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Let $\mathcal{A},\mathcal{B}$ be small categories, $\mathcal{C}$ a cocomplete category and $\mathcal{D}$ an arbitrary category. Consider functors $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\rightarrow\mathcal{C}$, $R:\mathcal{D}\rightarrow\mathcal{C}$ and $L:\mathcal{C}\rightarrow\mathcal{D}$, where $L$ is left adjoint to $R$. We want to show that $$L\circ\text{Lan}_F(G)=\text{Lan}_F(L\circ G).$$

The author provides the following proof:

For every functor $H:\mathcal{B}\rightarrow\mathcal{D}$, we get the following bijections:

$$ \begin{align} \text{Nat}\left(L\circ\text{Lan}_F(G),H\right) & \cong \text{Nat}\left(\text{Lan}_F(G),R\circ H\right) \\ & \cong \text{Nat}\left(G,R\circ H\circ F\right) \\ & \cong \text{Nat}\left(L\circ G,H\circ F\right)\\ & \cong \text{Nat}\left(\text{Lan}_F(L\circ G),H\right) .\end{align}$$

What is the justification for these isomorphisms; i.e. how does one deduce them? On the other hand, are these classes of natural transformations sets?

Edit:

Let $R_*:[\mathcal{B},\mathcal{D}]\rightarrow[\mathcal{B},\mathcal{C}]$ be the functor defined by $R_*(H)=R\circ H$ and $R_*(\alpha)=R*\alpha$. Then $R_*$ is left adjoint to $L_*$. But I am not sure how this helps. If $\mu:L\circ\text{Lan}_F(G)\Rightarrow H$, then $$R_*(\mu):R\circ L\circ\text{Lan}_F(G)\Rightarrow R\circ H$$ –but this doesn't give a natural transformation from $\text{Lan}_F(G)$ to $R\circ H$.

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If $L\dashv R$, then $L_*\dashv R_*$, not the other way around (the places are switched if you take precomposition, $L^*, R^*$)

A good way to see that $L_*\dashv R_*$ is via the formulation of adjoints with a unit and a counit satisfying the triangle identities.

You'll want $\epsilon_* : L_*R_*\to id$, which is simply given by $\epsilon G : LRG\to G$ for any $G$, and $\eta_* : id\to R_*L_*$ which is also given by $\eta G : G\to RLG$. That they satisfy the triangle identities is essentially obvious because $\epsilon,\eta$ do (if you're not convinced, write it down !)

Therefore all the isomorphisms in the proof are justified.

(with regards to your edit, maybe a more concrete way to see it : suppose you have $\theta : LT\to S$, then you get $R\theta : RLT\to RS$, and then you can precompose by $\eta T : T\to RLT$ to get $(R\theta)\circ \eta T : T\to RS$)

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  • $\begingroup$ Thank you. Just a quick question: Are these classes of natural transformations sets? I ask because I would like to know if any link can be drawn between this result and Yoneda. $\endgroup$ – alf262 Jul 30 '20 at 17:59
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    $\begingroup$ Since $\mathcal{A,B}$ are assumed to be small, they are sets indeed (for any small category $C$, and any category $D$ - which is assumed to be locally small, as is done most of the time -, $Fun(C,D)$ is locally small, that is, $Nat(F,G)$ is a set). But even if they were classes, you could use Yoneda, you just have to be careful in a sense $\endgroup$ – Maxime Ramzi Jul 30 '20 at 18:05

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