Enriched Category Theory
EnrichedCategoryTheory
2010-11-01 00:27:13
2010-11-01 01:21:03
Topic
Kan adjointness
enriched categories
weak Yoneda Lemma
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\section{Enriched Category Theory}
This is a new topic on enrichments of category theory, including a weak Yoneda Lemma, functor categories, 2-categories and representable V-functors.
\subsection{Monoidal Categories}
$2-category$ VCAT for a monoidal V category
$2-functors$, such as $F: VCAT \to CAT$
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}
\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 isomorphism $[A \times [B, C]] \cong [A,[B,C]]$
\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}
\textbf{more to come}