...Nice
Friends! I am returned from the 85th Peripatetic Seminar on Sheaves and Logic, which was held in balmy Nice. Well, the second day was held in balmy Nice - the first day was held in wet and slightly cold Nice. But we were inside talking about higher category theory, so that wasn't too much of a problem.
My talk, internally, felt like a disaster - the guy on immediately before me said a lot of the stuff I was going to say (although motivated by general 2-categorical concerns, rather than the ad-hoc "well, it kinda seems to work" approach I was taking), and this put me off-balance rather. He didn't state the theorem I was presenting, which still seems to be new, but there's a deeper way of understanding the definition I'd come up with ad-hoc. I could have taken advantage of this if I'd kept calmer, but I didn't. Gah. The doubly annoying thing is that I'd seen the 2-categorical stuff before and not understood it - maybe having re-invented that particular wheel will give me greater understanding in future. Anyway, I babbled, dropped things, forgot what I was saying, and generally didn't perform at my best. But people seemed to grasp the take-home message, so I guess it could have been worse. Some reviews:
"You need more stage presence. Put a bit more Shatner into your performance."
"You didn't make the usual graduate student mistake of preparing everything in far too much detail, going over the first two pages of notes slowly and rushing at the end. You've just got to have the confidence to let the words flow"
"Did you write everything out in detail? Because if not, you should consider doing that. Put everything on slides, so you don't get lost."
In fact, I did write everything out in detail, and it came to under two pages, but I kept getting lost on the page, largely due to nerves. My tiny handwriting doesn't help here. I think the thing to take from the last two comments (which are, as you can see, contradictory, and which came from equally good speakers) is "find out what works for you". What I'm doing now (blackboard + notes) doesn't, it seems.
BTW, michiexile, here's my current understanding of the relationship between operads and monads:
[Rows are classes of algebraic theories. Each row is included in the row below. For obvious reasons, this is presented from the perspective of universal algebra/category theory. Standard disclaimer applies.]
TheoriesSyntactic CharacterizationMonadsRepresenting ObjectsCan be interpreted inExamples
Strongly regular Equations have same variables in same order on each side, each appearing exactly once, eg (a.b).c = a.(b.c)Cartesian?Non-symmetric operadsMonoidal categories (more generally, multicategories)Monoids, semigroups, M-sets for some monoid M
Familially representable?Functor part is a coproduct of Hom(n,-)s; equivalently, functor part preserves connected limitsFamilies of finite sets?Involutive monoids
LinearSame variables on each side, each appearing exactly once, but not necessarily in the same orderAnalytic?Symmetric operadsSymmetric monoidal categories (more generally, symmetric multicategories)Commutative monoids
FinitaryOperations have finitely many argumentsFinitary, ie preserve filtered colimitsLawvere theories / finite product operads / clones / etc.Finite product categoriesGroups, rings, rigs, SKI combinator algebras, R-modules...
Non-finitary-General??Compact Hausdorff spaces
One of the questions I was asked was "How does this work in the symmetric case?", something I'd been intending to look at but didn't get round to. I've since looked at it, and it does seem to work - but interestingly, the 2-categorical justification vanishes! Now that would have been interesting to mention.
My talk, internally, felt like a disaster - the guy on immediately before me said a lot of the stuff I was going to say (although motivated by general 2-categorical concerns, rather than the ad-hoc "well, it kinda seems to work" approach I was taking), and this put me off-balance rather. He didn't state the theorem I was presenting, which still seems to be new, but there's a deeper way of understanding the definition I'd come up with ad-hoc. I could have taken advantage of this if I'd kept calmer, but I didn't. Gah. The doubly annoying thing is that I'd seen the 2-categorical stuff before and not understood it - maybe having re-invented that particular wheel will give me greater understanding in future. Anyway, I babbled, dropped things, forgot what I was saying, and generally didn't perform at my best. But people seemed to grasp the take-home message, so I guess it could have been worse. Some reviews:
"You need more stage presence. Put a bit more Shatner into your performance."
"You didn't make the usual graduate student mistake of preparing everything in far too much detail, going over the first two pages of notes slowly and rushing at the end. You've just got to have the confidence to let the words flow"
"Did you write everything out in detail? Because if not, you should consider doing that. Put everything on slides, so you don't get lost."
In fact, I did write everything out in detail, and it came to under two pages, but I kept getting lost on the page, largely due to nerves. My tiny handwriting doesn't help here. I think the thing to take from the last two comments (which are, as you can see, contradictory, and which came from equally good speakers) is "find out what works for you". What I'm doing now (blackboard + notes) doesn't, it seems.
BTW, michiexile, here's my current understanding of the relationship between operads and monads:
[Rows are classes of algebraic theories. Each row is included in the row below. For obvious reasons, this is presented from the perspective of universal algebra/category theory. Standard disclaimer applies.]
TheoriesSyntactic CharacterizationMonadsRepresenting ObjectsCan be interpreted inExamples
Strongly regular Equations have same variables in same order on each side, each appearing exactly once, eg (a.b).c = a.(b.c)Cartesian?Non-symmetric operadsMonoidal categories (more generally, multicategories)Monoids, semigroups, M-sets for some monoid M
Familially representable?Functor part is a coproduct of Hom(n,-)s; equivalently, functor part preserves connected limitsFamilies of finite sets?Involutive monoids
LinearSame variables on each side, each appearing exactly once, but not necessarily in the same orderAnalytic?Symmetric operadsSymmetric monoidal categories (more generally, symmetric multicategories)Commutative monoids
FinitaryOperations have finitely many argumentsFinitary, ie preserve filtered colimitsLawvere theories / finite product operads / clones / etc.Finite product categoriesGroups, rings, rigs, SKI combinator algebras, R-modules...
Non-finitary-General??Compact Hausdorff spaces
One of the questions I was asked was "How does this work in the symmetric case?", something I'd been intending to look at but didn't get round to. I've since looked at it, and it does seem to work - but interestingly, the 2-categorical justification vanishes! Now that would have been interesting to mention.