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The lozenge and compact-square forms of enriched naturality are equivalent

Statement

Let $T, S : A \to B$ be $V$ -functors and let $α_{A} : 1 \to B (T A, S A)$ be a family of components. Then the lozenge equation of Enriched natural transformation is equivalent to the commutativity, for every $A, B$ , of the square

$\begin{array}{ccc} A (A, B) & \overset{T_{A, B}}{\to} & B (T A, T B) \\ ↓ S_{A, B} & ↓ B (1, α_{B}) \\ B (S A, S B) & \overset{B (α_{A}, 1)}{\to} & B (T A, S B), \end{array}$

where $B (1, α_{B})$ and $B (α_{A}, 1)$ are the canonical maps induced by composition with the components of $α$ .

Facts & Assumptions

Given: $V$ -functors $T, S : A \to B$ and a family $α_{A} : 1 \to B (T A, S A)$ .

[L1]

A $V$ -natural transformation is exactly such a family satisfying the lozenge equation against every hom-object (Enriched natural transformation).

Proof

technique · direct

1.1L1given

By [L1], the lozenge compares two composites from $A (A, B)$ to $B (T A, S B)$ obtained by first applying $S_{A, B}$ or $T_{A, B}$ and then composing with the component of $α$ at the source or target. The canonical maps $B (α_{A}, 1)$ and $B (1, α_{B})$ are precisely those postcomposition and precomposition maps written without the unitors.

2.1L1step 1.1algebra

After identifying $1 \otimes A (A, B)$ and $A (A, B) \otimes 1$ with $A (A, B)$ by the unitors, the two composites in the lozenge become exactly $B (α_{A}, 1) \circ S_{A, B}$ and $B (1, α_{B}) \circ T_{A, B}$ . Hence the lozenge equation holds if and only if those two morphisms are equal, which is exactly the commutativity of the displayed square.

3.1step 2.1∎

Therefore the lozenge and compact-square formulations are equivalent.

Depends on

Enriched natural transformation

Used by

Dependency tree · two levels

2 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, equation (1.39) and Section 1.8 (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 3.5 (standard reference, not scraped)