Talk:PlanetPhysics/Enriched Category Theory
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[edit source]%%% 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}