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Dear nLab experts,
Even as a beginner in category theory, I was hoping to make some (simple, basic) contributions to the $n$Lab. To begin with, I thought of something simple and relatively safe–the diagonal functor. To make it interesting, I wanted also to show that the diagonal functor always preserves all limits, which could be useful as a starting point for proving cocompleteness by the AFT.
But then I noticed that I cannot find this result anywhere on the web. Moreover, looking at reflexive coequalizer, I saw that the preservation of colimits by $Delta$ was attributed to the fact that it was already known to be left adjoint (i.e., that certain limits exist). I am therefore not sure that my argument is right. I would be very thankful if the experts of $n$Lab could comment about it before I attempt to enter it as an new $n$Lab page. (I hope this is appropriate.)
So, here goes: Let $J$ and $C$ be arbitrary categories. The diagonal functor $\Delta=\Delta_J: C\to C^J$ is the functor sending each object $c$ to the constant functor $\Delta c$ (the functor having value $c$ for each object of $J$ and value $1_c$ for each arrow of $J$), and each arrow $f: c\to c'$ of $C$ to the the natural transformation $\Delta f : \Delta c \stackrel{.}{\to} \Delta c'$ which has the same value $f$ at each object $j$ of $J$.
Proposition. Let $P$ and $C$ be arbitrary categories. Then $\Delta_P: C\to C^P$ preserves all limits that exist in $C$.
Before the proof, recall that limits in functor categories are
calculated pointwise. In some detail, if for an object $p\in \mathrm{obj}(P)$
we write $E_p:X^P\to X$ for the ”evaluate at $p$” functor (with
$E_p(H: P\to X)=H(p)$ and $E_p(\sigma: H\stackrel{.}{\to} H')=\sigma_p: H(p)\to H'(p)$),
then we have the following fact (Theorem V.3.1 on p. 115 of CWM):
If $S: J\to X^P$ is such that for each object $p$ of $P$, $E_p S: J\to X$
has a limiting cone $\tau_p: L(p)\stackrel{.}{\to} E_p S$,
then there exists a unique functor $L$ with object function $p \mapsto L(p)$
such that $\tilde{\tau}=\{\tilde{\tau}_{j,p}\}$ with
$\tilde{\tau}_{j,p}:=\tau_{p,j}$ is a cone
$\tilde{\tau}: \Delta_J(L)\stackrel{.}{\to} S$; moreover, this
$\tilde{\tau}$ is a limiting cone from $L\in \mathrm{obj}(X^P)$ to
$S: J\to X^P$.
(Note: As far as I can see in the proof in CWM, there are no limitations whatsoever on the size of $J$ and $P$. If I recall correctly, in Borceux both are constraint to be small, but perhaps this is because in Borceux ”category” means ”locally small category” by definition. In the $n$Lab entry on limits, $P$ is arbitrary but $J$ is small. Would anyone agree to explain the reason for the constraint on $J$? Please note that I have a very lo-tech knowledge of foundations; I use what I found in CWM: ZFC $+$ one universe).
Back to the proof of the proposition, let $F: J\to C$ be a functor with a limiting cone $\nu: \Delta_J(\ell) \stackrel{.}{\to} F$. We would like to show that $\Delta_P\nu: \Delta_P\circ \bigl(\Delta_J(\ell)\bigr) \stackrel{.}{\to} \Delta_P\circ F$ is a limiting cone. Noting that $\Delta_P\circ \bigl(\Delta_J(\ell)\bigr)=\Delta_J(\Delta_P(\ell))$ (where the first $\Delta_J$ is $C\to C^J$ and the second is $C^P\to (C^P)^J$), the last cone may be written as $\Delta_P\nu: \Delta_J(\Delta_P(\ell)) \stackrel{.}{\to} \Delta_P\circ F$.
First, we note that for each object $p$ of $P$, $E_p\circ(\Delta_P\circ F)$ is just $F$, and therefore has the limiting cone $\nu:\ell \stackrel{.}{\to} F$ by assumption. Hence, it is clear that $\Delta_P\circ F$ has a limit, but we must verify that $\Delta_P\nu$ is a limiting cone.
One functor $P\to X$ with object function $p\mapsto \ell$ is just $\Delta_P(\ell)$. For this functor, we have our cone $\Delta_P\nu: \Delta_J(\Delta_P(\ell)) \stackrel{.}{\to} \Delta_P\circ F$. Since for all $j$ and $p$ we have $(\Delta_P\nu)_{j,p}=\nu_j=j\text{th component of the limiting cone of }E_p\circ(\Delta_P\circ F)$, we are done by the theorem on pointwise limits.
Is this wrong?
Thank you very much in advance! Yaron
Yaron – welcome aboard! There are lots of places on the Lab where you can help.
Your argument looks correct, and you’re right that it goes through without worrying about set-theoretic constraints. In other words, it is a theorem of the first-order theory of categories, which makes no reference to a background theory of sets. However, in practice, most categories (excepting complete posets) have only small limits, so the restriction to $J$ small is not too drastic, and there may be other reasons (such as being able to give more than one proof) that smallness assumptions are made.
As far as I could tell, at the page reflexive coequalizer, the only place where the diagonal functor was asserted to be a left adjoint is in the case of $C = Set$. Here, to get left adjointness, we actually do require that $P$ be small: $Set$ is only small-complete, and so there will generally be no limit functor $lim_P: Set^P \to Set$ if $P$ is large (although there are particular $P$ where it’s okay, such as where $P$ has two objects $a$, $b$ and all non-identity arrows forming a proper class which go from $a$ to $b$).
It would be fine to write up this result. Only I would like to see it on its own page (you could title it diagonal functors preserve limits or something), linked to as you see fit.
Thanks!
Thank you very much, Todd!
I will use your link to do this right away.
Yaron
OK, done. Still needs some formatting (for example, using the header format for theorems doesn’t look that good), but it’s getting a bit late here, so I’ll get back to it tomorrow.
Should I say anything in “latest changes?”
Thanks, Yaron
Thanks, Yaron! Yes, something in the Latest changes category would be good.
We didn’t actually have a page diagonal functor yet, and since half of what you wrote is a definition of that, I moved your page there and reformatted it slightly.
Welcome, Yaron! I will now proceed to welcome you further by pedantically nitpicking what Todd said… (-:
it is a theorem of the first-order theory of categories, which makes no reference to a background theory of sets
I’m not sure that’s quite the right way to phrase it, since without a background set theory, I don’t know how one would construct functor categories. I would say instead that the theorem makes sense in any reasonable set-theoretic background such that functor categories can be defined.
Also, most large categories do have some large limits. For instance, for any category C, the colimit of the forgetful functor $C/x \to C$ is the object $x$. They just don’t usually have all large limits, and we rarely ever use large limits for much. But they are sometimes convenient, cf. for instance total category.
I’m not sure that’s quite the right way to phrase it, since without a background set theory, I don’t know how one would construct functor categories.
D’oh! Silly me, of course that’s right.
Thanks for the inputs everyone!
Following Toby’s suggestion, I added an input to latest changes. Considering the current lab status, I can’t check the link…
Try the new address.
Thanks, Tim – It seems to work :)
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