Do I need to be careful when trying to talk about monoids in a 2-category? E.g. monoidal categories as monoids in CAT? I guess they're not exactly strict monoids, but only satisfy the axioms up to 2-cells? I'm feeling stressed about this, but I know it's really basic.

yeah, those 2-cells are provided by the associativity isomorphisms in the definition of a monoidal category

Ah sure, of course. Because when you want to say "the associativity diagram commutes," you can only say it "up to isomorphism."

right

Anyone got a reference for a 2-category structure on multicategories?

Probably not too hard to write down.

Also this: wslc.math.ist.utl.pt/s84.www/cs/claudio/articles/rep-mult.pdf

Reality check: is the space of E_∞-ring endomorphisms of HZ contractible?

I think commutative rings embeds fully faithfully into E_\infty rings

@DenisNardin Do you mean the connected component of the identity?

Map(S_0,HZ) = MAP(HZ,HZ) by adjunction of conctive E00 rings and truncated E00-rings

but S^0 is the intial ring

But nat is superior in every way

ubh

None of those spectra are 0-truncated...

HZ?

Ah sorry, I got it now.

@NatStapleton Do you have a reference for that?

@TomerSchlank Thanks that is a very slick argument :)

@Adeel No, I mean the whole space (that is, the connected component of the identity is contractible and the space is connected)

@DenisNardin What Nat said is true: taking n-truncations commutes with taking commutative monoids in any oo-category. This is clear for example when you look at the definition of commutative monoid as a presheaf on Segal's Gamma (since n-truncated presheaves are presheaves taking values in n-truncated objects). In particular, commutative rings are 0-truncated E_oo-ring spectra.

@Adeel I am very confused now. I believe Nat, but E_∞-rings are in no way presheaves over Gamma (as far as I know: if you have a way of seeing them that way it would be amazing) and not all 0-truncated E_∞-spectra are commutative rings (there are also the things with stuff in negative degree). You seem to be claiming that the inclusion of discrete objects in spectra is symmetric monoidal but that is false

Sorry, I meant connective E_oo-ring spectra!

And presheaves over Gamma (satisfying the usual Segal condition) is a standard definition of commutative monoid in an infinity-category. I believe Lurie proves this is equivalent to algebras over the E_oo-operad in HA.

@Adeel That is false. This works only if your symmetric monoidal structure is the cartesian product, and for spectra it isn't.

In fact I don't even know how to write down the Segal condition if the sms is not the cartesian product

Lurie's definition is much trickier and requires you to work with (symmetric) multicategories, using Fin_* as a model for the terminal (symmetric) multicategory

Fair enough, I was confused and was indeed thinking of cartesian monoidal structures.

But let me see if I understand your argument. We have a coreflective subcategory of discrete spectra in connective spectra. The inclusion of it is not symmetric monoidal though, it is just lax symmetric monoidal. Are you claiming that this is enough to conclude that E_∞-rings in discrete spectra embed fully faithfully in E_∞-rings in connective spectra?

The emmbding is lax but it's left adjoint (the truncation) is monoidal

Well, you can make what I said work by using James Cranch's model for E_oo-ring spectra as models for a certain algebraic theory (instead of presheaves on Gamma).

@Tomer Now I think I understand: you are saying that the colocalization Ab < Sp_{\ge0} descends to a colocalization E_∞(Ab)<E_∞(Sp_{\ge 0}) since the colocalization functor is symmetric monoidal.

(But yes, this shows that the truncation preserves commutative monoid objects)

Oh I had forgot connective E_∞-rings are algebras for a Lawvere theory. Yeah that seems to work, thanks! Somehow I was too concentrated thinking about nonconnective spectra...

Yeah sorry, I am coming up with a lot of BS today it seems

By the way, does anyone know any references for E_n-algebraic geometry other than John Francis's thesis?

e.g. I'm wondering whether Lurie has proved a version of his representability theorem for E_n-stacks.

@DenisNardin Exactly!

Does anyone know what the infinite loop space representing connective complex K-theory is, or at least where to find a description of this thing?

Isn't it BU * Z?

Hey @SaulGlasman and @DenisNardin either of y'all around, and maybe have a second to talk?

I'm here!

Okay, let me try to type this...

So, suppose we have a monoidal simplicial category C. Then we can construct a simplicial multicategory from it, and then a simplicial category of operators. Now, if C was fibrant, and the monoidal structure preserves cofibrancy, the category of operators will also be fibrant as a simplicial category.

(in fact the multicategory would have been a fibrant simplicial multicategory in a certain model structure)

Note that this final category of operators admits a functor down to $\Delta^{op}$ which, I'm guessing, is the right kind of categorical "fibration" to apply an enriched Grothendieck construction (I'm thinking of Dai Tamaki's construction here).

So, assume we do so.

Then we get a functor $\Delta^{op}\to sCat$, where by $sCat$ I mean simplicial categories and simplicial functors.

(really this is, I think, all happening 2-categorically)

Let's call that functor, I dunno $C':\Delta^{op}\to sCat$

Now, we can compose this with $op:sCat\to sCat$, which should preserve fibrancy, giving me a functor $C'^{op}:\Delta^{op}\to sCat$ which lands in fibrant simplicial categories.

And then take the coherent nerve, to get a monoidal quasicategory.

On the other hand, we could have started, back at the very beginning, with $C^{op}$, and then formed a functor $(C^{op})':\Delta^{op}\to sCat$ which is the same functor on objects, but the mapping spaces won't necessarily be Kan complexes anymore!

(because the tensor products don't necessarily preserve fibrant objects)

Do you think that $C'^{op}$ is a fibrant replacement of $(C^{op})'$?

And, so, the monoidal $\infty$-categories underlying $C'^{op}$ and $(C^{op})'$ should be equivalent, yes?

Sorry, that was kind of long winded.

my feeling is that those two objects should be equivalent in whatever model category you're considering them in

however, I don't know what a fibrant object of $\text{Fun}(\Delta^{op}, sCat)$ is

Hopefully you are using the projective model structure, so that fibrant objects are easy. But I agree that they should be equivalent

Hm, that's a good point. I guess one would probably assume the projective model structure. I'd want to check that the coherent nerve of $Fun(\Delta^{op},sCat)$ with this model structure recovers the functor quasicategory $Fun(N(\Delta)^{op},Cat_\infty)$ (up to equivalence, of course).

but the objects are pseudofunctors actually, right?

Yeah, at least, Tamaki's Grothendieck construction is really a 2-categorical construction.

So actually, erg, model structure on a 2-category of pseudofunctors. Lordy.

I'd still assume that exists somewhere in the literature

at least on a 1-category of pseudofunctors

Yeah. I don't even know what I'm supposed to think about model structures on 2-categories. I mean, there should be some way of saying "well, we've specified a little more of the homotopies, but still not all of them."

yeah

But I guess that doesn't matter actually, since once I GET the functor I can just forget the 2-cells anyway.

you could specialize Aaron's idea of an oo-model structure

but yeah, this is academic

In fact, I'm starting with a monoidal simplicial model category in which BOTH cofibrant and fibrant objects are closed under tensor product, so I think these two things might actually be EQUAL.

But I realized that going one way versus going the other way could create problems with fibrancy, and this asymmetry bothered me.

Lordy. All of this just to say "blah is a comodule."