EnrichedCategoryTheory 2010-11-01 00:27:13 2010-11-01 00:46:03 Topic Kan adjointness enriched categories weak Yoneda Lemma \section{Enriched Category Theory} \subsection{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}