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Let $C$ be a category with finite limits; that is, for any finite category $D$ and functor $F:D\to C$ the category $\mathrm{Cone} F$ of cones over $F$ is inhabited and has terminal objects (we could turn the morphisms around and call the limit an initial obect, but never mind).

That is, define $$ (A,\phi): \mathrm{Cone} F \iff \phi:Nat(\Delta_A, F) $$ where $\Delta_A(d) = A$ and $\Delta_A(\delta:d\to d')= 1_A$; and for $(A,\phi),(B,\psi):\mathrm{Cone} F$ define $$ \mathrm{Cone}F((A,\phi),(B,\psi)) = \{ f:A\to B \mid \phi_d = \psi_d\circ f\} .$$ A limit for $F$ is a terminal object $(H,\eta)$ in $\mathrm{Cone}F$.

Now there is an obvious forgetful functor $F^\perp: \mathrm{Cone} F \to C$ with $F^\perp(A,\phi)=A$. More: for any limit $(H,\eta)$ there is a co-cone $(H,\chi)$ with $\chi_\phi:F^\perp(\phi)\to H$, the unique one showing that $(H,\eta)$ IS a limit. For any co-cone $(A,c)$ under $F^\perp$, there is by hypothesis a morphism $c_\eta:H\to A$, again by hypothesis making everything commute that should; thus $\chi$ ( $\eta$ (or $H$)) is versal for cocones under $F^\perp$.

I'd like to eventually conclude that $H$ is universal --- that $c_\eta$ is the only thing making everything commute where it should, so that $\chi$ makes $H$ a colimit for $F^\perp$, but from the diagrams, I'm afraid it probably doesn't.

Are there supplementary assumptions that'll make everything work out nicely? Does it work out nicely and I just don't see it?

Or maybe I'm wrong about something!? That'd be OK, too.

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This is hard to follow. For example, what is "H"? –  David Carchedi Apr 7 '11 at 12:19
    
Hmmm... OK, will re-tune the notation; but $H$ is, I suppose, $F^\perp(\eta)$, the object that the cone $\eta$ shows is what is usually called a limit object. –  some guy on the street Apr 7 '11 at 12:50
    
Maybe I'm dumb, but, if I have a cocone for $F^\perp$, $$\rho:F^\perp \Rightarrow \Delta_C,$$ why must there be a map $l:H \to C$ such that $$\rho=F^\perp \stackrel{\chi}{\longrightarrow} \Delta_{H} \stackrel{\Delta_{l}}{\longrightarrow} \Delta_{C}?$$ You say this is "by hypothesis", by I don't see it. –  David Carchedi Apr 7 '11 at 16:13
    
Probably Todd says it better/obviates the issue in his Answer; but since you ask, the $l$ you seek is $\rho_\eta:F^\perp(H,\eta)\to C$; since $\rho$ is natural, $\rho_\eta F^\perp(\chi_\phi) = \Delta_C(\chi_\phi) \rho_\phi $ --- remember that morphisms of cones are essentially morhpisms in $\mathcal{C}$ satisfying properties, and $F^\perp$ just forgets those properties. –  some guy on the street Apr 7 '11 at 18:47
    
And, given how much re-writing what I try to say seems to need, no, there's no evidence of dumbness here. –  some guy on the street Apr 7 '11 at 18:54

3 Answers 3

up vote 6 down vote accepted

I first of all feel like rewriting the question, to give it a less cluttered look. Let $D$ be a finite category, and let $F: D \to C$ be a functor, where $C$ is finitely complete. Let $C^D$ denote the category of functors $D \to C$, and let $\Delta: C \to C^D$ denote the diagonal functor. Then, as usual, define the category of cones $\text{Cone}(F)$ to be the comma category

$$\text{Cone}(F) = \Delta \downarrow F.$$

A limit of $F$ is by definition a terminal object of $\text{Cone}(F)$. If $\pi: \text{Cone}(F) \to C$ is the projection functor, then the question asks whether (or under what conditions) $\lim F$ is a colimit of $\pi$.

A slightly different way of expressing the concept of limit is that the projection $\pi: \text{Cone}(F) \to C$ lifts to an equivalence

$$p: \text{Cone}(F) \simeq C/\lim F$$

which is to say that $\Sigma \circ p \cong \pi: \text{Cone}(F) \to C$, where $\Sigma: C/\lim F \to C$ is the standard projection or forgetful functor. So the question is whether $\Sigma (1_{\lim F}: \lim F \to \lim F) = \lim F$ is the colimit of $\pi$.

But notice that the colimit of the equivalence $p$ is the terminal object $1_{\lim F}$. Indeed, we may as well replace the equivalence by the identity functor, and make the key observation that the terminal object is always the colimit of the identity functor.

Now observe that $\Sigma: C/\lim F \to C$ preserves arbitrary colimits, even those over class-sized diagrams -- for example, under our hypotheses, $\Sigma$ has a right adjoint given by pulling back objects along $\lim F \to 1$, the projection to the terminal object, and left adjoints preserve arbitrary colimits. It follows that

$$\lim F \cong \Sigma(\text{colim } p) \cong \text{colim } \Sigma \circ p \cong \text{colim } \pi$$

so that the limit is indeed the colimit of the projection $\pi$.


I may as well write a little more on the "secret" significance of this sort of thing. Mac Lane in Categories for the Working Mathematician remarks (footnote, pp. 52-53) that "[comma categories] were for a time a sort of secret tool in the arsenal of knowledgeable experts". This is rather well seen by contemplating the adjoint functor theorem. The whole idea behind the adjoint functor theorem is that given a functor

$$R: C \to D$$

which is known to preserve limits, one would like to construct an initial object of the comma category $d \downarrow R$ for any object $d$ of $D$; this will be a pair $(c, d \to R c)$ which solves a universal mapping problem. How to construct the initial object? Well, it should be the limit of the identity functor, if that exists (this is dual to the key observation made above). But usually this is a large limit, whereas it is reasonable to ask only that $C$ has small limits. So the idea is to hope to find a small full subcategory inclusion $i: S \to d \downarrow R$ which is final (or cofinal, I forget which way the terminology should go), so that the limit of $i$ is the limit of the identity. (Finality means that the objects of $S$ form a weakly initial family.) So then construct the limit of $i$, and you're done. The existence of this small subcategory is called a solution set condition.


Edit: It might be clearer to some readers just to argue this directly, as follows. Let $(L, \Delta L \to F)$ denote the limit. Given a cocone $\gamma: \pi \to \Delta c$, there is a component at the limit which is just a map $L \to c$. This determines the whole cocone $\gamma$, because the component of $\gamma$ at any other object $(c', \Delta c' \to F)$ is a map $c' \to c$ which factors through the unique map $c' \to L$ in $\text{Cone}(F)$. In fact, any map $L \to c$ determines a cocone. Thus there is a natural bijective correspondence between maps $L \to c$ and cocones $\pi \to \Delta c$, making $L$ the colimit of $\pi$.

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Very nice Todd :) –  David Carchedi Apr 7 '11 at 16:18
    
Oh dear! Obviously, I don't speak "comma" yet. I'd like to "accept" this answer, but I'll have to think very carefully before I understand the stuff in between "what you meant to say was... " and "... so, yes". But thanks for taking the time! –  some guy on the street Apr 7 '11 at 18:52
    
It's all right -- what I wrote earlier was just a way of convincing myself via principles I am comfortable with. I've edited my answer to give an argument which is more nose-to-the-ground logic, without high falutin' categorical principles. But obviously I think the general principles are well worth knowing. :-) –  Todd Trimble Apr 7 '11 at 19:24

Todd already gave a beautiful answer. However, I will try to give a quick answer myself in simpler language:

Since $\eta:F \Rightarrow \Delta_H$ is a limiting cone, for each object $A$, the induced map

$$Hom\left(A,H\right) \to Cone\left(F,A\right)$$

which sends a morphism $w:A \to H$ to the cone $$\Delta_A \stackrel{\Delta_w}{\longrightarrow} \Delta_H \stackrel{\eta}{\longrightarrow} F$$ is a bijection.

Given a cone $\phi:\Delta_A \Rightarrow F$, denote by $$\chi\left(A,\phi\right):A \to H$$ the unique map which induces the cone $\left(A,\phi\right)$. This corresponds to the unique map from $\left(A,\phi\right)$ to $\left(H,\eta\right)$ that exists in the category of cones; i.e. $\left(H,\eta\right)$ is terminal in this category, as you stated.

It is clear that the morphisms $\chi\left(A,\phi\right)$ assemble into a cocone $$\chi:F^\perp \Rightarrow \Delta_{H}.$$ We want to show it is colimiting. To show this, we want to show that for each $C$, the induced map

$$Hom\left(H,C\right) \to Cocone\left(F^\perp,C\right)$$

which sends a morphism $g:H \to C$ to the cocone to $$F^\perp \stackrel{\chi}{\longrightarrow} \Delta_{H} \stackrel{\Delta_{g}}{\longrightarrow} \Delta_{C}$$ is a bijection.

As you were nice enough to point out in the comments, it is surjective:

Given a cocone $\rho:F^\perp \Rightarrow \Delta_C$, by the fact that $\chi$ is a natural transformation, it can easily be checked that $\rho$ is induced by the map $\rho\left(H,\eta\right)$.

Now for injectivity:

Suppose that $g$ and $g'$ are maps $H \to C$ such that the cocones $$F^\perp \stackrel{\chi}{\longrightarrow} \Delta_H \stackrel{\Delta_g}{\longrightarrow} \Delta_C$$ and $$F^\perp \stackrel{\chi}{\longrightarrow} \Delta_H \stackrel{\Delta_g'}{\longrightarrow} \Delta_C$$ are equal.

Then they have equal components on every cone $\left(A,\phi\right)$. But their components on $\left(H,\eta\right)$ are $g$ and $g'$ since $\chi\left(H,\eta\right)=id_H$. So $g=g'$.

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See also my edit in my answer. –  Todd Trimble Apr 7 '11 at 22:04
    
Ah, sorry Todd. I didn't notice you edited. –  David Carchedi Apr 7 '11 at 23:05

See the book http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html Prop.12.8 pag. 208.

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1  
gee, why didn't I think of that? –  some guy on the street Apr 7 '11 at 22:22

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