$ be an additive category, and suppose we have a diagram of the form
			\[ \begin{tikzcd}
			X \ar[r, bend left=20, "i_X"] & B \ar[l, bend left=20, "p_X"] \ar[r, bend right=20, "p_Y"'] & Y \ar[l, bend right=20, "i_Y"']
			\end{tikzcd} \]
		  satisfying the three equations
		    \[ p_X i_X = \id_X, \quad p_Y i_Y = \id_Y, \quad i_X p_X + i_Y p_Y = \id_B. \]
		  (This diagram and set of equations is a \textit{binary biproduct diagram}). Show that these maps then equip $B$ with the structure of a biproduct of $X$ and $Y$. Conversely, show that the equation $i_X p_X + i_Y p_Y = \id_B$ holds for any biproduct.
	\item Let $e_1, \ldots, e_n$ be a basis of the $k$-vector space $V$. This basis determines injections $i_j: k \to V$, $\lambda \mapsto \lambda e_j$ equipping $V$ with the structure of a coproduct of $n$ copies of $k$. There is a unique compatible product structure making $V$ into a biproduct $k^{\oplus n}$: what is it?
	\item \textbf{Important exercise}. Suppose $\cC$ is a pointed category with binary biproducts. Explain how a map $A \oplus B \xto{f} C \oplus D$ may be represented as a matrix of maps
		\[ [f] = \begin{pmatrix} f_{AC}& f_{BC} \\ f_{AD} & f_{BD}\end{pmatrix} = \begin{pmatrix} f_{AC}: A \to C & f_{BC}: B \to C \\ f_{AD}: A \to D & f_{BD}: B \to D\end{pmatrix}\]
		Write down formulas for each of the maps in the matrix. Show that the maps in the matrix uniquely determine $f$. Show that $[f \circ g] = [f][g]$, i.e. that composition is matrix multiplication.
	\item Write down the maps $i_X, i_Y, p_X, p_Y$ in the biproduct $A \oplus B$ in matrix form. Write down the diagonal $A \to A \oplus A$ and the codiagonal $A \oplus A \to A$ in matrix form.
	\item (\textit{Non-essential exercise: a category with addition but not subtraction}) The category $\cat{Rel}$ has sets as its objects, and relations as its morphisms: A morphism $R: A \to B$ is a subset of $B \times A$, with notation $bRa$ meaning $(b, a) \in R$. The composition rule for $R: A \to B$ and $S: B \to C$ is
		\[ S \circ R: A \to C, \quad c(S \circ R)a \iff \exists b \in B \text{ such that } cSb \text{ and } bRa. \]
		\begin{enumerate}
			\item Show this is a pointed category (identity morphism, composition is associative, zero object).
			\item Show that the disjoint union of sets can be equipped with a biproduct structure. (5) now implies that morphisms can be added. Can they always be subtracted?
		\end{enumerate}
\end{enumerate}

\clearpage
Some exercises on abelian categories:
\begin{enumerate}
	\item Show that if $u: B \to A$ and $v: C \to A$ are subobjects of $A$ in $\cat{Vect}_k$, that they are equivalent subobjects iff $\im u = \im v$ (where the image is a vector space image, not a categorical one).
	\item Let $\cC$ be a pointed category (so that kernels and cokernels are defined). Show the following:
		\begin{enumerate}
			\item Kernels are always monic.
			\item Cokernels are always epic.
		\end{enumerate}
	\item In an abelian category, a morphism which is both monic and epic is an isomorphism.
	\item In an abelian category, every arrow $f$ factors as $f = me$, where $m$ is monic and $e$ is epic. (\textit{Full disclosure: I have no idea how annoying this proof really is but it looks kinda annoying.})
	\item Show that the category of quiver representations $\cat{Rep}_k Q$ is abelian. (Either show it directly, or show $\cat{Rep}_k Q$ is isomorphic to the category $kQ\cat{-mod}$, where $kQ$ is the path algebra).
	\item Let $0 \to A \xto{f} B \xto{g} C \to 0$ be a sequence in an abelian category. Verify the usual stuff:
		\begin{enumerate}
			\item Exactness at $A$ iff $f$ is monic.
			\item Exactness at $C$ iff $g$ is epic.
			\item Exactness at $A, B$, and $C$ iff $f = \ker g$ and $g = \coker f$.
		\end{enumerate}
\end{enumerate}

Determine why each of the following categories fails to be additive/abelian:
\begin{enumerate}
	\item The category of groups and group homomorphisms.
	\item The full subcategory of $k$-vector spaces whose dimensions are powers of $2$.
	\item The full subcategory of even-dimensional $k$-vector spaces.
	\item The full subcategory of $\mathbb{Z}$-modules admitting a finite basis.
	\item $K^+(\bbZ\cat{-mod})$, the homotopy category of bounded-below complexes of $\bbZ$-modules. Hint: start with the nontrivial morphism $(\cdots \to 0 \to \bbZ \to 0 \to \cdots) \to (\cdots \to 0 \to \bbZ/(2) \to 0 \to \cdots)$. Since $K^+(\cA)$ may not be abelian, we care about its triangulated structure instead, where distinguished triangles would take the place of short exact sequences.
\end{enumerate}

\end{document}