# $\mathcal{V}$-naturality in enriched category theory

Let $\mathcal{V}$ be a monoidal category, in section 1.2 of "Basic concepts of enriched category theory" (http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) Max Kelly introduces the terms "$\mathcal{V}$-category", "$\mathcal{V}$-functor" and "$\mathcal{V}$-natural transformation" that make $\mathcal{V}$-CAT a 2-category. In section 1.7 he says:

... all the families of maps canonically associated to $\mathcal{V}$, or to a $\mathcal{V}$-Category $\mathcal{A}$, or to a $\mathcal{V}$-functor T, or to a $\mathcal{V}$-Natural $\alpha$, such as $a: (X \otimes Y) \otimes Z \rightarrow X \otimes ( Y \otimes Z)$, or $e: [Y,Z] \otimes Y \rightarrow Z$, or $M: \mathcal{A}(B, C) \otimes \mathcal{A}(A, B) \rightarrow \mathcal{A}(A, C)$ or $T:\mathcal{A}(A,B) \rightarrow \mathcal{B}(TA, TB)$, or $\alpha: I \rightarrow \mathcal{B}(TA, SA)$, are themselves $\mathcal{V}$-natural in every variable...

What does he mean by these families of maps "are themselves $\mathcal{V}$-Natural in every variable"? I thought the term $\mathcal{V}$-Naturality referred to $\mathcal{V}$-natural transformations $\alpha: T \rightarrow S: \mathcal{A} \rightarrow \mathcal{B}$ between $\mathcal{V}$-functors $T, S: \mathcal{A} \rightarrow \mathcal{B}$ (for $\mathcal{V}$-categories $\mathcal{A}$ and $\mathcal{B}$).

• Kelly uses "natural" for a more general notion, sometimes called extranatural or dinatural. – Zhen Lin Jan 5 '16 at 6:46
• If $\mathcal{V}$-naturality refers to a dinatural (extranatural) transformation between two functors, what does he mean by "the map $a: (X \otimes Y) \otimes Z \rightarrow X \otimes ( Y \otimes Z)$ is $\mathcal{V}$-natural in every variable"? Where are the two functors there? All I see is a map $a$ in the category $\mathcal{V}$. – Richard Jennings Jan 6 '16 at 0:25
• For fixed $Y$, $Z$, $X ↦ (X ⊗ Y) ⊗ Z$ is a functor, as is $X ↦ X ⊗ (Y ⊗ Z)$, and $a_{XYZ}$ is a natural transformation between these two functors. That it's $\mathcal V$-natural presumably means that the functors can be upgraded to $\mathcal V$-functors, and $a$ to a $\mathcal V$-transformation between them. – user54748 Jan 6 '16 at 1:31
• Sorry for taking so long, had a bit of a family emergency. So that's what he means by "$\mathcal{V}$-natural in each variable", so the functors $(- \otimes Y) \otimes Z$ and $- \otimes (Y \otimes Z)$ with the natural transformation $a_{X,Y,Z}$ define functors $(- \otimes Y) \otimes Z_{enr}$ and $- \otimes (Y \otimes Z)_{enr}$ and a $\mathcal{V}$-natural transformation $a_{enr}$ between them, right? I tried to prove this manually and got the corresponding $\mathcal{V}$-naturality condition in 1.7, but I used (1.28) to get there, not (1.34), so I don't get how it "follows easily from (1.34)"? – Richard Jennings Jan 13 '16 at 20:42

I believe you fall in the same problem I had while reading Kelly's book.

As user54748 suggested in a comment all the natural functors (the tensor product, internal hom, etc) lift to enriched $\mathcal V$-functors between the corresponding $\mathcal V$-categories. Similarly all the natural (apologize for the game of words) natural transformations between these functors lift to $\mathcal V$-natural transformations.

(A note: by the functors/natural transformations lift I mean that they are the images of $\mathcal V$-enriched functors/natural transformations through the underlying category functor, that should be named $\mathcal V_o \colon \mathcal V\text{-}\mathbf{Cat} \to \mathbf{Cat}$ if I remember correctly).

For what I remember, after the chapter on symmetric monoidal closed categories, Kelly basically exploits the categorical isomorphism between the category $\mathcal V$ and the underlying category to the $\mathcal V$-enriched category $\mathcal V$ (the category where $\text{hom}(A,B)=\mathcal V[I,[A,B]]$), and so it identifies $\mathcal V$-enriched functors and natural transformations with their not enriched counterparts.

That caused to me quite some problems, because in my personal opinion such identification is an abuse of notation, and the worst part is that the reader is not even warned (or at least I don't remember any warning).

Hope that this (not so short answer) could be of help (or at least that knowing that other people had the same problem could be of some confort).

• Thank you! Now I know I'm not the only one :), I thought Max Kelly's book was like the most basic and introductory book on the subject, are there other good introductory books out there that I can use as a reference too? – Richard Jennings Jan 13 '16 at 21:03