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Strong enriched Yoneda lemma as a particular end

Statement

Assume $V$ is symmetric monoidal right closed and locally small, and that its collection of objects is a set. Let $A$ be a small $V$ -category, let $K \in A$ , and let $F : A \to V$ be a $V$ -functor. Then the object $F K$ represents the enriched-wedge functor for the enriched end

$A ⟼ [A (K, A), F A],$

so there is a natural isomorphism in $V$

$F K ≅ \int_{A} [A (K, A), F A] .$

Thus this particular enriched end exists without assuming that every enriched end or the whole enriched functor category exists.

Facts & Assumptions

Given: A symmetric monoidal right-closed locally small base $V$ whose collection of objects is a set, a small $V$ -category $A$ , an object $K$ , and a $V$ -functor $F : A \to V$ .

[L1]

The weak enriched Yoneda lemma gives natural bijections ${Nat}_{V} (A (K, -), G) ≅ V (1, G K)$ for every $V$ -functor $G : A \to V$ (Weak enriched Yoneda lemma).

[L2]

In a right-closed base, a morphism $X \to [Y, Z]$ is equivalent to a morphism $X \otimes Y \to Z$ , and global elements of $[Y, Z]$ are morphisms $Y \to Z$ (The internal hom and its evaluation morphism, The tensor unit is an internal-hom unit).

[L3]

An enriched end is an object representing enriched wedges, whose dinaturality equations use the hom-objects of the enriching category rather than only the arrows of its underlying ordinary category.

[L4]

The base $V$ is a $V$ -category under its internal homs (A closed monoidal category is enriched in itself).

Proof

technique · direct

1.1

Fix an object $X$ of $V$ . An enriched wedge from $X$ to the displayed enriched end is, by [L2] and [L3], the same data as a family of morphisms $X \otimes A (K, A) \to F A$ satisfying the enriched dinaturality equations. After transposing by [L2], these are exactly the components and enriched naturality equations of a $V$ -natural transformation $A (K, -) \Rightarrow [X, F -]$ , where $[X, F -]$ is obtained by applying $[X, -]$ objectwise in the self-enrichment of [L4].

L2L3L4given

2.1

Apply [L1] to the functor $G = [X, F -]$ . This gives a bijection between the wedge data of step 1.1 and $V (1, [X, F K]) ≅ V (X, F K)$ , the second bijection coming from [L2]. The correspondence is natural in $X$ .

L1L2step 1.1

3.1

Step 2.1 says exactly that morphisms $X \to F K$ are in natural bijection with wedges from $X$ to the displayed diagram. By [L3], that is the universal property of the end, so $F K$ is the end $\int_{A} [A (K, A), F A]$ .

L3step 2.1

4.1

Therefore the particular end exists and is naturally isomorphic to $F K$ .

step 3.1∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (2.31) to (2.33) (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 7.3 (standard reference, not scraped)