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Manuel Rivera
Manuel Rivera
3:50 AM
hi! does anyone know if hopf algebras can be interpreted as (the?) fibrant objects of some homotopy theory for bialgebras? are there any precise statements of this sort in the literature? the analogy I have in mind is that one can model pointed homotopy types via a model category structure on simplicial monoids for which simplicial groups are fibrant.
 
 
5 hours later…
Shay Ben Moshe
Shay Ben Moshe
8:39 AM
Regarding @SaalHardali's question, and @DenisNardin's answer, I think this can be generalized and maybe simplified.
First of all, we can generalize from Cat to any SM category C, and ask if an invertible morphism in CAlg^lax(C) is in fact in CAlg(C).
We can use the model Fun^x(Span(Fin), C) to model CAlg(C) and CAlg^lax(C), where the morphisms are natural transformations and lax natural transformations respectively.
So I think that then the question becomes: is a lax natural transformation which is invertible a (strict) natural transformation.
(Note that by an invertible lax natural transformation, I really mean that we have lax nat. trans. in the other way with identification of the composition with identity and so on. Not merely that it is an isomorphism level-wise [which is how natural isomorphisms are usually defined]. This is exactly related to Shachar's comment, and Denis' reply to it.)
This then also generalizes to other lawvere theories I guess, or something like that (e.g. Alg in place of CAlg).
 
Tom Bachmann
Tom Bachmann
9:41 AM
This also sort of follows from doctrinal adjunction theory: the left adjoint of a lax monoidal functor is oplax. Thus if you have a lax monoidal equivalence, both functors are actually also oplax, and somehow being lax+oplax forces you to be strong. This is related to the asymmetry in the criteria for relative adjunctions, viz. HA.7.3.2.6 and 7.3.2.11.
In fact, I think a lax symmetric monoidal adjunction is a relative adjunction over some base (Span(Fin) or Fin_* presumably), and since the HA reference says that a functor is a relative left adjoint only if it is strong, you can deduce directly.
I guess I'm skirting over the fact why a lax symmetric monoidal equivalence upgrades to an adjoint lax monoidal equivalence, which presumably is the actual point...
 
 
6 hours later…
Saal Hardali
Saal Hardali
3:49 PM
I wonder if this is still true if one considers $\mathcal{O}$-monoidal categories. Denis's argument uses the fact that strictness of a lax symmetric monoidal functor can be checked only on the tensor bi-functor and the unit. For general operads i'm not sure whether this argument is easily generalizable...
On a different note: Does anyone know if any of the many "nice" topos-theoretic contexts to (derived)-differential geometry has some working framework for doing morse theory on infinite dimensional manifolds?
Persumably to get flows you eventually always have to go through some analysis, I was just wondering whether or not these general abstract frameworks can help with organizing everything else around that...
@TomBachmann i'm not sure i understood your argument but maybe it has potential to work for general $\mathcal{O}$
 
Tom Bachmann
Tom Bachmann
I don't have any intuition about this (right now), sorry!
 
Denis Nardin
Denis Nardin
@SaalHardali Yeah, it works. You just have to check on more compatibilities but the same principle apply
 
Saal Hardali
Saal Hardali
@DenisNardin I see so it's just a matter of formalizing it in a way that doesn't require you to go by hand and check all the diagrams
 
Denis Nardin
Denis Nardin
Morally there's one condition to check for each active arrow of the operad, but the same proof works
(recall: a lax O-monoidal functor is strict O-monoidal iff it sends cocartesian active arrows to cocartesian arrows)
In fact you can restrict to those arrows whose target is only one color
So for each arrow (o_1,\dots,o_n)→o you get a map F(-)⊗...⊗F(-)→F(-⊗-⊗..⊗-) and you run the same argument
 
Saal Hardali
Saal Hardali
4:05 PM
I'm convinced
thanks.
 
 
4 hours later…
Yuri Sulyma
Yuri Sulyma
7:40 PM
has anyone studied what happens if you invert all slice spheres ($W$ s.t. $W^{\Phi H}$ is a finite wedge of spheres for all $H\le G$) in $\mathrm{Sp}^G?$
 
Dylan Wilson
Dylan Wilson
8:27 PM
@YuriSulyma I thought about this at some point but couldn't make good sense of the category you get... Like, as a warm-up, what's the best way to get a handle on the category of abelian groups after inverting tensoring by Z\oplus Z, say? Or, closer to the case of interest, what happens to the category of C_p-modules if you invert tensoring by the augmentation ideal?
2
I'd be super excited if someone figured this out!
 
Denis Nardin
Denis Nardin
@DylanWilson Inverting $\mathbb{Z}⊕\mathbb{Z}$ gets you 0, as you can see by comparing the endomorphisms of $X$ with those of $X⊕X$
 
Nikitas
Nikitas
@DylanWilson Thanks !
 
Dylan Wilson
Dylan Wilson
8:50 PM
@DenisNardin oh yeah! I vaguely remember something like that... I think this doesn't kill the C_p-module example, since the augmentation ideal is invertible in the stable module category
 

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