1 Enriched Category Theory

This is a new, contributed topic on enrichments of category theory, including a weak Yoneda lemma, functor categories, 2-categories and representable V-functors.

1.1 Monoidal Categories

2 − category VCAT for a monoidal V category 2 − functors, such as F : V CAT → 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

1.2 The Weak Yoneda Lemma for VCAT

1.3 Adjunctions and equivalences in VCAT

1.4 2 − Functor categories

1.5 The functor category [A,B] for small A

1.6 The (strong) Yoneda lemma for VCAT and the Yoneda embedding

1.7 The free V category on a Set category

1.8 Universe enlargement V → enV : consider [A,B] as an enV category

The isomorphism [A × [B,C]][A, [B,C]]

1.9 Indexed limits and colimits

Indexing types; limits and colimits; Yoneda isomorphisms

Preservation of limits and colimits

1.10 Limits in functor categories: double limits and iterated limits

The connection with classical conical limits when V = Set

1.11 Full subcategories and limits: the closure of a full subcategory

1.12 Strongly generating functors

1.13 Tensor and Cotensor Products

1.14 Kan extensions

The definition of Kan extensions: their expressibility by limits and colimits

1.15 Iterated Kan extensions. Kan adjoints

1.16 Filtered categories when V = Set

1.17 General Representability and Adjoint Functor theorems

1.18 Representability and adjoint-functor theorems when V = Set

1.19 Functor categories, small Projective Limits and Morita Equivalence

more to come