6
$\begingroup$

I'm studying the notion of a a $\mathcal{V}$ category $\underline{\mathscr{A}}$ which is powered or compowered over $\mathcal{V}$. I'm having trouble finding a proof that powering/copowering gives a $\mathcal{V}$ functor $\underline{\mathcal{V}} \otimes \underline{\mathscr{A}} \to \underline{\mathscr{A}}$. (Add "op" as necessary.)

In Riehl's Categorical Homotopy Theory, she gives some indication of the proof, but I'm having trouble parsing it. In particular, see these two pages. The claim here appears to be the following:

Proposition: [Edit: I believe I have interpreted the text wrong, because this seems to be incorrect] Let $\underline{\mathscr{A}}$ and $\underline{\mathscr{B}}$ be $\mathcal{V}$ categories, with underlying categories $\mathscr{A}$ and $\mathscr{B}$. Let $L:\mathscr{A} \leftrightarrows \mathscr{B}: R$ be an adjunction with unit $\eta: 1_\mathscr{A} \Rightarrow RL$. Let there be a given isomorphism $\underline{\mathscr{A}}(a,Rb) \cong \underline{\mathscr{B}}(La, b)$ whose image in the underlying categories is the natural isomorphism of the adjunction. Let $R$ be the underlying functor of a $\mathcal{V}$ functor $\mathbf{R}:\underline{\mathscr{B}} \to \underline{\mathscr{A}}$. Then there is a $\mathcal{V}$ functor $\mathbf{F}:\underline{\mathscr{B}} \to \underline{\mathscr{A}}$ agreeing with $F$ on objects and with action on hom objects given by $$\mathbf{F}: \underline{\mathscr{A}}(a,a') \xrightarrow{(\eta_{a'})_*} \underline{\mathscr{A}}(a, RLa') \cong \underline{\mathscr{B}}(Fa, Fa')$$ $\mathbf{F}$ is the $\mathcal{V}$-adjunct of $G$.

I can't seem to show functoriality of $\mathbf{F}$. (See this link where I've drawn the diagrams out.) Maybe someone can help.

It could be that I'm misunderstanding the claim she's making.

Edit: Just considered the possibility that there's something special about $\underline{\mathcal{V}} = \underline{\mathscr{A}}$, which seems to be the case she's dealing with exclusively. Let me see if I can make that work.

Edit 2 The statement in Kelly is

When [the forgetful functor] is conservative--faithfulness is not enough--the existence of a left adjoint $S_0$ for $T_0$ implies that of a left adjoint $S$ for $T$.

So I think I must be misunderstanding Riehl's text here. Perhaps someone can help me.

$\endgroup$
  • $\begingroup$ If you are actually missing a hypothesis, I think it might be that the components $\underline{\mathscr{A}}(a,Rb) \cong \underline{\mathscr{B}}(La, b)$ satisfy the property of being enriched natural isomorphisms between the enriched (co)representable functors. $\endgroup$ – Vladimir Sotirov Aug 18 '16 at 15:50
  • $\begingroup$ @VladimirSotirov As you can see, a Yoneda Lemma argument is used to show $\underline{\mathcal{V}}(u \otimes v, w) \cong \underline{\mathcal{V}}(u, \underline{\mathcal{V}}(v,w))$. I can't see how to conclude from that that this is a natural (in $u$ and $w$) $\mathcal{V}$ isomorphism of bifunctors. Can you? $\endgroup$ – Eric Auld Aug 18 '16 at 16:27
  • $\begingroup$ I don't think she claims that it is a $\mathcal V$-natural isomorphism of functors. I think the claim is that powers and copowers are enriched functors that are ordinary adjoints to the appropriate representables. ($\mathcal V$-natural transformations aren't even defined at this point in the text, so I think when she says left/right $\mathcal V$-adjoint, she means a left or right adjoint that is $\mathcal V$-enriched, not a $\mathcal V$-adjunction which is what I think Kelly is talking about). $\endgroup$ – Vladimir Sotirov Aug 18 '16 at 17:55
  • $\begingroup$ @VladimirSotirov Thanks for the response. This is really helping. I guess I'm just not understanding how to show her claim that an enrichment of the (ordinary) right adjoint gives me an enrichment of the left adjoint. Even in the case of $-\otimes v \dashv [v,-]$, I can show that $-\otimes v$ is enriched directly, but I don't see how it follows from the enrichment of $[v,-]$ (even using her tip to postcompose with the unit). $\endgroup$ – Eric Auld Aug 18 '16 at 19:25
9
$\begingroup$

Let me start by stating clearly the unenriched version of the result whose enriched version I was trying to evoke here. This appears as Proposition 4.3.4 in Category Theory in Context which is available here:

Category Theory in Context

Prop. Consider a functor $G\colon B \to A$ so that for each $a \in A$ there exists an object $Fa \in B$ together with an isomorphism

$B(Fa,b) \cong A(a,Gb)$ that is natural in $b \in B$.

Then there is a unique way to extend the assignment $F \colon obA \to obB$ to a functor $F \colon A \to B$ so that these isomorphisms are also natural in $A$.

The proof is by the Yoneda lemma (see the cited notes).

Now I claim that the same result is true for $V$-categories $A$ and $B$, a $V$-functor $G \colon B \to A$, and a family of isomorphisms $B(Fa,b) \cong A(a,Gb)$ in $V$ that are $V$-natural in $b \in B$.

The given isomorphisms in $V$ can be used to define a canonical map in $V$

$A(a,a') \to V(B(Fa',b), B(Fa,b))$

for each $b \in B$. Not only is this canonical but it's uniquely determined if you want the iso to be $V$-natural in $A$ in the end.

Moreover these maps are (extra-ordinarily) $V$-natural in $B$. What this means is that you actually get a map into the enriched end

$A(a,a') \to \int_{b \in B} V(B(Fa',b), B(Fa,b))$.

(These are defined in section 7.3.) Now one version of the $V$-Yoneda lemma tells you that

$\int_{b \in B} V(B(Fa',b), B(Fa,b)) \cong B(Fa,Fa')$

so this map is the map you seek.

In the definition of tensors I should have asked that the isomorphisms

$M(v\otimes m,n)\cong V(v, M(m,n))$

were $V$-natural in $n$. (Vladimir's comment is correct.) In any case it was absolutely my intention to define tensored to mean that the hom bifunctor has a left $V$-adjoint. Apologies for the confusion!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.