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Stephan Alexander Spahn has created descent object, with some definitions from Street’s Categorical and combinatorial aspects of descent theory.
If I get the opportunity this weekend I’ll add details from Street’s Correction to ’Fibrations in bicategories’ and Lack’s Codescent objects and coherence. Anyone know of any other references?
Looking at Street’s paper again, what he describes as the ’$n=0$’ case of codescent objects looks to be just the notion of a coequalizer. I would have expected reflexive coequalizers, though, because the higher-$n$ case uses $n+2$-truncated simplicial objects. Is there a reason for this?
I think a page on descent objects should emphasize on the 2 and higher dimensional case; Street’s starting with 1-dimension is a bit idiosyncratic. By all means we should emphasize that it’s a generalization of coequalizers, but we shouldn’t give people the impression that “descent object” is just a fancy word for “coequalizer”. Perhaps this is what you were saying too.
I don’t know what Street would say if you asked him the question about reflexive coequalizers, but one answer I can think of is that 2-dimensional descent objects are often/usually defined using two levels of face maps, but only one level of degeneracies. Going down a level, you’d have one level of face maps but no degeneracies at all. I think the reason for this is that for defining the limit notion, at least, the top level of degeneracy is unnecessary. A coequalizer of a reflexive pair is the same as a coequalizer of the same pair with the common splitting ignored. Same for whether or not you include the second level of degeneracies in a descent object. But you can’t (in general) leave out the first level of degeneracies in a descent object, or you get a different kind of limit (I think).
Notice that Street’s definition of descent is lacking a condition to ensure that it gives the right answer. This is discussed at Verity on descent for strict omega-groupoid valued presheaves.
I have added to descent object an Idea-section, and a section on descent objects for ordinary presheaves (aka matching families).
There is much more to be said about examples, but much of it is already said at descent.
I think I’ll create a floating TOC for descent now, in order to provide a better overview of the topic.
Is it definitely true that
for defining the limit notion, at least, the top level of degeneracy is unnecessary
where (co)descent objects are concerned? I ask because I think I have a characterization of ’2-final’ 2-functors $F \colon I \to J$ as those for which the lax slice $j // F$ is non-empty, connected and locally connected for each j (where in a zigzag of triangles witnessing connectedness the backwards ones must contain an invertible 2-cell). It follows that if I and J are 1-categories then F is 2-final if and only if each ordinary slice $j/F$ is non-empty and codiscrete, i.e. a connected preorder.
If that’s right, then (fully weak) coequalizers are not the same thing as reflexive ones, because, taking F to be the inclusion of the free parallel pair into the free reflexive pair (a subcategory of $\Delta$), the morphisms $\delta_0 \sigma$ and $\delta_1 \sigma$ are idempotent and so are non-trivial endomorphisms of themselves in $[1]/F$. That was surprising, but not shocking.
But the same thing would seem to happen for the inclusion into $\Delta_{\leq 2}$ of itself without the top level of degeneracies, so it appears that colimits over the two categories need not coincide. Have I missed something, or does this show that my characterization is wrong?
Hmm… given the characterization of homotopy final functors as those whose slices $j/F$ all have (weakly) contractible nerves, I would expect a 2-final functor to be one where all the slices $j/F$ have nerves with trivial $\pi_0$ and $\pi_1$.
There are definitely a couple of mistakes in what I wrote – for a start, $\sigma$ doesn’t exist in the free parallel pair, so can’t be part of any morphism in the slice categories. Also, we’re talking about colimits over the opposites of these categories, so we want the slices $F/j$. But even with these errors corrected there are still unequal parallel morphisms in $F/[1]$, and the same problem as before arises for codescent objects, so there’s still something wrong.
The actual condition for a general 2-functor F to be final is that each $colim J(j,F)$ should be equivalent to the terminal category, i.e. non-empty and codiscrete. But I’ve clearly made some kind of mistake in translating this through the prescription for 2-colimits that I put at 2-limit recently. More thinking required…
Does $colim J(j,F)$ means a 2-categorical (pseudo) colimit of the diagram of hom-categories? That still doesn’t seem right to me; maybe if it were the lax colimit (whose nerve represents the homotopy colimit, I think).
Yes, I mean $\colim_i J(j,F i)$.
I’m following Kelly’s book, section 4.5. Suppose you have $F \colon I \to J$ as before and weights $W \colon J \to Cat$ and $V \colon I \to Cat$. Then you can follow Kelly’s arguments (replacing isomorphisms in V with equivalences in Cat) to show that $W \simeq Lan_F V$ if and only if $\{W, D\} \simeq \{V, D F\}$ for any functor D (where the Kan extension is defined pointwise as usual: $(Lan_F V) j = J(F-,j) \star V$). Setting both weights equal to the constant functor $\Delta \mathbf{1}$ at the terminal category you get the condition for F to be inital – that the identity transformation should exhibit $\Delta\mathbf{1} \colon J \to Cat$ as $Lan_F \Delta \mathbf{1}$ (where that $\Delta\mathbf{1} \colon I \to Cat$, of course). Then F is final if $F^{op}$ is initial, which is where the earlier condition comes from.
I’m pretty sure this all goes through OK in the weak/bicategorical setting. Unless I’ve missed something obvious here (which is not at all impossible), then the problem is with the translation of the condition on $\colim_i J(j,F i)$ into one on the bicategory of elements $\int J(j,F) = j//F$. Any thoughts?
I need to think about this some more, but just as a note: you can write \sslash
to get $\sslash$ (although apparently some people can’t view that character).
Argh, I had forgotten that descent objects are not conical limits – according to Street’s Correction to Fibrations in bicategories they are weighted by the cosimplicial object in Cat whose objects are the terminal category, the free isomorphism and the free composable pair of isomorphisms. The finality/Kan-extension condition in that case reduces to checking that any descent datum automatically satisfies the extra conditions wrt the coface maps, which I’m pretty sure is true.
With that in mind, the only slightly puzzling thing is that, in this fully weak setting at least, the coequalizer of a pair that has a common pseudo-section apparently need not be the colimit of the diagram with the section included. But I don’t think the condition I initially gave is quite equivalent to the one that says $colim J(j,F) \simeq \mathbf{1}$, so that could still be wrong.
Thanks for your responses so far. I’ll keep thinking.
Descent objects are not conical strict 2-limits, but unless I’m mistaken they are conical as bicategorical limits, which I thought was the setting you were working in. Specifically, the free isomorphism and the free composable pair of isomorphisms are both equivalent to the terminal category, so bicategorical limits weighted by that weight should be equivalent to conical ones.
Yes, I see. I hadn’t noticed that.
We don’t actually have a description/construction of a descent object in a 2-category at descent object, even though at 2-limit#2limits_over_diagrams_of_special_shape it points there in lieu of given a description, in the point about equalisers.
Well, someone should fix that. (-:
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