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1. • CommentRowNumber1.
   • CommentAuthorYaron
   • CommentTimeNov 27th 2010
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   Author: Yaron
   Format: MarkdownItexDear 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

   Dear nLab experts,

   Even as a beginner in category theory, I was hoping to make some (simple, basic) contributions to the 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 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 Lab could comment about it before I attempt to enter it as an new Lab page. (I hope this is appropriate.)

   So, here goes: Let and be arbitrary categories. The diagonal functor is the functor sending each object to the constant functor (the functor having value for each object of and value for each arrow of ), and each arrow of to the the natural transformation which has the same value at each object of .

   Proposition. Let and be arbitrary categories. Then preserves all limits that exist in .

   Before the proof, recall that limits in functor categories are calculated pointwise. In some detail, if for an object we write for the ”evaluate at ” functor (with and ), then we have the following fact (Theorem V.3.1 on p. 115 of CWM):
   If is such that for each object of , has a limiting cone , then there exists a unique functor with object function such that with is a cone ; moreover, this is a limiting cone from to .

   (Note: As far as I can see in the proof in CWM, there are no limitations whatsoever on the size of and . 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 Lab entry on limits, is arbitrary but is small. Would anyone agree to explain the reason for the constraint on ? 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 be a functor with a limiting cone . We would like to show that is a limiting cone. Noting that (where the first is and the second is ), the last cone may be written as .

   First, we note that for each object of , is just , and therefore has the limiting cone by assumption. Hence, it is clear that has a limit, but we must verify that is a limiting cone.

   One functor with object function is just . For this functor, we have our cone . Since for all and we have , we are done by the theorem on pointwise limits.

   Is this wrong?

   Thank you very much in advance! Yaron

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 27th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexYaron -- 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!

   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 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 . Here, to get left adjointness, we actually do require that be small: is only small-complete, and so there will generally be no limit functor if is large (although there are particular where it’s okay, such as where has two objects , and all non-identity arrows forming a proper class which go from to ).

   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!

3. • CommentRowNumber3.
   • CommentAuthorYaron
   • CommentTimeNov 27th 2010
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   Author: Yaron
   Format: MarkdownItexThank you very much, Todd! I will use your link to do this right away. Yaron

   Thank you very much, Todd!

   I will use your link to do this right away.

   Yaron

4. • CommentRowNumber4.
   • CommentAuthorYaron
   • CommentTimeNov 27th 2010
   • PermaLink
   Author: Yaron
   Format: MarkdownItexOK, 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

   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

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeNov 28th 2010
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   Author: TobyBartels
   Format: MarkdownItexThanks, 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 28th 2010
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   Author: Mike Shulman
   Format: MarkdownItexWelcome, 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]].

   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 is the object . 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.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 28th 2010
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   Author: Todd_Trimble
   Format: MarkdownItex> 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorYaron
   • CommentTimeNov 28th 2010
   • PermaLink
   Author: Yaron
   Format: MarkdownItexThanks 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...

   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…

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeNov 28th 2010
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   Author: Tim_Porter
   Format: MarkdownItexTry the new address.

   Try the new address.

10. • CommentRowNumber10.
    • CommentAuthorYaron
    • CommentTimeNov 28th 2010
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    Author: Yaron
    Format: MarkdownItexThanks, Tim -- It seems to work :)

    Thanks, Tim – It seems to work :)