Theorem 1. section 7 of chapter III(page 76), of Maclane's categories for working mathematicians, says any functor $K:\mathcal{D}\rightarrow Set$ can be represented as the colimit of a certain diagram of representable functors. I am a bit confused about this certain diagram, which in Maclane proof is given by $$J ^ \mathcal{D}\rightarrow Set^\mathcal{D}$$ where $J$ is the category of elements of $K$ (or $1 \downarrow K)$. What mkes me confused is that this functor is defined by $\left(d,x\right) \mapsto x$, where $d$ is an object of $\mathcal{D}$ and $x:* \rightarrow Kx$.

First, I would like to ask what is this diagram of representable functors which can recover the $K$.

Secondly, I would like to know what is the dual of this theorem (which supposedly establish an equivalence of any contravariant set valued functor with a limit of certain diagram of representable functors)

  • 2
    $\begingroup$ You dualised the statement incorrectly. The correct dualisation is obtained by replacing $\mathcal{D}$ with $\mathcal{D}^\mathrm{op}$. Unfortunately that means everything is still in terms of colimits. $\endgroup$
    – Zhen Lin
    Jan 15 '14 at 7:44

There is a typo in Maclane. Not $J^D$, but $J$, or, more precisely, $J^{op}$(or we have to say that $M$ is contravariant).

Let $D$ be a locally small category, $K\colon D\to\mathbf{Set}$ be a functor. Category of elements of $K$ is the comma category $(*\downarrow K)$. There exists a natural projection: $$ pr\colon(*\downarrow K)\to D, $$ which sends every pair $(d,x)\in(*\downarrow K)$ to the object $d\in D$. You can also consider it's dual: $$ pr^{op}\colon(*\downarrow K)^{op}\to D^{op}. $$ Another functor we need - Yoneda functor: $$ Y\colon D^{op}\to\mathbf{Set}^{D}, $$ which sends every object $d\in D$ to the representable functor $hom_D(d,-)$. Now we can take the composition: $$ (*\downarrow K)^{op}\xrightarrow{pr^{op}}D^{op}\xrightarrow{Y}\mathbf{Set}^{D}. $$ Thus, we have the functor $M=Y\circ pr^{op}$: $$ (*\downarrow K)^{op}\xrightarrow{M}\mathbf{Set}^D. $$ There exists a cocone $\varphi\colon M\to\Delta_K$, such that $\varphi(d,x)=\alpha_x$, where $\alpha_x$ is the image of $x$ by the Yoneda-lemma-isomorphism $K(d)\cong Nat(hom_D(d,-),K)$. It is an exercise to check that cocone $\varphi$ is limiting, and, consequently, $K$ is the colimit of $M$.

We can see that $M(d,x)$ is representable functor for any $(d,x)\in(*\downarrow K)$. Thus, functor $K$ is a colimit of a "diagram of representable functors".

  • 1
    $\begingroup$ It is interesting to note that this theorem is only present in the second edition of CWM. In the first edition there is no trace of it, as far as I can see. $\endgroup$
    – magma
    Jan 17 '14 at 0:45

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.