Say that R is a connective ring spectrum, M a connective R-module. Is flatness of M equivalent to the assertion that (M\otimes_R -) preserves homotopy pullbacks in the category of connective R-modules?

Where can I find the BP-homology of tmf ?

@Bogdan I wrote down an answer for MU_*tmf in my 512 notes, though based on assumptions about tmf that I didn't prove. Akhil's paper about homology of tmf might give a complete description and proof.

@CharlesRezk Oh thank you. I can't find it in Akhil's paper, but I see it in your 512 notes. You describe the MU-homology of tmf to be Z[a1, a2, a3, a4, a6, en] for n>3. Can you tell me if the next step is correct ?

To get to the BP-homology for p=2, we can just 2-localize the answer and then take an appropriate summand. I suppose the en's come from the homotopy of MU, and so to get the appropriate summand, I have to do the same step as I do to get v_n's from the x_n's ? (for some choice of v_n's)

Right: tensor it over MU_* along the projection MU_-->BP_. Figuring out what this means presumably amounts to untangling the definition of the formal group over BP_* (the universal p-typical formal group). This is possibly easier said than done.

Whoa, elementary question about operads... when we take a map from the commutative operad to the endomorphism operad of some set $X$, what's the unary operation we're picking out? The identity?

Yeah... must be.

Yes

Yeah, I guess this is typically just encoded into the definition.

Of the endomorphism operad? I think so

Well, at least classically it seems to be required that one's operads be unital in the sense that there's some pre-specified choice of identity element in $Hom(X,X)$ and that morphisms of your operads are unital morphisms, thus they have to pick out the identity.

Or, rather, the map from Comm has to pick out the identity. Anyway, dumb detail that one just has to put in the definition for stuff to be useful.

I'm explaining operads to someone as elementarily as possible!

I have another elementary question about operads, or rather models of algebraic theories in an infinity-category C. Is it the case that the algebras of a discrete Lawvere theory T are just T-algebras in Ho(C)? One the one hand that should be the case because Ho is right-adjoint to N : Cat -> Cat_oo, so functors T -> Ho(C) are the same as functors N(T) -> C, and the same should be true of finite-product functors.

On the other hand, I see in Joyal's notes on quategories that Mon is discrete and its algebras are E_1-objects. But certainly E_1-objects in Spaces are more than monoids in the Ho(Spaces), as those are A_3-objects. So where am I confused?

So in general rectification of diagrams in the homotopy category to actual homotopy coherent diagrams has a number of obstructions, if I recall correctly.

Which I know doesn't really answer your question.

Indeed, that's why there's a whole A_n-hierarchy before we get coherently associative objects. BTW, I'm referring to sec. 32.24 in Joyal's notes.

Hrm, yeah, this is puzzling. I think whatever is confusing you is also confusing me at the moment.

the adjunction goes the other way - Ho is left adjoint to N, not right adjoint

Oh yes of course! Haha, I literally just read those words.

Oh, here's another question: does anyone know what the Boardman-Vogt tensor product looks like if we think of it in terms of a tensor product on trees? For instance, the tensor product of two really simple binary trees with two inputs and one output?

Is it just a tree with with 4 leaves and a single vertex?

let's see, is that simple binary tree just the A_2-operad?

how are you representing operads as trees?

Well, yeah, at the moment basically totally intuitively.

The idea would be that I'm tensoring together two operads which are operads generated by trees.

Thanks, Saul, I knew I was thoroughly confused!

@UlrikBuchholtz The Lawvere theory for E_infty algebras is just the 1-category of finite sets, but taking E_infty algebras is far from a 1-categorical procedure.

So I guess the operad generated the tree Y is just going to be the free operad on a single binary operation, but I'm not sure if this is A_2 or not.

@JohnBerman, yes, knew that but then I got confused because I got that adjunction the wrong way. But now I'm confused: I thought CMon (the Lawvere theory for E_infty algebras) was 2-truncated, not discrete?! (Cf. Joyal's notes 32.25)

On the other hand, Ho(Cmon) should be the Lawvere theory of commutative monoids.

But it seems possible that... the resulting thing is not generated by a tree at all? I feel like I'm missing something.

Equivalently, if we take the Moerdijk-Weiss tensor product on dendroidal sets of two representable dendroidal sets A and B, is the tensor product representable...?

@JonBeardsley I think that's A_2, although I tend to get confused on the low-numbered As

Yeah, I've literally never thought about A_2 before.

That's not true I guess.

But not in terms of trees.

well, anyway, O_1 tensor O_2-algebras are supposed to be O_1-algebras in O_2-algebras, right?

tru

so if A is the free operad generated by one binary operation, then I guess A tensor A-algebras are objects with two binary operations f and g such that

g is a homomorphism with respect to f I guess

yeah

hm

I don't know how to say "homomorphism", though, because if M is an A-algebra, then there's no A-algebra structure on M x M making f into an A-algebra morphism, is there?

Er, I'm not sure I understand. I don't know how the A-algebra structure and the f-algebra structure are related.

oh I guess there is, you just use f x f

so I guess the condition is that

for a, b, c, d in M

f(g(a, c), g(b, d)) = g(f(a, c), f(b, d))

is that right?

@JohnBerman Yes.

Given a diagram of modules A -> B <- C, the map from the connective homotopy pullback to the homotopy pullback lives in a fiber sequence (connective fiber) -> (fiber) -> K where K is an Eilenberg-Mac Lane object concentrated in degree -1. so for sufficiency, tensoring with a flat thing preserves connective covers because tensoring K leaves it an eilenberg-mac lane object concentrated in degree -1.

for necessity, you can consider the pullback of a diagram 0 -> B <- 0 where B is an Eilenberg-Mac Lane object concentrated in degree 0. the connective homotopy fiber is 0 and the homotopy fiber is B[-1]. then you find that if tensoring-with-M preserves connective homotopy pullbacks then tensoring with M preserves eilenberg-mac lane objects -- this is equivalent to flatness.

@UlrikBuchholtz Sorry, you're right... The Lawvere theory for E_infty-algebras is the Burnside category, which is indeed 2-truncated. The counterexample I was looking for was the trivial Lawvere theory Fin^op.

@CharlesRezk Thank you!

@JonBeardsley The tensor product is not representable - for example if you take two corollas then if I remember correctly you get a colimit with all the possible ways you can stick one onto the other, glued along the inclusions of the two corollas.

Does anybody know a good reference for the homology spectral sequence of a simplicial space?

Guess I more or less figured it out. Would still like to know if there's a nice reference out there though!