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In a seminar I’m teaching, I wanted to share some of the relationships between some of the different sorts of objects that might be called “presentations of $\infty$-categories”. I ended up with this diagram. (Incidentally, I don’t know if “diagrams” is actually the correct category for this nforum discussion; if not, anyone should feel free to change it.)
But so, it occurred to me that this might be pedagogically useful to others. In particular, it might make sense to link on the nLab somewhere, though I don’t know where offhand. So I would appreciate any comments or suggestions on this front. (I’d be fine with hearing the opinion that it’s not worth putting on the nLab; I know it’s super rough, and I definitely wouldn’t be offended.)
Of course, it’d be better if this were a texed diagram instead of a picture, but I don’t have the time to do that right now. On the other hand, I think most aspects are more-or-less self-explanatory to someone who’s in the know, but definitely not everything. So I’ll at least make a few comments.
The lowercased objects such as $cat$, $relcat$, etc., are “strict” objects, i.e. their objects have sets of objects. For example, $cat$ is reflective inside of $sSet$, whereas $Cat \subset Cat_\infty$ is a full ($\infty$-)subcategory.
The “flag” hanging off to the left comes logically before the rest of the diagram. That is, the interpretation of the rest of the diagram is premised on the facts contained therein.
Things pretty much commute “as much as you would expect” (for a diagram involving a bunch of adjoints), with the following caveats.
The upper-left triangle (describing the two different ways of extracting an adjunction of $\infty$-categories from an enriched adjunction of simplicial model categories) isn’t (yet) known to commute.
The corresponding triangle for Quillen equivalences is known to commute. (I think? Actually I’m not sure how to prove this offhand…)
The pentagon with $modelcat^\delta_{sSet}$ at the top-left doesn’t commute.
However, the two ways of proceeding from $modelcat^\delta_{sSet}$ down to $Cat_\infty$ are naturally equivalent (although I guess “naturally equivalent” is kind of a weak assertion for a pair of functors off of a discrete category; maybe there’s something slightly better to say). This is a result of Dwyer–Kan.
This is all elaborated upon much more fully in the appendix to my “simplicial spaces” paper.
Nice! What about at (∞,1)-category?
Haha yes, I did at least take a quick look there. But it seems to me that this has a somewhat different bent, along the lines of the Barwick–Schommer-Pries unicity theorem. I googled those words plus “nLab” and came up empty, which was what led me to just ask here. I suspect there may not exist an nLab page explicitly dedicated to “model independence”, because the entire site is geared towards it already…?
Well, at (∞,1)-category we have
There are a number of different ways to make the idea of an (∞,1)-category precise… almost all the definitions of (∞,1)-category are known to form model categories that are Quillen equivalent.
so it seems to me that this (as well as a discussion of the unicity theorem) would fit perfectly there.
Regarding the “unicity theorem”: my colleagues back then complained about this terminology, arguing that it’s just another model with comparisons to other models.
@Mike and anyone else: Do you agree that it could make sense to have a separate page dedicated to “model independence”? I think that there seems to be a sort of yoga that those in the know are able to manipulate fluidly, but that people outside find somewhat confusing because the best they’ve ever seen is “an $\infty$-category is a quasicategory, and writing down anything else is basically equivalent to writing down a quasicategory”.
My own (certainly limited) view is that perhaps the best way to go about things is through the following steps (which are surely not original), which I think both make the minimal number of choices necessary (even “contractible” ones) and lead to a unified perspective on these various models. From a pragmatic point of view, I think the cleanest thing to do is to privilege quasicategories as an ambient framework, at least for now. And I’m not even going to pretend to worry about set-theoretic issues.
We know what a quasicategory is; these form a $Kan$-enriched category $\underline{QCat}$. Define a relative quasicategory to be a quasicategory equipped with a full sub-quasicategory of “weak equivalences” containing all equivalences. Note that using this definition, the natural map $\underline{hom}((C,W_C),(D,W_D)) \to \underline{hom}(C,D)$ (coming from the forgetful functor $\underline{RelQCat} \to \underline{QCat}$) is the inclusion of a disjoint union of connected components among Kan complexes (in the strictest possible sense). There’s an evident inclusion $min : \underline{QCat} \to \underline{RelQCat}$ of $Kan$-enriched categories.
Taking homotopy-coherent nerves, write $min : CAT_\infty \to RELCAT_\infty$ for the resulting map of quasicategories. This admits a quasicategorical left adjoint $L : RELCAT_\infty \to CAT_\infty$ (which presents “localization”), which is unique up to a contractible Kan complex worth of choices.
Note that any Quillen adjunction (including a Quillen equivalence) $C \rightleftarrows D$ determines a pair of relative functors $C^c \hookrightarrow C \to D$ and $C \leftarrow D \hookleftarrow D^f$. Using this, expand the Barwick–Schommer-Pries diagram into a diagram of relative functors between relative 1-categories. This defines a “diagram” in the quasicategory $RELCAT_\infty$, i.e. a morphism $F : K \to RELCAT_\infty$ of ssets.
Toen’s theorem implies that the composite $LF : K \to RELCAT_\infty \to CAT_\infty$ is “essentially contractible”. More precisely, for any for any cofibration into an acyclic object $i : K \to K' \approx pt$ in the Joyal model structure, there exists a contractible Kan complex of extensions of $LF$ over $i$.
Define $The \infty Cats \subset CAT_\infty$ to be the maximal sub-Kan complex generated by the image of $LF$. (Toen’s theorem says that after choosing a basepoint, this Kan complex is a model of $B(\mathbb{Z}/2)$.) I propose to write $Cat_\infty$ for any vertex of $The \infty Cats$, and I propose that to work “model independently” is to work within $Cat_\infty$.
So on the one hand we can reasonably say that any sort of object that might be considered as “a presentation of an $\infty$-category” (such as a relative category, or a Segal category, or whatever else) does indeed unambiguously define an object of $Cat_\infty$, but on the other hand everything is totally independent of which vertex of $The \infty Cats$ we choose. I prefer this to working in a specific model, because aside from having privileged quasicategories at the “outermost layer” from the start, this purposely refuses to prefer any model over any other.
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@Urs, I’m less familiar with the $(\infty,\geq 2)$ story than the $(\infty,1)$ story, so maybe let’s just stick to that if you don’t mind. (I’d expect the same objection would persist here?) I agree that in theory it’s possible to write down something else that you might like to call “a model of $(\infty,1)$-categories”. And there may well be a chicken-and-egg problem that the only way to solve is just come out and say once and for all what you want out of your theory. But are there any models that don’t satisfy the axioms that BSP give?
That’s a fine way to describe the situation, but I don’t see why it would need a separate page. It’s part of the story of (∞,1)-categories, so why not put it on the page (∞,1)-category?
My comment in #5 was a bit out of context, best to just ignore it. I was just reminded what happened to me back then after I had written the entry and used it in our group seminar. Some participants seemed to feel the terminology claimed more territory than was justified. But that’s really irrelevant, please just ignore it.
On your edits: I agree with Mike, just put the stuff into the existing entry, it will be a worthwhile addition!
Alright! I’ve updated that page. Further suggestions are of course welcome.
Does the nLab allow to upload images? This link to the diagram might not last forever…
Does the nLab allow to upload images?
You may upload any kind of file to the nLab, as long as it’s sufficiently small. Type into any entry symbols like
[[FileName.suf:file]]
Then save the page. This produces a grayish field “FileName” with a green question mark behind it. Clicking on that opens an upload dialogue. When done, the file sits at
http://ncatlab.org/nlab/files/FileName.suf
Then remove the
[[FileName.suf:file]]
from the source code and instead add the usual
<img src="http://ncatlab.org/nlab/files/FileName.suf" width="400">
where desired.
Alright, I’ve taken care of this. It seemed pretty unwieldy to actually embed the picture directly into the nLab page, so I just put a link to the (now more permanent) version. Thanks for the suggestion (Dmitri) and help (Urs)!
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