21,737 reputation
450170
bio website math.berkeley.edu/~theojf
location Berkeley
age 30
visits member for 5 years, 10 months
seen 8 mins ago

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.


1d
awarded  lie-algebras
Jul
31
comment Why do the model structures on dg-algebras and on dg-categories are not compatible?
At least they have the same weak equivalences. So in addition to your question --- does dgAlg have a model structure with the standard weak equivalences but the Dwyer--Kan fibrations --- one can ask whether dgCat has a model structure with its standard weak equivalences but the projective fibrations. Note that if the answers are "yes", then the identity functor will be a Quillen equivalence, which is often good enough.
Jul
30
accepted Does the following characterize local presentability?
Jul
29
asked Does the following characterize local presentability?
Jul
29
comment Frobenius $A_{\infty}$-bialgebras?
@ManuelRivera I haven't thought enough about cyclic algebras enough to confidently make any claims. Well, let me say that I would not immediately expect $u_k$ to be itself a map of bimodules (of some cohomological degree), but instead the first Taylor coefficient of a homotopy between maps of bimodules, which I think is different. Then somehow you're supposed to extract the remaining coefficients by considering signed combinations of compositions of $u_k$s and $m_j$s. At least, that's my expectation.
Jul
29
answered Frobenius $A_{\infty}$-bialgebras?
Jul
28
comment Reshetikhin-Turaev as a 3-2-1-theory
@SpicetheBird Or, you know, he could have used the Alexander invariant.
Jul
24
comment An example for a construction on monads/operads?
+1 for lots of details, definitions, theorems...
Jul
23
comment $Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]
The isomorphism Pin^+(4k) = Pin^-(4k) is documented in "Analysis, Manifolds and Physics. Part II" by Y. Choquet-Bruhat and C. De Witt-Morette (Elsevier, 2000). Is this what you're looking for? Note that it is not an isomorphism of groups over O(4k). (The isomorphism, as you point out, covers an interesting outer automorphism of O(4k).)
Jul
23
comment Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Incidentally, I asked a related question at mathoverflow.net/questions/184982/which-real-pin-groups-agree.
Jul
22
accepted When is the adjoint to a monoidal functor monoidal?
Jul
21
comment When is the adjoint to a monoidal functor monoidal?
Of course, the answer will probably be different for different ordered pairs in {lax,oplax,strong}. For example, if $F$ is lax, it is not too hard to believe that $F^L$ is oplax, based on the way that adjoints work (e.g. we do have a canonical map $F^L(1_D) \to 1_C$ corresponding to $1_D \to F(1_C)$).
Jul
21
asked When is the adjoint to a monoidal functor monoidal?
Jul
14
awarded  Nice Answer
Jul
13
comment Equivalent definition of a Kan fibration
Oh, I misunderstood. My apologies.
Jul
13
comment Equivalent definition of a Kan fibration
Am I right that when $i=1$, your lifting condition is trivial? So then you are asking whether a Kan fibration is characterized by having the right lifting property with respect to the maps $\Delta^n \hookrightarrow \Delta^1 \times \Delta^n$.
Jul
12
comment Natural transformations induce homotopies - Is this true in the “fat” world?
@DavidRoberts Well, it doesn't really matter to me. (And certainly I also often leave short answers --- too short, it seems, for a real answer --- on questions.) I used to think there was some benefit in allowing questions to go away, either by having an answer accepted or by closing for having an answer in the comments. I no longer think it matters, myself.
Jul
11
comment Natural transformations induce homotopies - Is this true in the “fat” world?
I don't know why there is a vote to close this (I mean, in MO 1.0 we could close as "no longer relevant", but that's been deprecated). It seems like a good question, research level, that was answered, albeit quickly.
Jul
9
comment Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?
... most natural is you think of $\mathbb A^1$ as a "space" (and not an abelian group). I don't see how pure category theory would ever come up with the fiberwise product from $\mathcal C^\circlearrowleft$ out of thin air --- it seems to require instead some external intervention. See, Tannakianism provides a functorial way to extract a space from its sheaves as a symmetric monoidal category, but not as a bare category (Gabriel's reconstruction theorem notwithstanding), just like you cannot extract an affine scheme from its vector space of functions.
Jul
9
comment Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?
The reason I find this less than a complete story, though, is the following. (This is closely related to the Pontrjagin vs Eilenberg-MacLane difference mentioned above.) Suppose that $\mathcal C$ is symmetric monoidal. Then the natural symmetric monoidal structure on $\mathcal C^{\circlearrowleft}$ corresponds, on the $\mathbb A^1$ side, to tensoring together spaces of global sections. This is the "convolution" tensor product for the additive group $\mathbb A^1$. The natural tensor structure on $\mathrm{Sheaves}(\mathbb A^1)$, on the other hand, is the fiberwise one --- at least, that's ...