From d4070b02f567aa3f9ab8b8f7cf2a8cd8fd882a58 Mon Sep 17 00:00:00 2001 From: Nathaniel Wesley Filardo Date: Wed, 17 Sep 2014 16:45:35 -0400 Subject: [PATCH] Tweak cheatsheet, add xref support --- ctcheat.tex | 230 ++++++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 197 insertions(+), 33 deletions(-) diff --git a/ctcheat.tex b/ctcheat.tex index 379bb06..62e5053 100644 --- a/ctcheat.tex +++ b/ctcheat.tex @@ -18,18 +18,27 @@ \usepackage{textcomp} \usepackage{url} \usepackage{multirow} -\usepackage{hyperref} \usepackage{breakurl} \usepackage{enumitem} +\usepackage{etoolbox} +\usepackage{hyperref} %\setlist{nolistsep} +%http://tex.stackexchange.com/questions/126750/how-can-i-number-paragraphs-without-higher-level-counters +\usepackage{chngcntr} +\counterwithout{paragraph}{subsubsection} +\renewcommand{\theparagraph}{{\tiny\P}{\small\arabic{paragraph}}} +\titleformat{\paragraph}[runin]{\normalfont\bfseries}{\theparagraph}{\wordsep}{} +\titlespacing{\paragraph}{0pt}{3.25ex plus 1ex minus .2ex}{\wordsep} +\setcounter{secnumdepth}{4} + \renewcommand{\baselinestretch}{0.9} %\newtheorem{thm}{Thm}[section] %\newtheorem{dfn}{Def}[section] \setlength{\parindent}{0pt} -\setlength{\parskip}{5pt} +\setlength{\parskip}{-2pt} % http://www.latex-community.org/forum/viewtopic.php?f=46&t=3837&start=0#p15112 \makeatletter @@ -70,7 +79,9 @@ \DeclareMathOperator{\mm}{\mid\mid} \newcommand{\natto}{\overset{\cdot}{\to}} -\newcommand{\defn}[1]{{\bf #1}} +\newcommand{\defn}[2][]{{\ifstrempty{#1}{\label{defn:#2}}{\label{defn:#1}}{\bf #2}}} +\newcommand\xrdefnhelper[1]{defn:#1} +\newcommand{\xrdefn}[1]{\ref{\forcsvlist{\xrdefnhelper}{#1}}} \begin{document} @@ -84,6 +95,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \section{Basics} + \paragraph{} + % A \defn{category} $\mathbf{C}$ (\S3.1) is a quadruple $(\mathcal{O},\mbox{hom},id,\circ)$ with \begin{itemize} @@ -95,15 +108,21 @@ Entries within each section are roughly sorted by definition, alphabetically. \item Identity arrows ($id_A$) on each object $A$, unit of $\circ$ \end{itemize} + \paragraph{} + % Categories may be described (Awodey:p21) as \[\xymatrix{ C_2 \ar[r]^\circ & C_1 \ar@<2ex>[r]_{cod} \ar@<-2ex>[r]_{dom} & C_0 \ar[l]^i }\] + \paragraph{} + % A category is (Awodey:p24-25,D1.11-12)\dots \begin{itemize} \item \defn{small} if $C_0$ and $C_1$ are sets and \defn{large} otherwise. \item \defn{locally small} if $\forall_{X,Y \in C_0} \mbox{hom}_C(X,Y) \subseteq C_1$ is a set. \end{itemize} + \paragraph{} + % A predicate $P$ is \defn{essentially unique} (\S7.3) if it is unique up to isomorphism: \begin{itemize} @@ -111,14 +130,20 @@ Entries within each section are roughly sorted by definition, alphabetically. \item If $PA$ and $A \simeq B$, then $PB$. \end{itemize} + \paragraph{} + % $\mathbf{B}$ is a \defn{subcategory} of $\mathbf{A}$ if it has subcollections of objects and morphisms with identical composition and identity (\S4.1.1). $\mathbf{B}$ is additionally \dots \begin{itemize} - \item \defn{full} if it has all morphisms from $\mathbf{A}$. (\S4.1.2) + \item \defn[fullcat]{full} if it has all morphisms from $\mathbf{A}$ + between objects in $\mathbf{B}$. (\S4.1.2) \item \defn{reflective} if each $B$ has an $\mathbf{A}$-reflection. (\S4.16.2) + \xrdefn{reflection} \end{itemize} + \paragraph{} + % A category is$\dots$ \begin{itemize} \item \defn{balanced} if all bi are iso (\S7.49.2) @@ -126,19 +151,55 @@ Entries within each section are roughly sorted by definition, alphabetically. \item \defn{thin} if $\forall_{A,B} \mbox{hom}(A,B) \simeq \set{*}$. (\S3.26.2) \end{itemize} +\section{Derived Categories} + + \paragraph{} + % + The \defn[conecat]{cone} category over a given diagram, $\mathbf{Cone}(D(J))$, has + as objects cones to that diagram and a morphism between cones is an arrow + $\phi : C \to C'$ s.t. $\forall_{D_j \in D(J)} c_j^\prime \circ \phi = + c_j$. \xrdefn{cone} + + \paragraph{} + % + The \defn{dual} (\S3.5;Awodey:p15,i2) category $\mathbf{A}^\text{op}$ + which exchanges domains and codomains of arrows in $\mathbf{A}$. Any + statement implies its dual. + + \paragraph{} + % + The \defn{arrow} (Awodey:p16,i3) category $\mathbf{C}^\to$ has arrows for + commutative squares in $\mathbf{C}$. There are two functors + $\mathbf{cod}, \mathbf{dom} : \mathbf{C}^\to \to \mathbf{C}$. + + \paragraph{} + % + The \defn{slice} (Awodey:p16,i4) category $\mathbf{C}/C$ has objects of + arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative + triangles. + \section{Object Properties} + \paragraph{} + % $C$ is a \defn{coseparator} if $\forall_{f,g : B \to A} . f \ne g \Rightarrow \exists_{h : A \to C} . h \circ f \ne h \circ g$. (\S7.17) (Contrast monomorphism.) + \paragraph{} + % An object $0$ is \defn{initial} if $\forall_B . \exists! f_B : 0 \to B$. (\S7.1) - A \defn{limit} (Awodey:D5.16) of a diagram $D(J)$ is a - terminal object in the category $\mathbf{Cone}(D(J))$. Written: - $c_i : (\varprojlim_{j} D_j) \to D_i$. - + \paragraph{} + % + A \defn{limit} (Awodey:D5.16) of a diagram $D(J)$ is a terminal object in + the category $\mathbf{Cone}(D(J))$. Written: $c_i : (\varprojlim_{j} D_j) + \to D_i$. A \defn{colimit} (Awodey:\S5.6) is an initial object in the + category of cocones; $c_i : D_i \to (\varinjlim_j D_j)$. \xrdefn{cone} + + \paragraph{} + % $(A \times B,\pi_1,\pi_2)$ is a \defn{product} iff (UMP) \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2} \exists!_u$\\$u\pi_1 = z_1 ~\wedge~ u\pi_2 = z_2$}\restore @@ -146,6 +207,8 @@ Entries within each section are roughly sorted by definition, alphabetically. & A & \ar[l]_{\pi_1} A \times B \ar[r]^{\pi_2} & B \\ }\] + \paragraph{} + % $(P,p_1,p_2)$ is a \defn{pullback} (Awodey:p80,D5.4) of $f,g$ iff (UMP) \[\xymatrix{ {}\save[]+<-1cm,0cm>*\txt<8pc>{$\forall_{Z,z_1,z_2 . fz_1 = gz_2}\exists!_u$\\$z_1 = p_1u ~\wedge~ z_2 = p_2u$}\restore @@ -154,23 +217,33 @@ Entries within each section are roughly sorted by definition, alphabetically. }\] $P$ may be denoted $A \times_C B$ when $f,g$ are clear. + \paragraph{} + % $S$ is a \defn{separator} if $\forall_{f,g : A \to B} . f \ne g \Rightarrow \exists_{h : S \to A} . f \circ h \ne g \circ h$. (\S7.10) (Contrast epimorphism.) $S$ is a separator iff $\mbox{hom}(S,-)$ is faithful. (\S7.12) + \paragraph{} + % A set of objects $\mathcal{T}$ is a \defn{separating set} if $\forall_{f,g : A \to B} . f \ne g \Rightarrow \exists{S \in \mathcal{T}, h : S \to A} . f \circ h \ne g \circ h$. (\S7.14) + \paragraph{} + % An object $1$ is \defn{terminal} if $\forall_A . \exists! f_A : A \to 1$. (\S7.4) + \paragraph{} + % An object that is both initial and terminal is called a \defn{zero}. (\S7.7) \section{Arrow Properties} + \paragraph{} + % $e$ is an \defn{epimorphism} (\S7.39) (the dual of a monomorphism) (equiv: is \defn{epic} (Awodey:D2.1)) if % @@ -180,6 +253,8 @@ Entries within each section are roughly sorted by definition, alphabetically. then so is $g$. (\S7.41) Epis generalize \defn{surjection} in $\mathbf{Set}$. + \paragraph{} + % $(E,e)$ is an \defn{equalizer} (\S7.51) of $f,g$ iff (UMP) $fe = ge$ and \[\forall_{Z,z . zf = zg} \exists!_u eu = z \quad \xymatrix{ @@ -187,11 +262,15 @@ Entries within each section are roughly sorted by definition, alphabetically. }\] Equalizers are essentially unique (\S7.53) and monic. % XXX Cite? + \paragraph{} + % A mono $m$ is a \defn{extremal} (\S7.61) if $e$ epic and $m = f \circ e$ implies that $e$ iso. + \paragraph{} + % Let $G: \mathbf{A} \to \mathbf{B}$ and $B \in \mathbf{B}$. A - \defn{$G$-structured arrow with domain $B$} is a pair $(f : B \to GA, A)$. + \defn[gstrarr]{$G$-structured arrow with domain $B$} is a pair $(f : B \to GA, A)$. (\S8.30) It is \begin{itemize} \item \defn{generating} if $\forall_{r,s : A \to A'} . Gr \circ f = Gs @@ -211,10 +290,14 @@ Entries within each section are roughly sorted by definition, alphabetically. When $G$ is a subcategory inclusion, a $G$-structured universal arrow is a \defn{reflection} (\S4.16). + \paragraph{} + % $f : A \to B$ is an \defn{isomorphism} if $\exists!_g . f \circ g = id_B ~\wedge~ g \circ f = id_A$. (\S3.8; ! in \S3.11). Every isomorphism is both monic and epic (Awodey:P2.6). + \paragraph{} + % $f$ is a \defn{monomorphism} (\S7.32) (equiv: is \defn{monic} (Awodey:D2.1)) if \[\xymatrix{\forall_{i,j} . mi = mj \Rightarrow i = j & C \ar@<1ex>[r]^{i} \ar@<-1ex>[r]_j & A \ar@{{ >}->}[r]^m & B} \] @@ -223,21 +306,31 @@ Entries within each section are roughly sorted by definition, alphabetically. $\mathbf{Set}$. Objects with monomorphisms to $X$ are called \defn{subobjects} of $X$ (Awodey:D5.1). + \paragraph{} + % A \defn{point} (Awodey:p32) of $C$ is any $c : 1 \to C$. + \paragraph{} + % $f$ is a \defn{regular monomorphism} (\S7.56) if it is an equalizer of some pair of morphisms. + \paragraph{} + % $f : A \to B$ is a \defn{retraction} if $\exists_g . f \circ g = 1_B$ (\S7.24) aka \defn{split epi} (Awodey:D2.7). If $f$ and $g$ are retractions, then so is $g \circ f$; if $g \circ f$ is a retraction, then so is $g$. (\S7.27) + \paragraph{} + % $f : A \to B$ is a \defn{section} if $\exists_g . g \circ f = 1_A$. (\S7.19) aka \defn{split mono} (Awodey:D2.7). If $f$ and $g$ are sections, then so is $g \circ f$; if $g \circ f$ is a section, then so is $f$. (\S7.21) + \paragraph{} + % Several morphism properties combine in useful ways: \begin{itemize} \item mono, epi $\Rightarrow$ \defn{bimorphism} (\S7.49) @@ -251,6 +344,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \section{Exponentials} + \paragraph{} + % (Awodey:p107,D6.1) In a category with binary products, given two objects $B$ and $C$, their \defn{exponential} is an object $C^B$ and arrow $\epsilon : C^B \times B \to C$ s.t. @@ -262,44 +357,64 @@ Entries within each section are roughly sorted by definition, alphabetically. }\] The arrows $f$ and $\tilde f$ are ``exponential transposes.'' + \paragraph{} + % Exponential transposition is self inverse (Awodey:p108). This implies \[ \mbox{hom}_{\bf C}(A \times B, C) \simeq \mbox{hom}_{\bf C}(A, C^B) \] + \paragraph{} + % A category is \defn{cartesian closed} (Awodey:p108,D6.2) if it has all finite products and exponentials. \section{Functors} + \paragraph{} + % Default notation here: functors $F,G : \mathbf{A} \to \mathbf{B}$. + \paragraph{} + % A \defn{(covariant) functor} $F$ (\S3.17;Awodey:D1.2) assigns to each $\mathbf{A}$-object a $\mathbf{B}$-object and to each $\mathbf{A}$-morphism a $\mathbf{B}$-morphism s.t. composition and identites are {\em preserved}. + \paragraph{} + % A \defn{contravariant functor} $F$ (\S3.20.5) is a (covariant) functor $\mathbf{A}^\text{op} \to \mathbf{B}$. + \paragraph{} + % A \defn{diagram} (Awodey:D5.15) is a functor $D : J \to C$ from some indexing category $J$. + \paragraph{} + % A \defn{endofunctor} has $\mathbf{A} = \mathbf{B}$. $F \circ F$ may be denoted $F^2$, etc. (\S3.23; ftn 15) + \paragraph{} + % Functors compose. (\S3.23) % XXX Cite + \paragraph{} + % A functor $F : C \to D$\dots \begin{itemize} - \item \defn{preserves limits of type $J$} if + \item \defn[fpresvlim]{preserves limits of type $J$} if \[ \forall_{D : J \to C}\forall_{\varprojlim_j D_j} F(\varprojlim_j D_j) \simeq \varprojlim_j F(D_j).\] - \item \defn{creates limits of type $J$} if $\forall_{D : J \to C}$ + \item \defn[fcreatlim]{creates limits of type $J$} if $\forall_{D : J \to C}$ and all limits $L = \varprojlim_j FD_j$ (i.e., bundle $p_j : L \to FD_j$ in $C'$), $\exists! (\bar{p_j} : \bar{L} \to D_j) \in C'$ with $F(\bar L) = L$, $F(\bar{p_j}) = p_j$, and $\bar L = \varprojlim_j D_j$. \end{itemize} + \paragraph{} + % A functor $F$ is (\S3.27, \S3.33) \begin{itemize} \item \defn{amnestic} if $f$ is an identity iff $Ff$ is an identity. @@ -309,10 +424,12 @@ Entries within each section are roughly sorted by definition, alphabetically. \item an \defn{embedding} if it is injective on morphisms. \item \defn{faithful} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')} \subseteq \mathbf{B}(FA, FA')$ is injective. - \item \defn{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective. + \item \defn[fullfunc]{full} if $\forall_{A,A'} . F\vert_{\mathbf{A}(A,A')}$ surjective. \item \defn{isomorphism-dense} if $\forall_B . \exists_A . F(A) \simeq B$. \end{itemize} + \paragraph{} + % A \defn{natural transformation} $\tau : F \natto G$ assigns each $A \in \mathbf{A}$ to $\tau_A : FA \to GA$ s.t. $\forall_{f : A \to A'} . G f \circ \tau_A = \tau_{A'} \circ F f$ @@ -324,14 +441,20 @@ Entries within each section are roughly sorted by definition, alphabetically. & FA \ar[r]^{\tau_A} \ar[d]_{Ff} & GA \ar[d]^{Gf} \\ & FB \ar[r]^{\tau_B} & GB} \] + \paragraph{} + % There is special notation for functors ($H$) applied to natural transformations and vice-versa (\S6.3): $H\tau : HF \natto HG$ defined by $(H\tau)_A = H(\tau_A)$ and $\tau H : FH \natto GH$ defined by $(\tau H)_A = \tau_{HA}$. + \paragraph{} + % All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$) isomorphisms (\S3.21), sections (\S7.22), and retractions (\S7.28). + \paragraph{} + % Some functors \defn{reflect} (in $\mathbf{B}$ implies in $\mathbf{A}$) useful properties: \begin{itemize} \item Full, faithful functors reflect sections (\S7.23) and retractions (\S7.29). @@ -340,12 +463,18 @@ Entries within each section are roughly sorted by definition, alphabetically. \subsection{Special Functors} + \paragraph{} + % The \defn{covariant representable functor} (Awodey:p44) at $A \in \mathbf{C}$ is defined by $\mbox{Hom}(A,\text{---}) : \mathbf{C} \to \mathbf{Sets}$. These functors are continuous (Awodey:P5.25). + \paragraph{} + % Representable functors preserve monos. (\S7.37.1) + \paragraph{} + % Pullback defines a functor \[ h^* : (A \stackrel{\alpha}{\to} C) \in \mathbf{C}/C \mapsto (C' \times_C A \stackrel{\alpha'}{\to} C') \in \mathbf{C}/C' \] @@ -354,36 +483,52 @@ Entries within each section are roughly sorted by definition, alphabetically. \section{Cones and Sources} + \paragraph{} + % A \defn{cone} (Awodey:D5.15) to a diagram $D(J)$ is a collection of arrows $c_j : C \to D_j$ s.t. $\forall_{D_\alpha \in D(J)} c_j = D_\alpha \circ c_i$. + \paragraph{} + % A \defn{source} in category $\mathbf{A}$ indexed by $I$ is a pair $(A, \set{f_i : A \to A_i}_{i \in I})$. This source has domain $A$ and codomain $\set{A_i}_{i\in I}$. (\S10.1) + \paragraph{} + % Given $(A,\set{f_i}_{i \in I})$ and $\{(A_i,\set{g_{ij}}_{j \in J_i})\}_{i \in I}$ all sources, their \defn{composite} is $(A, \set{g_{ij} \circ f_i}_{i\in I, j\in J_i})$. (\S10.3) + \paragraph{} + % A \defn{mono-source} (\S10.5) is $(A,\set{f_i})$ s.t. \\ $\forall r,s: B \to A. \brak{\forall_{i\in I} . f_i \circ r = f_i \circ s} \Rightarrow r = s$. \section{Concrete Categories} + \paragraph{} + % For this section, $\mathbf{A}$ is a \defn{concrete category} over $\mathbf{X}$ with \defn{forgetful} functor $U : \mathbf{A} \to \mathbf{X}$ faithful, denoted $(\mathbf{A}, U)$. (\S5.1.1) + \paragraph{} + % When $\mathbf{A} = \mathbf{X}$, $\mathbf{Alg}(U)$ has as objects $U$-\defn{algebra}s $(X \in \mathbf{X}, h : UX \to X)$ and morphisms $f : (X,h) \to (X',h')$ s.t. $f \circ h = h' \circ T(f)$. (\S5.37) + \paragraph{} + % If $\mathbf{X}$ is $\mathbf{Set}$, $\mathbf{A}$ is a \defn{construct}. (\S5.1.2) - $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn{is an $\mathbf{A}$-morphism} + \paragraph{} + % + $(UA \overset{f}{\to} UB) \in \mathbf{X}$ \defn[Amporphism]{is an $\mathbf{A}$-morphism} if $f$ has a unique $U$-preimage in $\mathbf{A}$. (\S5.3, \S6.22) %An object $A\in\mathbf{A}$ is @@ -393,6 +538,8 @@ Entries within each section are roughly sorted by definition, alphabetically. % \item \defn{indiscrete}, $(UB \to UA)$ (\S8.3) %\end{itemize} + \paragraph{} + % A \defn{free object} $A \in \mathbf{A}$ is one with a ($U$-structured) universal arrow $(u,UA)$ in $B$. (\S8.22+\S8.30) @@ -400,47 +547,38 @@ Entries within each section are roughly sorted by definition, alphabetically. %\overset{f \circ g}{\to} UB$ is an $\mathbf{A}$-morphism implies that $UC %\overset{g}{\to} UA$ is an $\mathbf{A}$-morphism. -\section{Derived Categories} - - The \defn{cone} category over a given diagram, $\mathbf{Cone}(D(J))$, has - as objects cones to that diagram and a morphism between cones is an arrow - $\phi : C \to C'$ s.t. $\forall_{D_j \in D(J)} c_j^\prime \circ \phi = - c_j$. - - The \defn{dual} (\S3.5;Awodey:p15,i2) category $\mathbf{A}^\text{op}$ - which exchanges domains and codomains of arrows in $\mathbf{A}$. Any - statement implies its dual. - - The \defn{arrow} (Awodey:p16,i3) category $\mathbf{C}^\to$ has arrows for - commutative squares in $\mathbf{C}$. There are two functors - $\mathbf{cod}, \mathbf{dom} : \mathbf{C}^\to \to \mathbf{C}$. - - The \defn{slice} (Awodey:p16,i4) category $\mathbf{C}/C$ has objects of - arrows in $\mathbf{C}$ with codomain $C$. Arrows are tops of commutative - triangles. - \section{Adjoints and Adjoint Situations} \subsection{Joy Approach} + \paragraph{} + % A functor $G : \mathbf{A} \to \mathbf{B}$ is \defn{adjoint} if $\forall_{B \in \mathbf{B}}$ there exists a $G$-structured universal arrow with domain $B$. (\S18.1) + \paragraph{} + % Adjoints compose (\S8.5), preserve mono sources (\S8.6), and preserve limits (\S8.9) + \paragraph{} + % Given adjoint $G$ with $\eta_B : B \to G(A_B)$ the $G$-structured universal arrow with domain $B$, $\exists!_F$ such that $FB = A_B$ and $\eta : id_B \natto G \circ F$ is natural; further, there is a unique, natural $\epsilon : F \circ G \natto id_A$ with $G\epsilon \circ \eta G = id_G$ and $\epsilon F \circ F \eta = id_F$. (\S19.1) + \paragraph{} + % $(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a \defn{adjoint situation} if the above relationships hold. (\S19.7) \subsection{Awodey Approach} + \paragraph{} + % An \defn{adjunction} (Awodey:D9.1) of $F : C \to D$ and $G : D \to C$ is a natural transformation $\eta : I_C \stackrel{\cdot}{\to} (G\circ F)$ s.t. \[\xymatrix{ @@ -454,6 +592,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \subsection{Moving Right Along} + \paragraph{} + % A \defn{monad} (\S20.1) on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, \eta : id_{\mathbf{X}} \natto T, \mu : T^2 \natto T)$ s.t. \[\forall_X \quad @@ -468,19 +608,27 @@ Entries within each section are roughly sorted by definition, alphabetically. \appendix \section{Miscellaneous Terminology} + \paragraph{} + % A category is \defn{finitely presented} (Awodey:p75) if it is the free category over a finite graph quotiented by a finite set of equations. + \paragraph{} + % The \defn{local membership relation} for generalized element $z : Z \to C$ and subobject $M$ (i.e., with monic $m : M \to C$), $z \in_X M$, holds iff $\exists_{f:Z \to M} . z = mf$. - An \defn{$\omega$-complete Partial Order} ($\omega$CPO) is a Poset which + \paragraph{} + % + An \defn[wCPO]{$\omega$-complete Partial Order} ($\omega$CPO) is a Poset which has all {\em co}limits of type $(\mathbb{N},\le)$. (All countably infinite ascending chains have a top.) (Awodey:p101,E5.33) \section{Miscellaneous Useful Properties} + \paragraph{} + % (Awodey:p84,L5.8) In the commuting diagram \[\xymatrix{ F \ar[r]_{f'} \ar[d]^{h''} & E \ar[r]_{g'} \ar[d]^{h'} & D \ar[d]^{h} \\ A \ar[r]^f & B \ar[r]^g & C @@ -490,8 +638,12 @@ Entries within each section are roughly sorted by definition, alphabetically. \item If $FDCA$ and $EDCB$ are pullbacks, so is $FEBA$. \end{enumerate} + \paragraph{} + % (Awodey:p84,C5.9) Pullbacks preserve commutative triangles. + \paragraph{} + % Universal Constructions (or Universal Mapping Properties, UMP) reduce to limits (Awodey:p91,E5.17-20): % @@ -506,21 +658,31 @@ Entries within each section are roughly sorted by definition, alphabetically. \hline \end{tabular} + \paragraph{} + % Objects defined by UCs are unique up to isomorphism. \section{Examples To Jog Your Memory} \subsection{$\mathbf{Mon}$} + \paragraph{} + % Mono-epics are not isos ($(\mathbf{N},+,0) \to (\mathbf{Z},+,0)$). (Pierce:\S1.6.3) + \paragraph{} + % $(\set{*},\cdot,*)$ is a (the) zero. + \paragraph{} + % Each monoid $M$ has only one point, $1 \to M$. \subsection{Adjoint Situations and Monads} + \paragraph{} + % Consider $(\eta, \epsilon) : F \dashv G : \mathbf{Mon} \to \mathbf{Set}$. $\eta_X : X \to GFX$ is insertion of generators: $\forall x \in X . \eta_X x = x$. $\epsilon_Y : FGY \to Y$ is the re-introduction of structure; @@ -530,6 +692,8 @@ Entries within each section are roughly sorted by definition, alphabetically. \quad \epsilon_Y (y \in GY) = y \] + \paragraph{} + % Further, $T = G \circ F$ is a monad. Generically, $\mu$... \begin{align*} \mu_X (TTX) &= (G \epsilon F)_X (TTX) = (G \epsilon_{FX}) (GFGFX) \\ -- 2.50.1