Looking at the entry infinity-cosmos, it could perhaps do with some more content, and it would be useful if someone could include more detail on what the theory gives one.

]]>Added cross link to infinity-cosmos under Related Entries.

]]>There is no claim here – we just added a kind note briefly indicating a colleague’s research.

Be our guest, hit “edit” and expand to your heart’s content! But it sounds like your comment would best fit into the Idea-section at *infinity-cosmos*.

Replaced “$\infty$-category theory” with “$(\infty, 1)$-category theory”; it does not seem appropriate to abbreviate here, as it will surely lead to confusion.

The theory is not just about $(\infty, 1)$-categories though. Rather, $\infty$-cosmoses are formed by models for $(\infty, n)$-categories for all $0 \le n \le \infty$, which is part of what motivates this program today. So claiming the theory is about foundations of $(\infty, 1)$-category theory specifically seems like a reduction to me. (Even though this is certainly an important specialization.) If one wants to avoid notational confusion, why not write e.g. “(weak) higher categories”? (To say more about the terminology initially used, in this specific context, “$\infty$-category” is just the name for the objects of a given $\infty$-cosmos, which could in the specific instances mean [certain models of] $(\infty, 1)$-categories, cartesian fibrations thereof, $(\infty, 2)$-categories, $\infty$-groupoids, or simply $1$-categories…)

]]>Replaced “$\infty$-category theory” with “$(\infty, 1)$-category theory”; it does not seem appropriate to abbreviate here

True, thanks.

one should prefer to avoid “omega-category”

But that’s how these authors, following the John Roberts that is mentioned in the sentence, tended to refer to their own subject.

]]>Replaced “$\infty$-category theory” with “$(\infty, 1)$-category theory”; it does not seem appropriate to abbreviate here, as it will surely lead to confusion. (More generally, I think one should prefer to avoid “omega-category”: I believe Tom Leinster had a pertinent comment on the n-Category Café on this topic, but I can’t find it now.)

]]>Thanks! I have replaced your “infinity category theory” by “omega-category theory”, but then slightly expanded the following sentence like so:

]]>More recently he has worked with Emily Riehl on foundations of $\infty$-category theory seen through their homotopy 2-category, and using the concept of ∞-cosmoi to capture common structure of different presentations of $\infty$-categories.

I have added a little summary of some of his work. It could include more but I have tried to link to the fuller entries rather than put a lot here.

]]>It seems sad to state nothing but employment status of a person, which is the least of interest to us here, unless you happen to be the one paying them. If you have the energy to edit, why not add a word on the person as a researcher.

]]>Removed broken website link. Added ’Emeritus’ status.

]]>Added a webpage link.

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