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1. • CommentRowNumber1.
   • CommentAuthorYaron
   • CommentTimeNov 27th 2010
   • PermaLink
   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 $n$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

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 27th 2010
   • PermaLink
   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 $J$ 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!

3. • CommentRowNumber3.
   • CommentAuthorYaron
   • CommentTimeNov 27th 2010
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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 $C/x \to C$ 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.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 28th 2010
   • PermaLink
   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 :)