Due to some conversations that were happening on Twitter, I feel compelled to tell the story of how I got into my research and the role that my advisor played. Note that I went to UC Riverside for my B.S., which is also where I’m doing my PhD now.

Choosing an advisor

In undergrad, I focused heavily on topology. After getting deep into it, I realized I really liked algebraic topology and homotopy theory, and not so much differential topology and geometry. But when it came time to choose an advisor, I realized there were no homotopy theorists available.

My choices were to either accept some more geometry and differential stuff, or to work with John Baez. By this point, I knew that John had done higher category theory in the past, and was now interested in applied category theory. I was fully indoctrinated in the idea that I must do pure math though.

Beginning research, first paper

Luckily, his group met every Wednesday and it was open to anybody stopping by and hanging out. So I hung out for a bit to see what it was like. I ended up really liking what I was hearing, though I was super lost on terminology.

One day he told me that if I wanted to keep hanging out, I was going to have to start proving things. Later that session, he was saying “…and now we just need to prove this really is an operad. Oh hey, Joe, you wanna do that?”, so I took it. By the way, it was about certain ways of combining simple graphs. I didn’t know what an operad was, but I said I’d prove it.

So I went home, started reading, came back the next week with a proof. Then he said “Great! Now can you prove it for directed graphs? multigraphs? directed multigraphs? Loops or no?”. I thought that would be no problem, because the proof really didn’t use anything about being a simple graph, really just some basic combinatorial properties of graphs generally.

So I came back the next week with the construction of the operad generalized to anything with this list of properties I cooked up. After I explained all the properties and the way they relate, he said “Oh, that sounds like a lax symmetric monoidal functor…”

So I came back the next week with the properties formulated as a lax symmetric monoidal functor (FinBij, +) -> (Mon, x). We called this gizmo a network model. I told John “The construction of the operad involves this weird process I don’t know how to explain cleanly, where you bust open all the monoids and the elements become the objects of this symmetric monoidal category.” He said I should check out something called the “category of elements”.

So I came back the next week with the construction of the operad described by the Grothendieck construction. I told him there was something funny about this though because I have a symmetric monoidal category, but the Grothendieck construction doesn’t mention monoidal categories at all, and I couldn’t find anything about their relationship. So he told me to work it out for myself. So I did.

All this was part of a project John was working on with another student, Blake Pollard, and a company called Metron which had a contract with DARPA, specifically an employee named John Foley. So this work I was doing culminated in a paper whose authors are John, John, Blake, and myself.

John Baez, John Foley, Joe Moeller, Blake Pollard, Network Models, arxiv:1711.00037, 2017.

Second and third papers, sole author and new collaboration

John and I agreed that it would be strange if nobody had considered the relationship between the Grothendieck construction and monoidal structures before, so he told me to send an email to the categories mailing list. I got one response, it was from Christina Vasilakopoulou. She said she had come up with something similar in her own research (completely unrelated to network theory by the way). By a very nice coincidence, unknown to me, she was about to start working at UCR the very next quarter!

Around the same time, I had been hanging around Metron for a few weeks, thinking about how to come up with more examples of network models to bring different types of networks and restrictions into our framework. In particular, I wanted a type of network which allowed you to remember the order in which the edges were added to the network. This turned out to be pretty tricky to fit into the framework because of the necessary symmetries involved. Of course I’d bounce ideas off of both Johns, and the other Baez students, but I basically figured this one out on my own. I give a construction for a free network model. I would describe this paper as idiosyncratic. This recently became my first published paper!

Joe Moeller, Noncommutative Network Models, arxiv:1804.07402, 2018. To appear in Mathematical Structures in Computer Science.

While I was writing that paper, Christina started working at UCR, and hanging out with the Baez group. We started talking about the monoidal Grothendieck construction, she taught me a ton of 2-category theory. After I sent Noncommutative Network Models out to a journal, we started working on a paper together. It turns out Mike Shulman had worked out a monoidal version of the Grothendieck construction before.

Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738. TAC

But our construction looked a little different. This was a pretty fun period for me, because every time we would meet, we would have a bunch of ideas about things that correspond by our constructions, or else examples where it would naturally show up in algebra and network theory. Eventually, we had two different monoidal versions of the GC, and conditions for when they were equivalent, which realizes Shulman’s version as a special case.

Joe Moeller, Christina Vasilakopoulou, Monoidal Grothendieck Construction, arxiv:1809.00727, 2018.

I have another paper out after that (with John Baez), and three more irons in the fire at the moment (with John Baez again, with Jade Master for the first time, and another sole-author). I wouldn’t say those count as part of the story of how I got into my research though, so I’ll stop here.

Some of the gang at SYCO 4: (left to right) Jason Erbele, Daniel Cicala, Christina Vasilakopoulou, John Baez, Jeffrey Morton, Joe Moeller, and Christian Williams