\documentclass[oneside,12pt]{article} \usepackage{latexsym,amscd, amsmath, amsthm, amssymb,epsfig,makeidx,tikz} \usepackage[nottoc,numbib]{tocbibind} \usepackage[top=1.4in, bottom=1.4in, left=1.4in, right=1.4in]{geometry} \def\calB{{\mathcal B}} \def\calC{{\mathcal C}} \def\calD{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}} \def\calG{{\cal G}} \def\calK{{\mathcal K}} \def\calQ{{\mathcal Q}} \def\calP{{\mathcal P}} \def\calS{{\mathcal S}} \def\RC{{R{\mathcal C}}} \def\RF{{R{\mathcal F}}} \def \Aut{\mathop{\rm Aut}\nolimits} \def\Cat{{\mathop{\rm Cat}\nolimits}} \def\cc{{\mathop{\rm cc}\nolimits}} \def \characteristic{\mathop{\rm char}} \def \End{\mathop{\rm End}\nolimits} \def \Ext{\mathop{\rm Ext}\nolimits} \def\Fun{\mathop{\rm Fun}} \def\Group{{\mathop{\rm Group}\nolimits}} \def \Hom{\mathop{\rm Hom}\nolimits} \def\Im{\mathop{\rm Im}} \def \Inn{\mathop{\rm Inn}\nolimits} \def \notdivide{\kern-2.1pt\not\kern2.1pt\bigm|} \def\Ob{{\mathop{\rm Ob}}} \def \Out{\mathop{\rm Out}\nolimits} \def \Rad{\mathop{\rm Rad}\nolimits} \def \rad{\mathop{\rm Rad}\nolimits} \def \rank{\mathop{\rm rank}} \def \restricted{\!\downarrow} \def \Soc{\mathop{\rm Soc}\nolimits} \def \soc{\mathop{\rm Soc}\nolimits} \def\Set{\mathop{\rm Set}\nolimits} \def \stab{\mathop{\rm Stab}} \def \Stab{\mathop{\rm Stab}} \def \Tor{\mathop{\rm Tor}\nolimits} \def\cod{\mathop{\rm cod}} \def\dom{\mathop{\rm dom}} \def\directlimit{{\lim\limits_{\textstyle \longrightarrow}}} \def\inverselimit{{\lim\limits_{\textstyle \longleftarrow}}} \def\AA{{\mathbb A}} \def\BB{{\mathbb B}} \def\CC{{\mathbb C}} \def\EE{{\mathbb E}} \def\FF{{\mathbb F}} \def\HH{{\mathbb H}} \def\NN{{\mathbb N}} \def\PP{{\mathbb P}} \def\QQ{{\mathbb Q}} \def\RR{{\mathbb R}} \def\SS{{\mathbb S}} \def\TT{{\mathbb T}} \def\ZZ{{\mathbb Z}} %% ------------------------------------------------------------------------------- %% Document begins here %% ------------------------------------------------------------------------------- \begin{document} {\bf\noindent Math 8300\hfill Homework 2\hfill PJW\break} {\bf\noindent Date due: Wednesday October 9, 2019} \medskip Assume all categories considered are small. A functor $F:\calC\to\calD$ is an \textit{equivalence} of categories if there is a functor $G:\calD\to\calC$ so that $GF$ is naturally isomorphic to the identity functor $1_\calC$ and $FG$ is naturally isomorphic to $1_\calD$ (meaning that there are invertible natural transformations $\tau:GF\to 1_\calC$ and $\sigma:FG\to 1_\calD$). \begin{enumerate} \item Suppose that $F:\calC\to\calD$ is an equivalence of small categories. \begin{enumerate} \item Show that, for all objects $x,y\in\Ob\calC$, $F$ provides a bijection $$ \Hom_\calC(x,y)\leftrightarrow\Hom_\calD(F(x),F(y)), $$ so that $\End_\calC(x)\cong \End_\calD(F(x))$ as monoids. \item Show that $x\cong y$ in $\calC$ if and only if $F(x)\cong F(y)$ in $\calD$. \item Let $\calE$ be a further category. Show that the functor categories $\Fun(\calC,\calE)$ and $\Fun(\calD,\calE)$ are naturally equivalent. \end{enumerate} \item Let $\calC$ be a category and let $x,y\in\Ob\calC$. Prove that if $x\cong y$ then $\Hom_\calC(x,-)$ and $\Hom_\calC(y,-)$ are naturally isomorphic functors. \item Let $F,G:\calC\to\calD$ be functors and $\eta:F\to G$ a natural transformation. \begin{enumerate} \item Show that if, for all $x\in\Ob\calC$, the mapping $\eta_x:F(x)\to G(x)$ is an isomorphism in $\calD$, then $\eta$ is a natural isomorphism (meaning that it has a 2-sided inverse natural transformation $\theta:G\to F$). \item Suppose that $F$ is an equivalence of categories and that $F$ is naturally isomorphic to $G$, so $F\simeq G$. Show that $G$ is an equivalence of categories. \end{enumerate} \item Let $\calC$ be a small category. A \textit{self-equivalence} of $\calC$ is an equivalence of categories $F:\calC\to\calC$. Show that the set of natural isomorphism classes of self equivalences of $\calC$ is a group, with multiplication induced by composition of functors. \item Let $G$ be a group, which we regard as a category $\calG$ with a single object, and with the elements of $G$ as morphisms. Let $F:\calG\to\calG$ be a functor. \begin{enumerate} \item Show that $F$ is naturally isomorphic to the identity functor $1_\calG:\calG\to \calG$ if and only if the mapping $F:G\to G$, induced by $F$ on the set of morphisms, is an inner automorphism; that is, an automorphism of the form $c_g: G\to G$ for some $g\in G$, where $c_g(h)=ghg^{-1}$ for all $h\in G$. \item Show that self equivalences of $\calG$ are automorphisms of $\calG$. \item Optional: do not hand in. Show that the group of natural isomorphism classes of self equivalences of $\calG$ is isomorphic to $\Aut(G)/\Inn(G)$. (In the context of group theory, $\Inn(G)$ denotes the set of inner automorphisms of $G$, and $\Aut(G)/\Inn(G)$ is called the group of \textit{outer} (or \textit{non-inner}) automorphisms.) \end{enumerate} \item Let $I$ be the poset with two elements 0 and 1, and with $0<1$. If $P$ and $Q$ are posets we can regard them as categories $\calP$ and $\calQ$ whose objects are the elements of the posets, and where there is a unique morphism $x\to y$ if and only if $x\le y$. \begin{enumerate} \item Show that if $P$ and $Q$ are posets then a functor $\calP\to\calQ$ is `the same thing as' an order-preserving map. \item Now consider two order-preserving maps $f,g:P\to Q$, and regard them as functors $F,G:\calP\to\calQ$. Show that the following three conditions are equivalent: \begin{enumerate} \item there is a natural transformation $\tau: F\to G$, \item $f(x)\le g(x)$ for all $x\in P$, \item there is an order-preserving map $h:P\times I\to Q$ such that $h(x,0)=f(x)$ and $h(x,1)=g(x)$ for all $x\in\calP$. Here $P\times I$ denotes the product poset with order relation $(a_1,b_1)\le (a_2,b_2)$ if and only if $a_1\le a_2$ and $b_1\le b_2$, where $a_i\in P$ and $b_i\in I$. \end{enumerate} \end{enumerate} \item Let $K$ be a field. Show that the category Vec of finite dimensional vectors spaces over $K$ is equivalent to the category $\calC$ with objects $\NN:=\{0, 1, 2,\ldots\}$, where $\Hom_\calC(n,m)$ is the set $M_{m,n}(K)$ of $m\times n$ matrices with entries in $K$, and where composition of morphisms is matrix multiplication. In case $m$ or $n$ is zero, give a definition of $\Hom_\calC(n,m)$ that will make this question make sense. \end{enumerate} \end{document}