Talk:PlanetPhysics/Enriched Category Theory

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Enriched Category Theory %%% Primary Category Code: 00. %%% Filename: EnrichedCategoryTheory.tex %%% Version: 15 %%% Owner: bci1 %%% Author(s): bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

\section{Enriched Category Theory} This is a new, contributed topic on enrichments of \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, including a weak \htmladdnormallink{Yoneda lemma}{http://planetphysics.us/encyclopedia/AbelianCategoryEquivalenceLemma.html}, \htmladdnormallink{functor categories}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{2-categories}{http://planetphysics.us/encyclopedia/2Category2.html} and representable V-functors.

\subsection{Monoidal Categories}

$2-category$ VCAT for a monoidal V \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $2-functors$, such as $F: VCAT \to CAT$

\htmladdnormallink{Tensor}{http://planetphysics.us/encyclopedia/Tensor.html} products and \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} Closed and bi-closed bimonoidal categories

Representable V \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} 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}

\subsection{The (strong) Yoneda lemma for VCAT and the Yoneda embedding}

\subsection{The free V category on a Set category}

\subsection{Universe enlargement $V \to enV$ : consider $[A,B]$ as an enV category}

The \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} $[A \times [B, C]] \cong [A,[B,C]]$

\subsection{Indexed limits and colimits}

Indexing \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html}; 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}

\textbf{more to come}

\end{document}