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.

]]>It’s kind of strange to have separate pages for compactly generated (∞,1)-category and locally presentable (∞,1)-category, since they are really terminological variants of each other. There’s a grammatical difference in how Lurie uses them: for him, “[locally] presentable ∞-category” leaves the degree of accessibility unspecified, whereas “compactly-generated” sets the degree of accessibility at $\kappa = \omega$. When he wants to specify the degree of accessibility, he has a choice between “[locally] $\kappa$-presentable” and “$\kappa$-compactly-generated”; he opts for the latter. At the very least, these articles should be modified to better reflect this.

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