bio  website  math.berkeley.edu/~theojf 

location  Berkeley  
age  30  
visits  member for  5 years, 9 months 
seen  2 hours ago  
stats  profile views  15,532 
I am a recent graduate from UC Berkeley. Starting Fall 2013, I am a Boas Assistant Professor at Northwestern University. My research is broadly centered on quantum field theory — my interests include category theory, representation theory, homological algebra, algebraic topology, Poisson geometry, and theoretical physics.
2h

comment 
Is the $\infty$category of presentable $\infty$categories presentable?
... are wellstudied in some areas of computer science, because they allow for coinductive, rather than inductive, reasoning. A more downtoearth reason not to be afraid of Cantor's paradox is the wellknown theorem that the homotopy category of topological spaces is not concretizable (because every nonempty object has a proper class, not a set, of subobjects). 
2h

comment 
Is the $\infty$category of presentable $\infty$categories presentable?
I disagree: in principle the answer should be yes. Indeed, $Pr^L$ contains all limits and colimits (the strict version of this statement is due to Greg Bird in his unpublished '76 thesis, if my memory is correct), and feels like it is generated under colimits by some basic building blocks. The reason Cantor's paradox does not apply is that we are in the homotopical world, not the settheoretic world. Indeed, there are good homotopical models (although now I am speaking outside my expertise) in which all functors, and in particular the "power set" functor, have fixed points. These worlds ... 
8h

awarded  Popular Question 
Jul 1 
awarded  Nice Question 
Jul 1 
accepted  Does projective imply flat? 
Jun 30 
comment 
Does projective imply flat?
@EricWofsey Ah, good point. 
Jun 30 
comment 
Does projective imply flat?
Well, grumble. And this gives an example where I don't even have flat resolutions, since by the same logic $(N,1)$ fails to be flat for every $N$. (Tensoring with $(M,0)$ and then projecting onto the $\mathcal B$ factor still gives the functor $F$.) 
Jun 30 
comment 
Does projective imply flat?
@AlexDegtyarev Do you mind spelling out your abstract nonsense? Eric Wofsey in an answer below seems to provide a counterexample. 
Jun 30 
comment 
Does projective imply flat?
@AlexDegtyarev The thing I know how to do is to use the fact that projective implies flat to conclude that Tor groups can be computed by projectively resolving only one of the two variables. Please explain what you have in mind? All the categories I care about have enough projectives, so I don't mind assuming that as an axiom. 
Jun 30 
comment 
Does projective imply flat?
@FernandoMuro Yes, all the categories I care about have enough injectives, so I'm happy to add that as an axiom. If you have a proof available, I'll be happy to accept it as an answer. 
Jun 30 
comment 
Intuition behind the definition of quantum groups
... the "quantum plane" above I imagine as having spectrum $\mathbb R$ (or maybe $\mathbb C$). Then again, coordinates satisfying $XY = qYX$ have funny behavior along the axes... I guess the main difference is that $xp  px = i\hbar$ is natural if you have some translationinvariance, and for QM on a line we do want $x \mapsto x+x_0$ to be a symmetry. Whereas $XY = qYX$ is natural if you want an action by scaling  if you're trying to do "quantum linear algebra". 
Jun 30 
comment 
Intuition behind the definition of quantum groups
@AndreKornell Good question. The word "plane" is, I hope, selfexplanatory  this is some kind of 2dimensional linear space. Probably this use of "quantum" is nothing better than an abbreviation for "noncommutative". In the quantum mechanics of a particle on the line, the phase space is a plane with coordinates $x,p$, but those satisfy $xp  px = i\hbar$. Setting $q = e^{i\hbar}$, $X = e^x$ and $Y = e^Y$ does give coordinates satisfying $XY = qYX$ (or perhaps I'm off by a sign), but those should have spectrum $\mathbb R_+$ (or maybe $\mathbb C^\times$), whereas the coordinates on ... 
Jun 30 
asked  Does projective imply flat? 
Jun 28 
awarded  Nice Question 
Jun 23 
revised 
Why does the bitxor function appear in Nim?
error in an equation 
Jun 23 
comment 
Why does the bitxor function appear in Nim?
@Halbort: Yes; I will correct it. 
Jun 23 
comment 
Why does the bitxor function appear in Nim?
Huh. I started writing this last night, but only posted it this morning, and only then saw that Will Sawin has given essentially the same argument. 
Jun 23 
answered  Why does the bitxor function appear in Nim? 
Jun 22 
comment 
Is there a symmetric monoidal 2category “SuperDuperVect”?
@MikeShulman I meant only that I thought I remember at least some discussion of symmetric monoidal bicategories in one of your papers, perhaps on double categories? 
Jun 21 
comment 
Lurie's Endomorphism Space vs. Endomorphisms
In other examples I've thought about, to produce the map $A \to Map_{Top}(M,M)$ qua algebras requires more than just the action $A \times M \to M$, but also the associativity data of that action. Does that work here? 