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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMoving to here some very old discussions from [[preorder]]: _[[Todd Trimble|Todd]] says:_ It's not clear to me how one avoids the axiom of choice. For example, any equivalence relation $E$ on a set $X$ defines a preorder whose posetal reflection is the quotient $X \to X/E$, and it seems to me you need to split that quotient to get the equivalence between the preorder and the poset. _[[Toby Bartels|Toby]] says:_ In the absence of the axiom of choice, the correct definition of an equivalence of categories $C$ and $D$ is a span $C \leftarrow X \rightarrow D$ of full, faithful, essentially surjective functors. Or equivalently, a pair $C \leftrightarrow D$ of [[anafunctor]]s (with the usual natural transformations making them inverses). _[[Todd Trimble|Todd]] says:_ Thanks, Toby. So if I understand you aright, the notion of equivalence you have in mind here is not the one used at the top of the entry [[equivalence]], but is more subtle. May I suggest amplifying a little on the above, to point readers to the intended definition, since this point could be confusing to those inexperienced in these matters? _[[Urs Schreiber|Urs]] says_: as indicated at [[anafunctor]] an equivalence in terms of anafunctors can be understood as a span representing an [[isomorphism]] in the [[homotopy category]] of $Cat$ induced by the [[folk model structure]] on $Cat$. _Toby says_: I think that this should go on [[equivalence]], so I\'ll make sure that it\'s there. People that don\'t know what 'equivalence' means without choice should look there. _Mike_: Wait a minute; I see why every preorder is equivalent to a poset without choice, but I don't see how to show that every preorder has a [[skeleton]] without choice. So unless I'm missing something, the statement that every preorder is equivalent to a poset isn't, in the absence of choice, a special case of categories having skeletons. _Toby_: Given the definition there that a skeleton must be a subcategory (not merely any equivalent skeletal category), that depends on what [[subcategory]] means, doesn\'t it? If a subcategory can be any category equipped with a [[pseudomonic functor]] and if functor means [[anafunctor]] in choice-free category theory, then it is still true. On the other hand, since we decided not to formally define 'subcategory', we really shouldn\'t use it to define 'skeleton' (or anything else), in which case ‹equivalent skeletal category› is the guaranteed non-evil option. You *still* need choice to define a skeleton of an arbitrary category, but not of a proset. _Mike_: We decided not to formally define a non-evil version of "subcategory," but [[subcategory]] currently is defined to mean the evil version. However, I see that you edited [[skeleton]] to allow any equivalent skeletal category, and I can't really argue that that is a more reasonable definition in the absence of choice.

   Moving to here some very old discussions from preorder:

   Todd says: It’s not clear to me how one avoids the axiom of choice. For example, any equivalence relation $E$ on a set $X$ defines a preorder whose posetal reflection is the quotient $X \to X/E$, and it seems to me you need to split that quotient to get the equivalence between the preorder and the poset.

   Toby says: In the absence of the axiom of choice, the correct definition of an equivalence of categories $C$ and $D$ is a span $C \leftarrow X \rightarrow D$ of full, faithful, essentially surjective functors. Or equivalently, a pair $C \leftrightarrow D$ of anafunctors (with the usual natural transformations making them inverses).

   Todd says: Thanks, Toby. So if I understand you aright, the notion of equivalence you have in mind here is not the one used at the top of the entry equivalence, but is more subtle. May I suggest amplifying a little on the above, to point readers to the intended definition, since this point could be confusing to those inexperienced in these matters?

   Urs says: as indicated at anafunctor an equivalence in terms of anafunctors can be understood as a span representing an isomorphism in the homotopy category of $Cat$ induced by the folk model structure on $Cat$.

   Toby says: I think that this should go on equivalence, so I'll make sure that it's there. People that don't know what ’equivalence’ means without choice should look there.

   Mike: Wait a minute; I see why every preorder is equivalent to a poset without choice, but I don’t see how to show that every preorder has a skeleton without choice. So unless I’m missing something, the statement that every preorder is equivalent to a poset isn’t, in the absence of choice, a special case of categories having skeletons.

   Toby: Given the definition there that a skeleton must be a subcategory (not merely any equivalent skeletal category), that depends on what subcategory means, doesn't it? If a subcategory can be any category equipped with a pseudomonic functor and if functor means anafunctor in choice-free category theory, then it is still true. On the other hand, since we decided not to formally define ’subcategory’, we really shouldn't use it to define ’skeleton’ (or anything else), in which case ‹equivalent skeletal category› is the guaranteed non-evil option. You still need choice to define a skeleton of an arbitrary category, but not of a proset.

   Mike: We decided not to formally define a non-evil version of “subcategory,” but subcategory currently is defined to mean the evil version. However, I see that you edited skeleton to allow any equivalent skeletal category, and I can’t really argue that that is a more reasonable definition in the absence of choice.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex_[[Eric Forgy|Eric]] says_: It seems to me that any category $C$ gives rise to a preorder in a natural way, i.e. $$a\to b\in Mor(C)\implies a\le b.$$ Is there a slick arrow theoretic way to say that? Something like, "There is a [[forgetful functor]] $F:Cat\to Ord$"? I hope to write/read some arrow theoretic definition of a [[Hasse diagram]] too. _Mathieu says_: It's rather the other way round: there is a “forgetful” 2-functor $Ord \to Cat$ (which forgets that the category is in fact an order (it's a full inclusion, so it forgets a property rather than a structure)), and the left adjoint to this 2-functor maps a category $C$ to the order with the same objects as $C$ and $a\leq b$ iff there exists an arrow $a\to b$. _[[John Baez|John]] says_: You might try to define a Hasse diagram to be a [[directed graph]] whose [[quiver]] is a [[partial order|poset]]. In other words, you take a directed graph, consider the free category on that graph, and demand that this category be a poset. However, maybe it's a rule that a [[Hasse diagram]] with directed edges $x \to y$ and $y \to z$ is not allowed to have a directed edge $x \to z$. If so, this is a further restriction on what directed graphs count as Hasse diagrams. _Toby_: Eric, there *is* a functor $\Cat \to \Ord$ as you describe. If you call it a forgetful functor, then a category is a proset with extra stuff (not just structure), since this functor is not faithful (although it is essentially surjective). In contrast, a proset is a category with extra property (as you can literally see from John\'s slick definition above), since the functor $\Ord \to \Cat$ is fully faithful (as Matthieu points out). >(Recall that [[essentially surjective functor]]s, [[full functor]]s, and [[faithful functor]]s form the three anti-homogenous parts of the yoga of stuff, structure, and property, where in general you can call *any* functor a 'forgetful functor' and see what that gets you. I am treating only the $1$-category of categories, since otherwise this discussion would be at [[partial order]] by rights.) In fact, as Mike points out below, these two functors form an adjunction that makes $\Ord$ into a [[reflective subcategory]] of $\Cat$. For a Hasse diagram, I like John\'s definition. It is true that a Hasse diagram shouldn\'t have an edge $x \to z$ if there are edges $x \to y \to z$, but this follows; it\'s not a further restriction. (Similarly, there can\'t be any loops, even though directed graphs in general should allow them.) _Mike_: I don't think the preorder reflection $Cat\to Ord$ is full. Let $C$ be the walking commutative square (aka $\mathbf{2}\times\mathbf{2}$, where $\mathbf{2}$ is the walking arrow), and let $D$ be the walking non-commutative square (the free category on the directed graph that looks like a square with no diagonals). Then $C$ and $D$ have isomorphic preorder reflections, but I don't believe there is a functor $C\to D$ mapping to this isomorphism. _Toby_: Yeah, you\'re right; I made a level slip while thinking about this. I fixed my comment above.

   Eric says: It seems to me that any category $C$ gives rise to a preorder in a natural way, i.e.

   $$a\to b\in Mor(C)\implies a\le b.$$

   Is there a slick arrow theoretic way to say that? Something like, “There is a forgetful functor $F:Cat\to Ord$”?

   I hope to write/read some arrow theoretic definition of a Hasse diagram too.

   Mathieu says: It’s rather the other way round: there is a “forgetful” 2-functor $Ord \to Cat$ (which forgets that the category is in fact an order (it’s a full inclusion, so it forgets a property rather than a structure)), and the left adjoint to this 2-functor maps a category $C$ to the order with the same objects as $C$ and $a\leq b$ iff there exists an arrow $a\to b$.

   John says: You might try to define a Hasse diagram to be a directed graph whose quiver is a poset. In other words, you take a directed graph, consider the free category on that graph, and demand that this category be a poset.

   However, maybe it’s a rule that a Hasse diagram with directed edges $x \to y$ and $y \to z$ is not allowed to have a directed edge $x \to z$. If so, this is a further restriction on what directed graphs count as Hasse diagrams.

   Toby:

   Eric, there is a functor $\Cat \to \Ord$ as you describe. If you call it a forgetful functor, then a category is a proset with extra stuff (not just structure), since this functor is not faithful (although it is essentially surjective). In contrast, a proset is a category with extra property (as you can literally see from John's slick definition above), since the functor $\Ord \to \Cat$ is fully faithful (as Matthieu points out).

   (Recall that essentially surjective functors, full functors, and faithful functors form the three anti-homogenous parts of the yoga of stuff, structure, and property, where in general you can call any functor a ’forgetful functor’ and see what that gets you. I am treating only the $1$-category of categories, since otherwise this discussion would be at partial order by rights.)

   In fact, as Mike points out below, these two functors form an adjunction that makes $\Ord$ into a reflective subcategory of $\Cat$.

   For a Hasse diagram, I like John's definition. It is true that a Hasse diagram shouldn't have an edge $x \to z$ if there are edges $x \to y \to z$, but this follows; it's not a further restriction. (Similarly, there can't be any loops, even though directed graphs in general should allow them.)

   Mike: I don’t think the preorder reflection $Cat\to Ord$ is full. Let $C$ be the walking commutative square (aka $\mathbf{2}\times\mathbf{2}$, where $\mathbf{2}$ is the walking arrow), and let $D$ be the walking non-commutative square (the free category on the directed graph that looks like a square with no diagonals). Then $C$ and $D$ have isomorphic preorder reflections, but I don’t believe there is a functor $C\to D$ mapping to this isomorphism.

   Toby: Yeah, you're right; I made a level slip while thinking about this. I fixed my comment above.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeApr 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex[[Tim Porter]]: I see a potential problem with this terminology. Proset has an already accepted meaning as a pro-object in Sets and that terminology dates back to early days of Grothendieck on Descent, so is well established. There are several situations in the various pages and entries that will lead to pro-objects (i.e. projective systems in ...) so feel it might be better to avoid the term proset here. _Toby_: Does one write that 'proset', 'pro-set', or both? [[Tim Porter|Tim]]: It depends on how lazy one is feeling! I have used both. Pro-set is probably more correct, but the other does occur. It is perhaps unfortunate that poset is so well established otherwise 'porset' might be used with 'torset' etc as a consequence, but I wander!!!! _Mike_: Good point, Tim! Can't we just use "preorder" to mean "set equipped with a preorder"? That usage seems to be fairly common. _Toby_: Well, you can if you want, but I don\'t. Seriously, 'proset' is not something that I pulled out of thin air, and in fact I\'ve seen it more often than 'toset' (although that probably doesn\'t mean much given my interests). Google `+proset preorder` to keep out the false positives and see its occasional independent usage for yourselves. It\'s as good a term as 'poset'. But don\'t let this matter be conflated with the weighty question of whether things should be kept at separate pages! The distinction between [[linear order]] and [[loset]] is probably the best example of how they would look if kept separate, while [[apartness relation]] is probably the closest example of something where there is even now only one page. (By 'things' here, I mean [[poset]]/[[partial order]] and [[toset]]/[[total order]] in addition to ones already considered. Since a [[quasiorder]] cannot define [[equality]], I don\'t think that [[quoset]] is necessary, and of course a [[woset]] is the same as (one kind of) [[ordinal number]], so it may not come up either; anyway, we haven\'t even written [[well-order]] yet.) [[Tim Porter|Tim]]: On [[proset]] I suggest that we use the word 'synonym'! Something like, 'proset' is often used in the literature as a synonym for 'preordered set'. This meaning should not be confused with that of a projective system in the category of sets, that is a pro-object in Sets. I am not that worried about this, but do feel that we are hitting 'Disambiguation' problems at various points and may need a discussion on how we should handle those technically see the Wikipedia page on Disambiguation, which you all probably know well. _Toby_: [[simplicial category]] is pretty much a disambiguation page, although with some content too. There a couple of others like it. [[Tim Porter|Tim]]: I had noticed that as a good example of how one might proceed. We don't necessarily need a single way of doing this Disambiguation, but rather one or two examples of what we collectively think of as 'good practice'. Perhaps also I should create an entry on pro-objects which would link with profinite groups. (More work!) Your other point about separate pages needs thought. My view would be that we need to think both of the main users (i.e. us an co!) and external users since as we make reference to the nLab more and more people are likely to use it as one entry point for definitions etc. [[Mike Shulman|Mike]]: Toby, I wasn't saying you pulled "proset" out of thin air; I've never seen it in the literature but clearly I read different literature than you do. (-: My point was rather about what terminology we should use here on the nLab. I think having "proset" be a disambiguation page is a good idea, but since there is potential ambiguity I would prefer that we avoid its use elsewhere on the nLab. I'm all in favor of having lots of pages, but I'm not really in favor of duplicating _content_. I'd rather have [[toset]] just link to [[total order]], which in turn mentions that "toset" is a sometimes-used terminology for "totally ordered set." _Toby_: Mike, except for briefly summarising a definition, there is no content duplicated between [[linear order]] and [[loset]]. (Some of the other pairs haven\'t been cleaned up so well, but I will happily do that.) So my question is, is that a better way of organising things, or is [[poset]] (from which I have removed nothing to [[partial order]] yet) a better way? I am torn. If the original page names [[poset]] and [[preorder]] had been consistent (one way or the other), I\'d have just gone with it. (I have a better formed opinion that *if* we put everything on one page, then [[preorder]] is better and [[poset]] is worse.) If we have to make an except for [[proset]], then that\'s a different matter. (Although I think that we should be able to distinguish [[proset]] from [[pro-set]] if we want.) Tim, I would like to see [[pro-object]] (which I see you have already written!) and [[pro-set]], but I don\'t think that you have to write them just to convince me that 'proset' can be ambiguous. I\'ll take your word for it that these are useful concepts. (cont...)

   Tim Porter: I see a potential problem with this terminology. Proset has an already accepted meaning as a pro-object in Sets and that terminology dates back to early days of Grothendieck on Descent, so is well established. There are several situations in the various pages and entries that will lead to pro-objects (i.e. projective systems in …) so feel it might be better to avoid the term proset here.

   Toby: Does one write that ’proset’, ’pro-set’, or both?

   Tim: It depends on how lazy one is feeling! I have used both. Pro-set is probably more correct, but the other does occur. It is perhaps unfortunate that poset is so well established otherwise ’porset’ might be used with ’torset’ etc as a consequence, but I wander!!!!

   Mike: Good point, Tim! Can’t we just use “preorder” to mean “set equipped with a preorder”? That usage seems to be fairly common.

   Toby: Well, you can if you want, but I don't. Seriously, ’proset’ is not something that I pulled out of thin air, and in fact I've seen it more often than ’toset’ (although that probably doesn't mean much given my interests). Google +proset preorder to keep out the false positives and see its occasional independent usage for yourselves. It's as good a term as ’poset’.

   But don't let this matter be conflated with the weighty question of whether things should be kept at separate pages! The distinction between linear order and loset is probably the best example of how they would look if kept separate, while apartness relation is probably the closest example of something where there is even now only one page. (By ’things’ here, I mean poset/partial order and toset/total order in addition to ones already considered. Since a quasiorder cannot define equality, I don't think that quoset is necessary, and of course a woset is the same as (one kind of) ordinal number, so it may not come up either; anyway, we haven't even written well-order yet.)

   Tim: On proset I suggest that we use the word ’synonym’! Something like, ’proset’ is often used in the literature as a synonym for ’preordered set’. This meaning should not be confused with that of a projective system in the category of sets, that is a pro-object in Sets.

   I am not that worried about this, but do feel that we are hitting ’Disambiguation’ problems at various points and may need a discussion on how we should handle those technically see the Wikipedia page on Disambiguation, which you all probably know well.

   Toby: simplicial category is pretty much a disambiguation page, although with some content too. There a couple of others like it.

   Tim: I had noticed that as a good example of how one might proceed. We don’t necessarily need a single way of doing this Disambiguation, but rather one or two examples of what we collectively think of as ’good practice’.

   Perhaps also I should create an entry on pro-objects which would link with profinite groups. (More work!)

   Your other point about separate pages needs thought. My view would be that we need to think both of the main users (i.e. us an co!) and external users since as we make reference to the nLab more and more people are likely to use it as one entry point for definitions etc.

   Mike: Toby, I wasn’t saying you pulled “proset” out of thin air; I’ve never seen it in the literature but clearly I read different literature than you do. (-: My point was rather about what terminology we should use here on the nLab. I think having “proset” be a disambiguation page is a good idea, but since there is potential ambiguity I would prefer that we avoid its use elsewhere on the nLab.

   I’m all in favor of having lots of pages, but I’m not really in favor of duplicating content. I’d rather have toset just link to total order, which in turn mentions that “toset” is a sometimes-used terminology for “totally ordered set.”

   Toby: Mike, except for briefly summarising a definition, there is no content duplicated between linear order and loset. (Some of the other pairs haven't been cleaned up so well, but I will happily do that.) So my question is, is that a better way of organising things, or is poset (from which I have removed nothing to partial order yet) a better way?

   I am torn. If the original page names poset and preorder had been consistent (one way or the other), I'd have just gone with it. (I have a better formed opinion that if we put everything on one page, then preorder is better and poset is worse.) If we have to make an except for proset, then that's a different matter. (Although I think that we should be able to distinguish proset from pro-set if we want.)

   Tim, I would like to see pro-object (which I see you have already written!) and pro-set, but I don't think that you have to write them just to convince me that ’proset’ can be ambiguous. I'll take your word for it that these are useful concepts.

   (cont…)

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex(continued) [[Mike Shulman|Mike]]: No, there is no actual content duplicated between [[linear order]] and [[loset]], but it seems completely arbitrary to me which content is on which of the two pages. Shouldn't the examples be on the same page that gives the definition and other discussion? I actually don't think that [[poset]] and [[preorder]] are inconsistent, since as I said, "preorder" can mean the same thing as "set equipped with a preorder." This is actually fairly common, especially in set theory one speaks of "a linear order" or "a well-ordering" without specifying explicitly the set on which it lives (since, of course, it has to live on some set). It's a similar sort of abuse to when we say "let $G$ be a group" rather than "let $(G,\cdot,e)$ be a group" (only maybe in reverse). _Toby_: Ha, it\'s funny that you say that 'let $G$ be a group' is in reverse, since I think of that as perfectly correct and later discussion of $G$ as a mere set to be the abuse. But anyway ... The difference between [[loset]] and [[linear order]] is that the latter is about all of the linear orders that can be put on a set, while the former talks about only one set at time (at least one set per linear order, there should also be stuff about morphisms). As the set of linear orders on $S$ forms a subcategory of the category of all linearly ordered sets, there is a certain amount of arbitrariness there, but I think no more than in other fine distinctions. Do you really not see how having both [[preorder]] and [[poset]] was inconsistent? Yes, you can speak of a 'preorder' or a 'linear order' or a 'well ordering' without specifying the set ahead of time, but the same thing is true of a 'partial order'! All in all, I see your arguments as suggesting that we should put everything on the same page, and that page should be [[preorder]]/[[partial order]]/etc rather than [[proset]]/[[poset]]/etc, but you make an arbitrary exception for [[poset]] (and maybe [[toset]], that\'s not clear). _Mike_: Okay, there's a fine distinction between what's on [[loset]] and what's on [[linear order]]. But I don't think its a fine distinction worth perpetuating; I have trouble imagining someone wanting to read what's on one of those pages but not what's on the other one. Is there a specific reason not to put all the content on one page and have the other one be a redirect? I admit that having both preorder and poset is not completely uniform terminology, but I draw a fine distinction between nonuniformity and inconsistency. (-:O However, if you really want uniform terminology, I would be okay with putting everything on [[preorder]], [[partial order]], [[total order]], etc. with [[proset]], [[poset]], [[toset]], etc. as redirects. _Toby_: The way I envision someone reading [[linear order]] but not [[loset]] (the converse is unlikely) is if they\'re reading about some subject on the wiki and someone remarks that such and such is a linear order. They don\'t know what that is, or just want to review the definition, so they read [[linear order]], read the definition, learn that classically they can think of it as the same as a total order, etc. Obviously the definition of [[loset]] is there (with a link if they want more), but they don\'t need to see examples or learn about morphisms between them, since they\'ve already got an example on the referring page, and that\'s the only that\'s relevant. I can see the other side too, however. Just put it all on one page and let people read whichever sections they want. Given the links and pages on the wiki when I decided to write all of this, it was most natural to write things separately, but if you\'re convinced that it\'s better to have them combined, then I can accept that too. _Mike_: I do think it's better to have them combined. It might be nice if we could get a third opinion, but apparently no one else is paying much attention to our petty disputes. (-: [[Tim Porter|Tim]] I could not ignore that last comment so here is my pennyworth! Too many unnecessary links may be a bad thing. Why not have something of the form 'Most common terminology' then small list of synonyms after the definition i.e. near the top. If someone then want to use loset rather than linear order they can then use linear order|loset88 (where 8 stands for a square bracket either right or left. Clicking on loset will bring up the definition and the word loset will be bold (or whatever) and they can see it more or less immediately. There can also be warnings such as I felt were needed about proset i.e. 'proset' can also refer to a pro-object in the category of sets. For more on that meaning click ****** . _Mike_: Yes, I like that. I think it would be even better if we could do like Wikipedia does and have loset automatically redirect to linear order, but instiki doesn't appear to have that capability. _Toby_: Well, I guess that I\'ll move things around then. We can still discuss whether [[proset]] should simply redirect to [[preorder]] or should be a more explicit disambiguation page. (For now, I\'ll move things so that it redirects, but I\'ll do that last, so feel free to make more comments in the meantime.)

   (continued)

   Mike: No, there is no actual content duplicated between linear order and loset, but it seems completely arbitrary to me which content is on which of the two pages. Shouldn’t the examples be on the same page that gives the definition and other discussion?

   I actually don’t think that poset and preorder are inconsistent, since as I said, “preorder” can mean the same thing as “set equipped with a preorder.” This is actually fairly common, especially in set theory one speaks of “a linear order” or “a well-ordering” without specifying explicitly the set on which it lives (since, of course, it has to live on some set). It’s a similar sort of abuse to when we say “let $G$ be a group” rather than “let $(G,\cdot,e)$ be a group” (only maybe in reverse).

   Toby: Ha, it's funny that you say that ’let $G$ be a group’ is in reverse, since I think of that as perfectly correct and later discussion of $G$ as a mere set to be the abuse. But anyway …

   The difference between loset and linear order is that the latter is about all of the linear orders that can be put on a set, while the former talks about only one set at time (at least one set per linear order, there should also be stuff about morphisms). As the set of linear orders on $S$ forms a subcategory of the category of all linearly ordered sets, there is a certain amount of arbitrariness there, but I think no more than in other fine distinctions.

   Do you really not see how having both preorder and poset was inconsistent? Yes, you can speak of a ’preorder’ or a ’linear order’ or a ’well ordering’ without specifying the set ahead of time, but the same thing is true of a ’partial order’! All in all, I see your arguments as suggesting that we should put everything on the same page, and that page should be preorder/partial order/etc rather than proset/poset/etc, but you make an arbitrary exception for poset (and maybe toset, that's not clear).

   Mike: Okay, there’s a fine distinction between what’s on loset and what’s on linear order. But I don’t think its a fine distinction worth perpetuating; I have trouble imagining someone wanting to read what’s on one of those pages but not what’s on the other one. Is there a specific reason not to put all the content on one page and have the other one be a redirect?

   I admit that having both preorder and poset is not completely uniform terminology, but I draw a fine distinction between nonuniformity and inconsistency. (-:O However, if you really want uniform terminology, I would be okay with putting everything on preorder, partial order, total order, etc. with proset, poset, toset, etc. as redirects.

   Toby: The way I envision someone reading linear order but not loset (the converse is unlikely) is if they're reading about some subject on the wiki and someone remarks that such and such is a linear order. They don't know what that is, or just want to review the definition, so they read linear order, read the definition, learn that classically they can think of it as the same as a total order, etc. Obviously the definition of loset is there (with a link if they want more), but they don't need to see examples or learn about morphisms between them, since they've already got an example on the referring page, and that's the only that's relevant.

   I can see the other side too, however. Just put it all on one page and let people read whichever sections they want. Given the links and pages on the wiki when I decided to write all of this, it was most natural to write things separately, but if you're convinced that it's better to have them combined, then I can accept that too.

   Mike: I do think it’s better to have them combined. It might be nice if we could get a third opinion, but apparently no one else is paying much attention to our petty disputes. (-:

   Tim I could not ignore that last comment so here is my pennyworth! Too many unnecessary links may be a bad thing. Why not have something of the form ’Most common terminology’ then small list of synonyms after the definition i.e. near the top. If someone then want to use loset rather than linear order they can then use linear order|loset88 (where 8 stands for a square bracket either right or left. Clicking on loset will bring up the definition and the word loset will be bold (or whatever) and they can see it more or less immediately. There can also be warnings such as I felt were needed about proset i.e. ’proset’ can also refer to a pro-object in the category of sets. For more on that meaning click ****** .

   Mike: Yes, I like that. I think it would be even better if we could do like Wikipedia does and have loset automatically redirect to linear order, but instiki doesn’t appear to have that capability.

   Toby: Well, I guess that I'll move things around then. We can still discuss whether proset should simply redirect to preorder or should be a more explicit disambiguation page. (For now, I'll move things so that it redirects, but I'll do that last, so feel free to make more comments in the meantime.)

5. • CommentRowNumber5.
   • CommentAuthorEric
   • CommentTimeApr 2nd 2012
   • PermaLink
   Author: Eric
   Format: MarkdownItexWhen you guys remove an old discussion from a page and move it here, would it be possible to leave a link from the page to the nForum so it isn't lost in history (which no one will ever look at)?

   When you guys remove an old discussion from a page and move it here, would it be possible to leave a link from the page to the nForum so it isn’t lost in history (which no one will ever look at)?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 2nd 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI support that when the discussion is worth preserving (i.e. in the same situations where we would *add* a link to a current discussion on the nForum). But in this case, all of the discussion had already had their conclusions incorporated into the pages in question, so there's no more reason to link to the forum from the nLab page than there would be to any other random discussion we have here about an nLab page.

   I support that when the discussion is worth preserving (i.e. in the same situations where we would add a link to a current discussion on the nForum). But in this case, all of the discussion had already had their conclusions incorporated into the pages in question, so there’s no more reason to link to the forum from the nLab page than there would be to any other random discussion we have here about an nLab page.

7. • CommentRowNumber7.
   • CommentAuthorEric
   • CommentTimeApr 3rd 2012
   • (edited Apr 3rd 2012)
   • PermaLink
   Author: Eric
   Format: MarkdownItexEvery WIkipedia page has a "Talk" page, e.g. [Talk:Category_thoery](http://en.wikipedia.org/wiki/Talk:Category_theory). The discussion there may seem random, but exists and contains information some people may find interesting/useful. I don't see why an nLab page can't contain a link to an nForum discussion even if that discussion seems random. That would also serve to collect any future random discussions of the page into one location. I personally find the discussions useful even if the tidy result is incorporated in the page. The above discussions are now effectively lost from the nLab (since no one will think to visit the history pages).

   Every WIkipedia page has a “Talk” page, e.g. Talk:Category_thoery. The discussion there may seem random, but exists and contains information some people may find interesting/useful. I don’t see why an nLab page can’t contain a link to an nForum discussion even if that discussion seems random. That would also serve to collect any future random discussions of the page into one location. I personally find the discussions useful even if the tidy result is incorporated in the page. The above discussions are now effectively lost from the nLab (since no one will think to visit the history pages).

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeApr 3rd 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI'm not against such links existing, but I would prefer that they not exist at random or have to be added by hand. I'd be in favor (and I think I've said this before) of a link at the top or bottom of every page, analogous to Wikipedia's Talk pages, taking one to a canonical Forum discussion page (or a list of such).

   I’m not against such links existing, but I would prefer that they not exist at random or have to be added by hand. I’d be in favor (and I think I’ve said this before) of a link at the top or bottom of every page, analogous to Wikipedia’s Talk pages, taking one to a canonical Forum discussion page (or a list of such).