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\textit{Categories for the Working Mathematician} by S. Mac Lane. +This is notes for problems on \textit{Category for the Working Mathematician} by S. Mac Lane. \section{} @@ -325,38 +325,79 @@ In $\mathbf{Matr}_K$, the object set is all positive integers $\{1, 2, 3, ...\} Let $R_T \subseteq (T \rightarrow \mathbb{R})$ be a ring whose elements are continuous functions from a topological space $T$ to real number. We construct $R_T$ as follows: \begin{enumerate} - \item Additive identity: $0_{R_T} = x \mapsto 0$. - \item Multiplicative identity: $1_{R_T} = x \mapsto 1$. - \item Addition: $f + g = x \mapsto f(x) + g(x)$. - \item Multiplication: $f \times g = x \mapsto f(x) \times g(x)$. + \item Additive identity. $0_{R_T} : x \mapsto 0$. + \item Multiplicative identity. $1_{R_T} : x \mapsto 1$. + \item Addition. $f + g : x \mapsto f(x) + g(x)$. + \item Multiplication. $f \times g : x \mapsto f(x) \times g(x)$. \end{enumerate} -Let $X$ and $Y$ be any topological spaces. If we have a continuous function $f : Y \rightarrow X$, we can construct a ring homomorphism $H(f) = h : R_X \rightarrow R_Y$. We define $h(r) = r \circ f$. Then $h(0_{R_X}) = (x \mapsto 0) \circ f = 0_{R_Y}$, $h(1_{R_X}) = (x \mapsto 1) \circ f = 1_{R_Y}$, $(h(s + t))(x) = (s + t)(f(x)) = s(f(x)) + t(f(x))$, $(h(s \times t))(x) = (s \times t)(f(x)) = s(f(x)) \times t(f(x))$. Therefore $H(f)$ is a ring homomorphism. +Let $X$ and $Y$ be any topological spaces. If we have a continuous function $f : Y \rightarrow X$, we can construct a ring homomorphism $H(f) = h : R_X \rightarrow R_Y$. We define $h(r) = r \circ f$. Then $h(0_{R_X}) = 0_{R_Y}$, $h(1_{R_X}) = 1_{R_Y}$, $(h(s + t))(x) = (s + t)(f(x)) = s(f(x)) + t(f(x))$, $(h(s \times t))(x) = (s \times t)(f(x)) = s(f(x)) \times t(f(x))$. Therefore $H(f)$ is a ring homomorphism. -Now we construct a functor $F : \mathbf{Top}^{\mathrm{op}} \rightarrow \mathbf{Rng}$. Let the object function be $F(A) = R_A$ and the arrow function be $F(g) = H(g^{\mathrm{op}})$, except for any $T \in \mathbf{Top}^{\mathrm{op}}$, $F(\mathrm{id}(T)) = \mathrm{id}(R_T)$. For all arrow $a, b$ in $\mathbf{Top}^{\mathrm{op}}$ such that $a \neq \mathrm{id}\ \wedge\ b \neq \mathrm{id}$, $F(b \circ a) = H((b \circ a)^{\mathrm{op}}) = H(a^{\mathrm{op}} \circ b^{\mathrm{op}}) = H(b^{\mathrm{op}}) \circ H(a^{\mathrm{op}}) = F(b) \circ F(a)$. For all arrow $a$ in $\mathbf{Top}^{\mathrm{op}}$, $F(\mathrm{id} \circ a) = F(a \circ \mathrm{id}) = F(\mathrm{id}) \circ F(a) = F(a) \circ F(\mathrm{id}) = F(a)$. Therefore $F$ is a functor and $\overline{F}$ is a contravariant functor from $\mathbf{Top}$ to $\mathbf{Rng}$. +Now we construct a functor $F : \mathbf{Top}^{\mathrm{op}} \rightarrow \mathbf{Rng}$. Let the object function be $F(A) = R_A$ and the arrow function be $F(g) = H(g^{\mathrm{op}})$. For all arrow $a, b$ in $\mathbf{Top}^{\mathrm{op}}$, $F(b \circ a) = H((b \circ a)^{\mathrm{op}}) = H(a^{\mathrm{op}} \circ b^{\mathrm{op}}) = H(b^{\mathrm{op}}) \circ H(a^{\mathrm{op}}) = F(b) \circ F(a)$. For all topological space $T \in \mathbf{Top}^{\mathrm{op}}$, $F(\mathrm{id}(T)) = H(\mathrm{id}(T)) = \mathrm{id}(R_T)$. Therefore $F$ is a functor and $\overline{F}$ is a contravariant functor from $\mathbf{Top}$ to $\mathbf{Rng}$. \subsection{} \subsubsection{} -\textbf{Show that for any ring $R$, $R\text{-}\mathbf{Mod}$ is a full subcategory of $\mathbf{Ab}^R$.} +\textbf{Show that for any ring $R$, $R\textbf{-}\mathbf{Mod}$ is a full subcategory of $\mathbf{Ab}^R$.} + +TODO + +% First, we prove for any objects $A, B \in R\textbf{-}\mathbf{Mod}$, a functor $f : A \rightarrow B$ exists. Let $f(a) = 0$. Then for any $r \in R$ and $a \in A$, $f(r \times a) = r \times f(a)$. Also, for any $a_1 \in A$ and $a_2 \in A$, $f(a_1 + a_2) = f(a_1) + f(a_2)$. Thus $f$ is an arrow on $A$ to $B$. + +%We define a functor $F : R\textbf{-}\mathbf{Mod} \rightarrow \mathbf{Ab}^R$ by $F() = $. + +%We proof $F$ is full. + +% First, we can see $R\textbf{-}\mathbf{Mod}$ as a subcategory of $\mathbf{Ab}^R$. + +% Second, we prove for any abelian groups $A, B \in R\textbf{-}\mathbf{Mod}$, $\hom_{R\textbf{-}\mathbf{Mod}}(A, B)=\hom_{\mathbf{Ab}^R}(A, B)$. \subsubsection{} \textbf{For a finite discrete category $X$, describe $B^X$.} +For any functor $T : X \rightarrow B$, if its object function is $T(a) = b$, its arrow function only maps $\mathrm{id}(a)$ to $\mathrm{id}(b)$. Such a functor $T$ is an object of $B^X$. + +Let $R, S : X \rightarrow B$ be functors and $\tau$ be a map on an object of X to an arrow in $B$. $(\tau : R\ \dot{\rightarrow}\ S) \Leftrightarrow (\forall x \in X,\ \tau_x(R(x)) = S(x))$. Therefore $\hom(R, S) = \{\tau\ |\ \forall x \in X,\ \tau_x(R(x)) = S(x)\}$. Therefore an arrow on $R$ to $S$ exists iff $(\forall x, y \in X,\ e_R(x, y) \rightarrow e_S(x, y))\ \wedge\ (\forall x \in X,\ \hom(R(x), S(x)) \neq \emptyset)$, where $e_T(x, y) \Leftrightarrow (\exists a \in B, \{x, y\} \subseteq \{ w\ |\ a = T(w) \})$. + \subsubsection{} \textbf{Let $\mathbf{N}$ be a discrete category of natural numbers. Describe $\mathbf{Ab}^{\mathbf{N}}$.} +An object of $\mathbf{Ab}^{\mathbf{N}}$ is a map on $\mathbb{N}$ to $\mathbf{Ab}$. Same as above, $\hom(R, S) = \{\tau\ |\ \forall n \in N,\ \tau_n(R(n)) = S(n)\}$. In other words, a map $\tau : \mathbb{N} \rightarrow (\mathbf{Ab} \rightarrow \mathbf{Ab})$ is an arrow iff, for every $n \in \mathbb{N}$, there is a corresponding group homomorphism $\tau_n$ on $R(n)$ to $S(n)$, and, for every $m \in \mathbb{N}$ such that $R(n) = R(m)$, $S(n) = S(m)$. + \subsubsection{} \textbf{Let $P$ and $Q$ be preorders. Describe $Q^P$ and show it is a preorder.} +Let $R, S : P \rightarrow Q$. Then $R$ and $S$ are objects of $Q^P$. Let $\tau$ be a natural transform $\tau : R\ \dot{\rightarrow}\ S$. $\tau$ is an arrow on $R$ to $S$ in $Q^P$. Since $\tau$ is natural and $P$ is preorder, following diagram is commute for every pair of objects $p, p^\prime \in P$, where $f(p, p^\prime)$ is the only arrow on $p$ to $p^\prime$. Since $Q$ is preorder, $\tau x = g(Rx, Sx)$, where $g(Rx, Sx)$ is the only arrow on $Rx$ to $Sx$. + +\begin{center} +\begin{tikzcd}[sep=9em] +Rp \arrow[d, "{Rf(p,\,p^\prime)}"] \arrow[r, "{\tau p\,=\,g(Rp,\,Sp)}"] & Sp \arrow[d, "{Sf(p,\,p^\prime)}"]\\ +Rp^\prime \arrow[r, "{\tau p^\prime\,=\,g(Rp^\prime,\,Sp^\prime)}"] & Sp^\prime +\end{tikzcd} +\end{center} + +From the two downward arrows in the diagram, we can say that $\mathrm{Im}(R)$ and $\mathrm{Im}(S)$ contain the preorder structure of $P$. There are two functors $P \rightarrow \mathrm{Im}(R)$ and $P \rightarrow \mathrm{Im}(S)$, where $\mathrm{Im}(T)$ is a category from the image of the object function of $T$ and all arrows between any two pairs in the image. + +As explained above, $\sigma p = g(Rp, Sp)$ for all $\sigma : R\ \dot{\rightarrow}\ S$. Thus $|\hom(R, S)| \leq |\{\sigma\ |\ \forall p \in P,\ \sigma p = g(Rp, Sp)\}| = 1$. Therefore $Q^P$ is preorder. + \subsubsection{} \textbf{Let $\mathbf{Fin}$ be a category of all finite sets and $G$ be a finite group. Describe $\mathbf{Fin}^G$.} + + +\subsection{} + +\subsection{} + +\subsection{} + +\subsection{} + \section{} \section{} -- GitLab