EnrichedCategoryTheory 2010-11-01 00:27:13 2010-11-01 00:44:39 Topic Kan adjointness enriched categories weak Yoneda Lemma % this is the default PlanetPhysics preamble. as your \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm 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Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Enriched Category Theory} \emph{Monoidal Categories} 2--category VCAT for a monoidal V--category 2--functor $F: VCAT \to CAT$ Tensor products and duality Closed and bi--closed bimonoidal categories Representable V--functors Extraordinary V--naturality and the V--naturality of the canonical maps \subsection{The Weak Yoneda Lemma for VCAT} \subsection{Adjunctions and equivalences in VCAT} \subsection{2--Functor categories} \subsection{The functor category $[A, B]$ for small A} The isomorphism $[A \times [B, C]] isomorphic~ with [A, [B,C]]$ \subsection{The (strong) Yoneda lemma for VCAT and the Yoneda embedding} \subsection{The free V category on a Set category} \subsection{Universe enlargement $V \to enV$; $[A,B]$ as a enV category$ \subsection{Indexed limits and colimits} Indexing types; limits and colimits; Yoneda isomorphisms Preservation of limits and colimits \subsection{Limits in functor categories: double limits and iterated limits} The connection with classical conical limits when $V = Set$ \subsection{Full subcategories and limits: the closure of a full subcategory} \subsection{Strongly generating functors} \subsection{Tensor and Cotensor Products} \subsection{Kan extensions} The definition of Kan extensions; their expressibility by limits and colimits \subsection{Iterated Kan extensions; Kan adjoints} \subsection{Filtered categories when $V = Set$} \subsection{General Representability and Adjoint Functor theorems} \subsection{Representability and adjoint-functor theorems when $V = Set$} \subsection{Functor categories, small Projective Limits and Morita Equivalence}