Enriched Category Theory

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.

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

The Weak Yoneda Lemma for VCAT

Adjunctions and equivalences in VCAT

$ 2-Functor$ categories

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

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

The free V category on a Set category

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

The isomorphism $ [A \times [B, C]] \cong [A,[B,C]]$

Indexed limits and colimits

Indexing types; limits and colimits; Yoneda isomorphisms

Preservation of limits and colimits

Limits in functor categories: double limits and iterated limits

The connection with classical conical limits when $ V = Set$

Full subcategories and limits: the closure of a full subcategory

Strongly generating functors

Tensor and Cotensor Products

Kan extensions

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

Iterated Kan extensions. Kan adjoints

Filtered categories when $ V = Set$

General Representability and Adjoint Functor theorems

Representability and adjoint-functor theorems when $ V = Set$

Functor categories, small Projective Limits and Morita Equivalence

more to come



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As of this snapshot date, this entry was owned by bci1.