1 September 2017:
Here is this week's progress, as far as I know:
1) I made some final changes to a paper with Brandon Coya and Franciscus Rebro, Props in network theory
, and we submitted it to Theory and Application of Categories
The hardest part was proving a result Jason Erbele wanted for his thesis: if we have a prop P presented by some generators and relations, and we add some extra generators, we get a new prop P' having P as a sub-prop.
There's clearly a morphism P → P'. The challenge is to show it's monic. This seems obvious, but I needed Todd Trimble to help me prove it, and eventually Charles Rezk explained why the proof is subtle: while this result is true for props, it's not true for all algebraic gadgets described by multi-sorted Lawvere theories.
Yes, sometimes adding an extra generator can make an algebraic gadget "collapse" and become smaller in some ways! This doesn't happen for the familiar gadgets that have just one sort of element, like groups or rings, so it seems bizarre, but Rezk came up with a cool example of what can go wrong when we have more sorts.
The story is tersely told in Appendix A.2 of our paper, but you can see the whole saga in this discussion on the n-Café:
* A puzzle on multi-sorted Lawvere theories
Moral: it's good to solve problems in public, on blogs.
More brains make things go faster, it's more fun, and people get to know you, which is essential for getting jobs. If you have questions you want help with, write a blog article and I'll post it.
2) Brendan Fong visited Blake Pollard at Princeton, where Blake is interning with Arquimedes Canedo at the engineering firm Siemens. Brendan gave a talk. Blake wrote a nice description of what happened, but the most important part for you grad students is:
Canedo wants to hire more interns in the future: ideally people who know some applied category theory and can do some scientific programming! This could be YOU.
The word "Princeton" looks really good on your CV - believe me. Blake writes:
> Brendan's visit and accompanying talk went really well. Afterwards (over beers), Arqui proposed that the Siemens Princeton office become the first corporate 'Applied Category Theory Hub.' He is also interested in coming to the Netherlands workshop next summer, or at least to contribute some new problems from the Siemens domain.
> It was great to get Brendan, Dimitris, and myself together in a room with Arqui to discuss the potential role that applied category theory could play in helping Siemens realize this vision of a kind of `universal knowledge base' for their engineering and manufacturing processes and systems.
> It is clear that their vision is still a ways out, but in addition to trying to formalize such a knowledge base, riding the current wave of interest in applied category theory might help the Siemens folk continue to secure both internal and government funding to realize such a vision. Apparently there is a long-time Siemens guy in Germany who is also interested in exploring potential applications of category theory.
> I think we've done a good job helping shift the perceived benefits of category theory from 'provides magic wand' to 'provides new, potentially unifying perspective.'
> Some good analogies were thrown around with respect to the question: What is category theory buying us?
> This is like asking: What has set theory done for us?
> Or if you headed back and showed calculus to the folks designing and building the Roman aqueducts, they'd most likely respond, 'So what? Check out those aqueducts.'
> Brendan's talk went really well. I convinced him to include his result translating the condition of controllability for linear time invariant systems to a diagrammatic condition and I think people liked that a lot. Feeling that you understand something more intuitively by looking at pictures is a big initial draw for category theory.
> Afterwards we all got to chat a bit about categories and where exactly they might be useful.
> I like the perspective that we're trying to find the analogue of the functional programming style in engineering.
> Arqui hasn't had any 'ah ha' moments yet, but he's a dreamer, he sees the potential, and now he understands better the work required to get to a more realistic vision of how categories can help corporate technology. I really hope we can get beyond 'category theory as a way to secure DARPA funds.'
> Arqui wants more interns in the future exploring this stuff. So tell the gang! Ideally they should know basic scientific programming for prototyping ideas. I'm trying to provide evidence that folks with my kind of background are the ones you want around, not just to dream about categories, but also to push forward more practical research projects which are only tangentially related (hence my current interlude with machine learning). I'm giving a talk tomorrow about my work so far on my internship as kind of an interview for a full-time position.
> I invited Jason to give a talk next week since he'll be in Philly. The Siemens gang is up for a weekly category theory seminar!
> Also Brendan and I revived the idea of a Banff Oaxaca workshop 'Sticking things together in Mexico.' I think the applications are due in late September.
I forgot about that workshop. I'd be happy to help a bit with applying, though I'm getting insanely busy.
Finally, some non-progress. Last week Lisa and I took a vacation in Ubud, the "cultural capital" of Bali. We listened to music:
hiked through rice paddies:
and much more
. We'd like to rent a house and spend a month there sometime.