%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% This top matter is essentially a combination of Graber-Vakil %%% %%% and Li-Liu-Liu-Zhou %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{amsart} %\usepackage{amsmath,amsthm,amsfonts,verbatim,amstex} %\usepackage{diagrams} %\usepackage{eepic,epic} \usepackage{ccfonts} \usepackage[euler-digits]{eulervm} \renewcommand{\familydefault}{ppl} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\marginparwidth}{0in} \setlength{\marginparsep}{0in} \setlength{\marginparpush}{0in} \setlength{\topmargin}{0in} \setlength{\headheight}{0pt} \setlength{\headsep}{0pt} \setlength{\footskip}{.3in} \setlength{\textheight}{9.2in} \setlength{\textwidth}{6.5in} \setlength{\parskip}{4pt} \title{\large{Math 171 Fall 2010: HW 18}} \author{Due Mon Nov 15} \date{\today} % \address{ % Department of Mathematics\\ % Harvey Mudd College\\ % Claremont, CA 91711 % } % \email{dk@math.hmc.edu} %\newtheorem{theorem}{Theorem}[section] \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{warning}[theorem]{Warning} \newtheorem{questions}{Question} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{assumption}{Assumption} \newtheorem{example}[theorem]{Example} \newtheorem{rem1}[theorem]{Remark} \newenvironment{remark}{\begin{rem1}\em}{\end{rem1}} %% Math Blackboard \newcommand{\A}{{\mathbb{A}}} \newcommand{\CC} {{\mathbb C}} \newcommand{\EE}{\mathbb{E}} \newcommand{\NN} {{\mathbb N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\QQ} {{\mathbb Q}} \newcommand{\RR} {{\mathbb R}} \newcommand{\ZZ} {{\mathbb Z}} %% Operators \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\End}{\operatorname{End}} \newcommand{\Tr}{\operatorname{tr}} \newcommand{\Coker}{\operatorname{Coker}} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\stab}{stab} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\ext}{\cE \it{xt}} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\image}{Im} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\val}{val} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\spec}{Spec} \DeclareMathOperator{\Symm}{Symm} \DeclareMathOperator{\inv}{{inv}} \newcommand{\dd}[1]{\frac{d}{d{#1}}} \newcommand{\delx}[1]{\frac{\partial {#1}}{\partial x}} \newcommand{\dely}[1]{\frac{\partial {#1}}{\partial y}} \newcommand{\delf}[1]{\frac{\partial f}{\partial #1}} \newcommand{\ithpart}[2]{\frac{\partial {#1}}{\partial x_{#2}}} \newcommand{\arbpart}[2]{\frac{\partial {#1}}{\partial {#2}}} \newcommand{\arbd}[2]{\frac{d {#1}}{d {#2}}} %% Mbar \newcommand{\M}{\overline{{M}}} \newcommand{\smargin}[1]{\marginpar{\tiny{#1}}} %% mathbf \newcommand{\bL}{\mathbf{L}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bk}{\mathbf{k}} \newcommand{\bm}{\mathbf{m}} \newcommand{\bn}{\mathbf{n}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bq}{\mathbf{q}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bt}{\mathbf{t}} \newcommand{\bw}{\mathbf{w}} \newcommand{\bx}{\mathbf{x}} \newcommand{\kk}{\bk} \newcommand{\bi}{{\mathbf{i}}} \newcommand{\bj}{{\mathbf{j}}} %% mathcal \newcommand{\cal}{\mathcal} \def\cA{{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}} \def\cD{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}} \def\cH{{\cal H}} \def\cK{{\cal K}} \def\cL{{\cal L}} \def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}} \def\cP{{\cal P}} \def\cQ{{\cal Q}} \def\cR{{\cal R}} \def\cT{{\cal T}} \def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cZ{{\cal Z}} %% mathfrak \def\fB{\mathfrak{B}} \def\fC{\mathfrak{C}} \def\fD{\mathfrak{D}} \def\fE{\mathfrak{E}} \def\fF{\mathfrak{F}} \def\fM{\mathfrak{M}} \def\fV{\mathfrak{V}} \def\fX{\mathfrak{X}} \def\fY{\mathfrak{Y}} \def\ff{\mathfrak{f}} \def\fm{\mathfrak{m}} \def\fp{\mathfrak{p}} \def\ft{\mathfrak{t}} \def\fu{\mathfrac{u}} \def\fv{\mathfrak{v}} \def\frev{\mathfrak{rev}} %% tilde, Greek \newcommand{\tPhi }{\tilde{\Phi} } \newcommand{\tGa}{\tilde{\Gamma}} \newcommand{\tbeta}{\tilde{\beta}} \newcommand{\trho }{\tilde{\rho} } \newcommand{\tpi }{\tilde{\pi} } %% tilde, English \newcommand{\tC}{\tilde{C}} \newcommand{\tD}{\tilde{D}} \newcommand{\tE}{\tilde{E}} \newcommand{\tF}{\tilde{F}} \newcommand{\tK}{\tilde{K}} \newcommand{\tL}{\tilde{L}} \newcommand{\tT}{\tilde{T}} \newcommand{\tY}{\tilde{Y}} \newcommand{\tf}{\tilde{f}} \newcommand{\tih}{\tilde{h}} \newcommand{\tz}{\tilde{z}} \newcommand{\tu}{\tilde{u}} \def\tilcW{{\tilde\cW}} \def\tilcM{\tilde\cM} \def\tilcC{\tilde\cC} \def\tilcZ{\tilde\cZ} %% hat \newcommand{\hD}{\hat{D}} \newcommand{\hL}{\hat{L}} \newcommand{\hT}{\hat{T}} \newcommand{\hX}{\hat{X}} \newcommand{\hY}{\hat{Y}} \newcommand{\hZ}{\hat{Z}} \newcommand{\hf}{\hat{f}} \newcommand{\hu}{\hat{u}} \newcommand{\hDe}{\hat{Delta}} %% check \newcommand{\eGa}{\check{\Ga}} %% moduli \newcommand{\Mbar}{\overline{\cM}} \newcommand{\MX}{\Mbar^\bu_\chi(\Gamma,\vec{d},\vmu)} \newcommand{\GX}{G^\bu_\chi(\Gamma,\vec{d},\vmu)} \newcommand{\Mi}{\Mbar^\bu_{\chi^i}(\Po,\nu^i,\mu^i)} \newcommand{\Mv}{\Mbar^\bu_{\chi^v}(\Po,\nu^v,\mu^v)} \newcommand{\GYfm}{G^\bu_{\chi,\vmu}(\Gamma)} \newcommand{\Gdmu}{G^\bu_\chi(\Gamma,\vd,\vmu)} \newcommand{\Mdmu}{\cM^\bu_\chi(\hatYrel,\vd,\vmu)} \newcommand{\tMd}{\tilde{\cM}^\bu_\chi(\hatYrel,\vd,\vmu)} \def\mzd{\cM^\bu_{\chi}(Z,d)} \def\mzod{\cM^\bu_{\chi}(Z^1,d)} \def\mhzd{\cM^\bu_{\chi}(\hat Z,d)} \def\Mwml{\cM^\bu_{\chi,\vd,\vmu}(W\urel,L)} \def\Mfml{\cM_{\chi,\vd,\vmu}^\bu(\hatYrel,\hL)} \def\Mgw{\cM_{g,\vmu}(W\urel,L)} \def\mxdm{\cM_{\chi,\vd,\vmu}} \def\Mfmlz{\mxdm(\hY_{\Gamma^0}\urel, \hL)} \def\Mgz{\cM_{g,\vmu}(\hY_{\Ga^0}\urel,\hL)} \def\Mmu{\cM^\bu_{\chi,\vmu}(\Gamma)} \def\mapright#1{\,\smash{\mathop{\lra}\limits^{#1}}\,} \def\twomapright#1{\,\smash{\mathop{-\!\!\!\lra}\limits^{#1}}\,} \def\mapdown#1{\ \downarrow\!{#1}} \def\Mw{\cM_{\chi,\vd,\vmu}^\bu(W\urel,L)} \def\Myy{\cM_{\chi,\vd,\vmu}^\bu(Y_{\Gamma^0}\urel,L)} \def\mw{\cM_{\chi,\vd,\vmu}^\bu(W\urel,L)} \def\My{\cM_{\chi,\vd,\vmu}^\bu(\hat Y_{\Gamma^0}\urel,\hat L)} \def\my{\cM^\bu(\hat Y\urel,\hat L)} \def\myy{\cM^\bu(Y\urel,L)} \def\mwtdef{\cM^\soe\ldef} \def\MY{\cM^\bu_\chi(\hat{\cY},\vd,\vmu)} %% vector \newcommand{\two }[1]{ ({#1}_1,{#1}_2) } \newcommand{\three}[1]{ ({#1}_1,{#1}_2,{#1}_3) } %% vec \newcommand{\pair}{(\vx,\vnu)} \newcommand{\vmu}{{\vec{\mu}}} \newcommand{\vnu}{{\vec{\nu}}} \newcommand{\vx}{\vec{\chi}} \newcommand{\vz}{\vec{z}} \newcommand{\vd}{\vec{d}} \newcommand{\vsi}{\vec{\sigma}} \newcommand{\vn}{\vec{n}} \newcommand{\va}{\vec{a}} \newcommand{\vb}{\vec{b}} \newcommand{\vr}{\vec{r}} \newcommand{\vect}[1]{{\overrightarrow{#1}}} %% bold \newcommand{\ba}{{\bf{a}}} \newcommand{\bb}{{\bf{b}}} \newcommand{\bc}{{\bf{c}}} %% superscript \newcommand{\up}[1]{ {{#1}^1,{#1}^2,{#1}^3} } \def\dual{^{\vee}} \def\ucirc{^\circ} \def\sta{^\ast} \def\st{^{\mathrm{st}}} \def\virt{^{\mathrm{vir}}} \def\upmo{^{-1}} \def\sta{^{\ast}} \def\ori{^{\mathrm{o}}} \def\urel{^{\mathrm{rel}}} \def\pri{^{\prime}} \def\virtt{^{\text{vir},\te}} \def\virts{^{\text{vir},\soe}} \def\virz{^{\text{vir},T}} \def\mm{{\mathfrak m}} \def\sta{^*} \def\uso{^{S^1}} %% subscript \newcommand{\lo}[1]{ {{#1}_1,{#1}_2,{#1}_3} } \newcommand{\hxn}[1]{ {#1}_{\hat{\chi},\hat{\nu}} } \newcommand{\xmu}[1]{{#1}^\bu_{\chi,\up{\mu}} } \newcommand{\xnu}[1]{{#1}^\bu_{\chi,\up{\nu}} } \newcommand{\gmu}[1]{{#1}_{g,\up{\mu}}} \newcommand{\xmm}[1]{{#1}^\bu_{\chi,\mm}} \newcommand{\xn}[1]{{#1}_{\vx,\vnu}} \def\lbe{_{\beta}} \def\lra{\longrightarrow} \def\lpri{_{\text{pri}}} \def\lsta{_{\ast}} \def\lvec{\overrightarrow} \def\Oplus{\mathop{\oplus}} \def\las{_{a\ast}} \def\ldef{_{\text{def}}} %% Greek \newcommand{\Del}{\Delta} \newcommand{\Si}{\Sigma} \newcommand{\Ga}{\Gamma} \newcommand{\ep}{\epsilon} \newcommand{\lam}{\lambda} \newcommand{\si}{\sigma} \begin{document} \pagestyle{plain} \maketitle \begin{enumerate} \item Prove the free category $\widehat{\Gamma}$ of the directed graph $\Gamma$ is indeed a category. \item Verify every preordered set $I$ can be regarded as a category. \item Let $\mathcal{G}ps$ be the category of groups and $\mathcal{A}b \mathcal{G}ps$ be the category of abelian groups. Consider the assignment $F$, \[ F: \mathcal{G}ps \rightarrow \mathcal{A}b \mathcal{G}ps \] given as follows. For any group $G$, \[ F(G) = G/G', \] where $G'$ is the commutator subgroup of $G$. Also, if $\phi: G \rightarrow H$ is a group homomorphism, then \[ F(\phi): G/G' \rightarrow H/H' \] is given by \[ F(\phi) (xG') = \phi (x) H', \] for all $xG' \in G/G'$. Prove $F$ is a functor. \end{enumerate} \end{document}