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Prove that for any small category $\mathcal C$ and any functor $F:\mathcal C^\text{op}\to\textbf{Set}$, $F$ can be written as a colimit of representable functors $h_x=\text{Hom}_{\mathcal C}(-,x)$.

I am aware that there is already this post. However, I am not sure if I understand Maclane's notation and terminology. Furthermore, his question was for covariant functors, while my questions is for contravariant functors. So please consider this question as a new question.

Let $\hat{X}$ denote $\text{Mor(-,X)}$ and for $f: X \to Y$ let $\hat{f}$ denote the map $f \circ -$ from $\hat{X}$ to $\hat{Y}$. I am told that we can consider the functor $G: I \to \textbf{Set}^{\mathcal C^{op}}$ where the objects of the category $I$ consists of the pairs $(X, \alpha: \text{Hom}(-,X) \implies F)$ where $X$ is any object in $Obj(\mathcal C)$ and $\alpha$ is any natural transformation between $\text{Hom}(-,X)$ and $F$, and the morphisms in $I$ are morphisms $f: X \to Y$ in $\mathcal C$ such that the following induced diagram commutes (in case the graph shows weirdly, the relation should be $\alpha = \beta \circ \hat{f}$ where $\alpha: \hat{X} \to F, \beta: \hat{Y} \to F, \hat{f}: \hat{X} \to \hat{Y}$:

[ \begin{array}{c} \hat{X} \\ \downarrow \alpha \quad \downarrow \hat{f} \\ \hat{Y} \quad \xrightarrow{\beta} \quad F \end{array} ]

Since the category of presheaves on $\mathcal C$ is complete and cocomplete, we know that the colimit of $G$ exists. I have shown that there is a unique morphism between from $\text{colim} G$ to $F$, and what remains to show is that there is a uniuqe morphism from $F$ to $\text{colim} G$, and that the composition of the two morphisms is in fact identity on $F$ or identity on $\text{colim} G$. So suppose that $H$ is a functor from $\mathcal C^{op}$ to $\textbf{Sets}$ such that for any morphism $f : X \to Y$ in $\mathcal C$ we have that $\delta_X = \delta_Y \circ \hat{f}$ where $\delta_X, \delta_Y$ are morphisms of our cone with object $H$ in $\textbf{Sets}^{\mathcal C^{op}}$. By Yoneda's lemma, there is a concrete object $a_X, a_Y \in H(X)$ that uniquely represents $\delta_X$ and $\delta_Y$, respectively. However, I am stuck after here. What do I do?

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  • $\begingroup$ There's no difference between the covariant and contravariant cases, you just replace $C$ with $C^{op}$ to translate between one and the other. $\endgroup$ Nov 1, 2023 at 20:17

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I don't want to go into details here since these can be found in many places. I just present the idea and some sense of how you can find the idea for yourself. Basically, you do the only thing you can do, which generally is what works in elementary category theory.


Rather than try to say, $\varinjlim G$ exists for abstract reasons and we should figure out whether or not $\varinjlim G\cong F$, let's just directly witness a cocone $\lambda_i:G(i)\to F$ and show this cocone is universal.

So, the object "$i$" is really just a pair $(X,\alpha)$ where $X$ is an object of $\mathsf{C}$ and $\alpha$ is an element of $F(X)$ (or a natural transformation $\hat{X}\implies F$, there is no real difference) and we want to find some arrow $\lambda:\hat{X}\to F$ in the presheaf category... but we are already given the natural transformation $\alpha$ so of course we should choose $\lambda=\alpha:\hat{X}\to F$.

Is this a cocone? Yes, by an easy check. Is it universal? If $H$ also has some $\mu_{(X,\alpha)}:\hat{X}\to H$, these correspond to elements $\mu'_{X,\alpha}$ of $H(X)$ and we want to find a natural transformation $F\to H$ and show this is unique in a certain sense. Well, that means giving a function, for every $X$, $F(X)\to G(X)$. Ok, well I have been given my elements $\alpha\in F(X)$ ($\sim$ maps $\hat{X}\to F$) and I've also been given my elements $\mu'_{X,\alpha}\in H(X)$... it stands to reason I should be mapping $\alpha\mapsto\mu'_{X,\alpha}$. I have literally no other information about the elements of $H(X)$, so this is basically the only reasonable choice.

Then you can check this really is well-defined and a natural transformation, and further check it turns the cocone to $F$ into the cocone to $H$ and is also unique in this respect. I leave those to you.

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