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1. • CommentRowNumber1.
   • CommentAuthorFinnLawler
   • CommentTimeJan 23rd 2012
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   Author: FinnLawler
   Format: MarkdownItex(This is the sort of problem that one would typically wander over to a colleague with, but since I have no category-theorist colleagues I've wandered over here instead. I apologise if it's too trivial or localized, but I've been banging my head against it for the better part of a fortnight and would really appreciate an outside perspective.) I've [been trying](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3396&Focus=28410#Comment_28410) to characterize those 2-functors $F \colon I \to J$ (everything is as weak as possible here) that are '2-final' in the sense that for any $D \colon J \to K$ the conical 2-colimit $\colim D$ is equivalent to $\colim F D$. The usual reasoning for enriched categories (see section 4.5 of Kelly's book) carries over to show that F is final in this sense if and only if $\colim_i J(j,F i) \simeq \mathbf{1}$ for each $j \in J$, where $\mathbf{1}$ is the terminal category. According to the prescription at [[2-limit]] for 2-colimits in Cat, $\colim_i J(j,F i)$ is obtained by * taking the bicategory of elements of $i \mapsto J(j,F i)$, that is, the lax slice category $j\sslash F$, * applying the 'local $\pi_0$' functor to get a category, and * [[localization|formally inverting]] the (equivalence classes of the) opcartesian morphisms, in this case the triangles in $j\sslash F$ that contain an invertible 2-cell. The upshot is that F is final if and only if for each j these categories are equivalence relations, aka simply-connected groupoids. That happens when each _pseudo_ slice is nonempty and has each object connected to any other by a _unique_ zigzag of triangles. The examples I'm interested in have F an ordinary 1-functor between 1-categories, and this is where the trouble starts. Let's take coequalizers: let $P = [0] \rightrightarrows [1]$ be the free parallel pair, $\Delta_{\leq [1]}$ be the free reflexive pair and F the inclusion. We want to show that F is initial, i.e. that $F^{op}$ is final, meaning that coequalizers are the same thing as reflexive ones, just as for ordinary colimits. It suffices to show that for any functor $D \colon \Delta_{\leq[1]} \to Cat$ (i.e. a coreflexive pair) there is a bijection between (pseudo)cones $\mathbf{1} \Rightarrow D$ (coreflexive equalizer diagrams) and cones $\mathbf{1} \Rightarrow D F$ (equalizer diagrams), which I am now almost certain is true. The problem comes when we look at the finality condition above: there are two objects in $F/[0]$ (recall that we are working with $F^{op}$), namely the identity $[0] \to [0]$ and $\sigma \colon [1] \to [0]$, and two non-trivial morphisms, namely $\delta_0, \delta_1 \colon 1_{[0]} \to \sigma$. These two morphisms are not equal, hence the slice category would appear not to be simply connected. The effect of the local-$\pi_0$ functor $\pi_*$ on a bicategory of the form $j \sslash F$ is to identify two triangles if there is a 2-cell between the morphisms involved, whose F-image forms a commutative diagram with the 2-cells in the triangle. So if there are no such 2-cells (such as when the 2-categories involved are 1-categories) it would appear that the functor $\pi_*$ is just the identity. The only solution that has occurred to me, which doesn't seem very plausible, to the problem above is that perhaps this fact about $\pi_*$ is not true: notice, for example, that although there is no 2-cell between $\delta_0$ and $\delta_1$, the (identity) 2-cell in the triangles $\sigma\delta_0 = 1$ and $\sigma\delta_1 = 1$ is the same, and this seems to be what enables us to construct a cone over a coreflexive pair from one over the pair without its common retraction. So perhaps this is the right way to construct $\pi_*(F/[0])$, by taking a morphism to be determined by a 2-cell in a triangle. So my question is this: what's going on here? Which is the right prescription for $\pi_*$ = local-$\pi_0$ of a (lax, pseudo or ordinary) slice category? If I was right the first time and this functor is trivial on 1-categories of this form, how do I reconcile the apparent initiality of the functor F above with the fact that the slice $F/[0]$ is not simply connected? (Thanks for taking the time just to read all this. You can probably tell that I've got a bit lost and rather confused...)

   (This is the sort of problem that one would typically wander over to a colleague with, but since I have no category-theorist colleagues I’ve wandered over here instead. I apologise if it’s too trivial or localized, but I’ve been banging my head against it for the better part of a fortnight and would really appreciate an outside perspective.)

   I’ve been trying to characterize those 2-functors $F \colon I \to J$ (everything is as weak as possible here) that are ’2-final’ in the sense that for any $D \colon J \to K$ the conical 2-colimit $\colim D$ is equivalent to $\colim F D$. The usual reasoning for enriched categories (see section 4.5 of Kelly’s book) carries over to show that F is final in this sense if and only if $\colim_i J(j,F i) \simeq \mathbf{1}$ for each $j \in J$, where $\mathbf{1}$ is the terminal category. According to the prescription at 2-limit for 2-colimits in Cat, $\colim_i J(j,F i)$ is obtained by

   • taking the bicategory of elements of $i \mapsto J(j,F i)$, that is, the lax slice category $j\sslash F$,

   • applying the ’local $\pi_0$’ functor to get a category, and

   • formally inverting the (equivalence classes of the) opcartesian morphisms, in this case the triangles in $j\sslash F$ that contain an invertible 2-cell.

   The upshot is that F is final if and only if for each j these categories are equivalence relations, aka simply-connected groupoids. That happens when each pseudo slice is nonempty and has each object connected to any other by a unique zigzag of triangles.

   The examples I’m interested in have F an ordinary 1-functor between 1-categories, and this is where the trouble starts. Let’s take coequalizers: let $P = [0] \rightrightarrows [1]$ be the free parallel pair, $\Delta_{\leq [1]}$ be the free reflexive pair and F the inclusion. We want to show that F is initial, i.e. that $F^{op}$ is final, meaning that coequalizers are the same thing as reflexive ones, just as for ordinary colimits. It suffices to show that for any functor $D \colon \Delta_{\leq[1]} \to Cat$ (i.e. a coreflexive pair) there is a bijection between (pseudo)cones $\mathbf{1} \Rightarrow D$ (coreflexive equalizer diagrams) and cones $\mathbf{1} \Rightarrow D F$ (equalizer diagrams), which I am now almost certain is true. The problem comes when we look at the finality condition above: there are two objects in $F/[0]$ (recall that we are working with $F^{op}$), namely the identity $[0] \to [0]$ and $\sigma \colon [1] \to [0]$, and two non-trivial morphisms, namely $\delta_0, \delta_1 \colon 1_{[0]} \to \sigma$. These two morphisms are not equal, hence the slice category would appear not to be simply connected.

   The effect of the local-$\pi_0$ functor $\pi_*$ on a bicategory of the form $j \sslash F$ is to identify two triangles if there is a 2-cell between the morphisms involved, whose F-image forms a commutative diagram with the 2-cells in the triangle. So if there are no such 2-cells (such as when the 2-categories involved are 1-categories) it would appear that the functor $\pi_*$ is just the identity. The only solution that has occurred to me, which doesn’t seem very plausible, to the problem above is that perhaps this fact about $\pi_*$ is not true: notice, for example, that although there is no 2-cell between $\delta_0$ and $\delta_1$, the (identity) 2-cell in the triangles $\sigma\delta_0 = 1$ and $\sigma\delta_1 = 1$ is the same, and this seems to be what enables us to construct a cone over a coreflexive pair from one over the pair without its common retraction. So perhaps this is the right way to construct $\pi_*(F/[0])$, by taking a morphism to be determined by a 2-cell in a triangle.

   So my question is this: what’s going on here? Which is the right prescription for $\pi_*$ = local-$\pi_0$ of a (lax, pseudo or ordinary) slice category? If I was right the first time and this functor is trivial on 1-categories of this form, how do I reconcile the apparent initiality of the functor F above with the fact that the slice $F/[0]$ is not simply connected?

   (Thanks for taking the time just to read all this. You can probably tell that I’ve got a bit lost and rather confused…)

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2012
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   Author: Mike Shulman
   Format: MarkdownItexCan you explain > The upshot is that F is final if and only if for each j these categories are equivalence relations, aka simply-connected groupoids. That happens when each *pseudo* slice is nonempty and has each object connected to any other by a unique zigzag of triangles. more? Which categories does "these categories" refer to? I guess it would be the result of all three steps, right? By "equivalence relations" do you mean "cliques" (an equivalence relation need not be connected)? And finally, how does the last sentence follow? What happened to the non-opcartesian arrows?

   Can you explain

   The upshot is that F is final if and only if for each j these categories are equivalence relations, aka simply-connected groupoids. That happens when each pseudo slice is nonempty and has each object connected to any other by a unique zigzag of triangles.

   more? Which categories does “these categories” refer to? I guess it would be the result of all three steps, right? By “equivalence relations” do you mean “cliques” (an equivalence relation need not be connected)? And finally, how does the last sentence follow? What happened to the non-opcartesian arrows?

3. • CommentRowNumber3.
   • CommentAuthorFinnLawler
   • CommentTimeJan 23rd 2012
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   Author: FinnLawler
   Format: MarkdownItexI apologise -- that paragraph is badly mixed up. What I should have said is that F is final if and only if each $\colim_i J(j,F i)$ (the result of all three steps, as you say) is codiscrete, that is, there is a unique morphism (necessarily an isomorphism) between any two objects. An isomorphism in the localization at a class S is, I think, a zigzag of morphisms that are either isomorphisms or S-morphisms, but here all the isomorphisms are already opcartesian = S-morphisms, so the correct condition is that $\pi_*$ of the pseudo slice $j / F$ is simply connected.

   I apologise – that paragraph is badly mixed up. What I should have said is that F is final if and only if each $\colim_i J(j,F i)$ (the result of all three steps, as you say) is codiscrete, that is, there is a unique morphism (necessarily an isomorphism) between any two objects. An isomorphism in the localization at a class S is, I think, a zigzag of morphisms that are either isomorphisms or S-morphisms, but here all the isomorphisms are already opcartesian = S-morphisms, so the correct condition is that $\pi_*$ of the pseudo slice $j / F$ is simply connected.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2012
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   Author: Mike Shulman
   Format: MarkdownItexIf you're right about isomorphisms in a localization, then it seems to me that the condition would be what you say, plus that every zigzag is equivalent to one consisting only of S-morphisms. Without some condition that refers to morphisms that aren't S-morphisms, how can you fully characterize the localization?

   If you’re right about isomorphisms in a localization, then it seems to me that the condition would be what you say, plus that every zigzag is equivalent to one consisting only of S-morphisms. Without some condition that refers to morphisms that aren’t S-morphisms, how can you fully characterize the localization?

5. • CommentRowNumber5.
   • CommentAuthorFinnLawler
   • CommentTimeJan 24th 2012
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   Author: FinnLawler
   Format: MarkdownItexYes, good point. My condition says that there is an isomorphism to which every other isomorphism is equal, but what I want to say is that there is an isomorphism to which every other morphism is equal, right? Whatever the right condition is, I suspect it will be complicated to use in practice. What I'm really interested in is whether (or when, if not always) a final ordinary functor is also 2-final. If F is an ordinary functor, then the lax and pseudo slices are the same as the ordinary slice, and $\pi_*$ would seem to be the identity on them, so as I said above the inclusion of the free parallel pair into the free coreflexive pair seems not to be initial; but working it out in terms of a bijection (or perhaps more correctly an equivalence of categories) between cones seems to indicate that it is. So I wonder if I'm wrong about $\pi_*$; it seems unlikely, I know, but maybe there's something I've missed. I think that to answer that question it would be enough to know whether, given an ordinary functor $P \colon C \to Set$, its strict colimit, given as $\pi_0(\int P)$, is equivalent to its 2-colimit (when P is considered as a diagram of discrete categories). The latter is the groupoid reflection of $\pi_*(\int P)$, so again the question hinges on whether $\pi_*$ is the identity on a locally discrete bicategory of elements, or ends up taking its preorder reflection by identifying triangles/morphisms that contain/overlie the same (identity) 2-cell.

   Yes, good point. My condition says that there is an isomorphism to which every other isomorphism is equal, but what I want to say is that there is an isomorphism to which every other morphism is equal, right?

   Whatever the right condition is, I suspect it will be complicated to use in practice. What I’m really interested in is whether (or when, if not always) a final ordinary functor is also 2-final. If F is an ordinary functor, then the lax and pseudo slices are the same as the ordinary slice, and $\pi_*$ would seem to be the identity on them, so as I said above the inclusion of the free parallel pair into the free coreflexive pair seems not to be initial; but working it out in terms of a bijection (or perhaps more correctly an equivalence of categories) between cones seems to indicate that it is. So I wonder if I’m wrong about $\pi_*$; it seems unlikely, I know, but maybe there’s something I’ve missed.

   I think that to answer that question it would be enough to know whether, given an ordinary functor $P \colon C \to Set$, its strict colimit, given as $\pi_0(\int P)$, is equivalent to its 2-colimit (when P is considered as a diagram of discrete categories). The latter is the groupoid reflection of $\pi_*(\int P)$, so again the question hinges on whether $\pi_*$ is the identity on a locally discrete bicategory of elements, or ends up taking its preorder reflection by identifying triangles/morphisms that contain/overlie the same (identity) 2-cell.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 24th 2012
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   Author: Mike Shulman
   Format: MarkdownItex> I think that to answer that question it would be enough to know whether, given an ordinary functor $P \colon C \to Set$, its strict colimit, given as $\pi_0(\int P)$, is equivalent to its 2-colimit (when P is considered as a diagram of discrete categories). That's certainly not true: let $C= B G$ be the delooping of a group and let $P$ be constant at a terminal set. Then its colimit is again terminal, but its 2-colimit is $B G$.

   I think that to answer that question it would be enough to know whether, given an ordinary functor $P \colon C \to Set$, its strict colimit, given as $\pi_0(\int P)$, is equivalent to its 2-colimit (when P is considered as a diagram of discrete categories).

   That’s certainly not true: let $C= B G$ be the delooping of a group and let $P$ be constant at a terminal set. Then its colimit is again terminal, but its 2-colimit is $B G$.

7. • CommentRowNumber7.
   • CommentAuthorFinnLawler
   • CommentTimeJan 24th 2012
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   Author: FinnLawler
   Format: MarkdownItexOK, that makes sense, thanks -- I wasn't holding out much hope, but it seems as though $\pi_*$ is indeed the identity in the cases I was considering. That means that a 1-functor F is 2-final if each of the slices j/F has a codiscrete groupoid reflection. Going all the way back to codescent objects, if it is indeed true that they can be computed over the subdiagram with the top level of face maps left out, then the inclusion $i \colon D \to \Delta_{\leq[2]}$ (D being the obvious subcategory) must be initial. In particular, there are maps in i/[1], given by $\delta_0$ and $\delta_1$ (or $\delta_1$ and $\delta_2$), from the identity on [1] to $\sigma_0 \colon [2] \to [1]$ (or $\sigma_1$), that must become equal in the groupoid reflection of i/[1]. But I can't see how. Do you know of a published proof of that fact about codescent objects, so that I could try to work backwards from it?

   OK, that makes sense, thanks – I wasn’t holding out much hope, but it seems as though $\pi_*$ is indeed the identity in the cases I was considering. That means that a 1-functor F is 2-final if each of the slices j/F has a codiscrete groupoid reflection.

   Going all the way back to codescent objects, if it is indeed true that they can be computed over the subdiagram with the top level of face maps left out, then the inclusion $i \colon D \to \Delta_{\leq[2]}$ (D being the obvious subcategory) must be initial. In particular, there are maps in i/[1], given by $\delta_0$ and $\delta_1$ (or $\delta_1$ and $\delta_2$), from the identity on [1] to $\sigma_0 \colon [2] \to [1]$ (or $\sigma_1$), that must become equal in the groupoid reflection of i/[1]. But I can’t see how. Do you know of a published proof of that fact about codescent objects, so that I could try to work backwards from it?

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 25th 2012
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   Author: Mike Shulman
   Format: MarkdownItex> That means that a 1-functor F is 2-final if each of the slices j/F has a codiscrete groupoid reflection. Okay, great! I think that means that I was also right in my guess [back here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3396&Focus=28416#Comment_28416), since $\pi_0$ and $\pi_1$ of the nerve of a category are the same as those of its groupoid reflection. > Do you know of a published proof of that fact about codescent objects? Try B3.4.11 in *Sketches of an Elephant*?

   That means that a 1-functor F is 2-final if each of the slices j/F has a codiscrete groupoid reflection.

   Okay, great! I think that means that I was also right in my guess back here, since $\pi_0$ and $\pi_1$ of the nerve of a category are the same as those of its groupoid reflection.

   Do you know of a published proof of that fact about codescent objects?

   Try B3.4.11 in Sketches of an Elephant?

9. • CommentRowNumber9.
   • CommentAuthorFinnLawler
   • CommentTimeJan 27th 2012
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   Author: FinnLawler
   Format: MarkdownItexRight, I think we have this figured out now. The Elephant (p. 15) says that a category has a codiscrete groupoid reflection iff it is simply connected, iff any functor out of it into a groupoid is essentially constant. In the reflexive-coequalizer example, the category i/[0] has two objects, the identity $1_{[0]}$ and $\sigma$, and two non-trivial morphisms, $\delta_0$ and $\delta_1$. You can send i/[0] to $B \mathbb{Z}_2$ by $\delta_i \mapsto i$, and this is constant iff 0 is conjugate to 1, which it isn't, so i/[0] is definitely not simply connected, so i is definitely not initial. I don't know exactly when two morphisms become equal in a groupoid reflection, but they do if there is a morphism that (co)equalizes them; in the codescent-object example, the morphisms $\delta^2_0, \delta^2_1 \colon [1] \to [2]$ are equalized by $\delta^1_0 \colon [0] \to [1]$, so they are equal in the groupoid reflection of i/[1], and I think that takes care of that. Thanks for your help!

   Right, I think we have this figured out now. The Elephant (p. 15) says that a category has a codiscrete groupoid reflection iff it is simply connected, iff any functor out of it into a groupoid is essentially constant. In the reflexive-coequalizer example, the category i/[0] has two objects, the identity $1_{[0]}$ and $\sigma$, and two non-trivial morphisms, $\delta_0$ and $\delta_1$. You can send i/[0] to $B \mathbb{Z}_2$ by $\delta_i \mapsto i$, and this is constant iff 0 is conjugate to 1, which it isn’t, so i/[0] is definitely not simply connected, so i is definitely not initial.

   I don’t know exactly when two morphisms become equal in a groupoid reflection, but they do if there is a morphism that (co)equalizes them; in the codescent-object example, the morphisms $\delta^2_0, \delta^2_1 \colon [1] \to [2]$ are equalized by $\delta^1_0 \colon [0] \to [1]$, so they are equal in the groupoid reflection of i/[1], and I think that takes care of that.

   Thanks for your help!

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 27th 2012
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    Author: Mike Shulman
    Format: MarkdownItexYou're welcome!

    You’re welcome!