This might be silly and/or off topic, but here it goes: given a sufficiently nice symmetric monoidal category C, I can (?) Left kan extend the monoidal product on C along the free monoid functor C -> CMon(C) to produce a new symmetric monoidal category. Now I can iterate. What happens? E.g. CMon(Set) is abelian semigroups, and CMon(CMon(Set)) is commutative semirings. Does the new monoidal operation on commutative semirings and/or the category CMon^3(Set) have a name?

@WilliamBalderrama Thank you :)

What kind of pullback diagrams are preserved by $\Sigma_+^{\infty} : S \to ComCoAlg(Sp}$?

@TomBachmann do we expect this process to terminate? Similar to how Grp(Grp(Set))= Grp(Grp(Grp(Set))).

@ReubenStern I was thinking this was related to the stabilization hypothesis, but not sure

@TomBachmann left kan extensions of monoidal products are not usually monoidal

if the target is locally presentable, then it's only monoidal when the extended bifunctor satisfies Day's theorem

So I think where this goes wrong is at your "?" step

well, I dunno about 'only', that might be false

but that's a sufficient condition

@HarryGindi my statement is just Eckmann-Hilton, so I highly doubt something analogous is true here.... trying to think math on little sleep is dangerous

Analogous meaning an analogous statement proved analogously

@ReubenStern Yeah, that's related to teh stabilization hypothesis

any E_2 monoid in Set is E_infty

any E_3 monoid in cat is E_infty

any E_4 monoid in 2-cat is E_infty

@HarryGindi ahh, I see! Neat

an n+2-tuply-monoidal n-category is commutative

or symmetric however yo uwant to say it

@ReubenStern that’s a different procedure than what Tom described. In the stabilization story you use the ‘same’ monoidal structure each time. Tom is suggesting something weirder: in the case of commutative rings, the new tensor product of free things would be the free thing on the tensor product... but I guess I agree with Harry that this doesn’t seem like a monoidal structure any more?

@DylanWilson yeah, also the cat of commutative monoids is totally diff from the category of commutative semirings

comm. monoids are pointed

comm semirings are just not

That’s just notation I think and not really the issue- the monoidal structures he’s using aren’t Cartesian but he’s calling algebras monoids, which is fine

That's not notation

I mean the categories are pointed

the terminal object in comm. semirings is the 0 semiring

but the initial object is N

so that's not pointed

my point was that there's no obvious way to iterate this, the categories you get vary wildly

even if it were monoidal

if you take the monoidal product in comm. semirings, the tensor product there is the coproduct

Anyway, Tom, yeah, you basically can't do that it doesnt' work that way

I see. Thanks.

np!

I am confused about Steenrod squaring operations. I want to ask two questions. The first is maybe rather stupid: we know that $\operatorname{Sq}^i$ are given by $H\mathbb F_2\to H\mathbb F_2[i]$, which is not $\mathbb F_2$ linear. However, I know that given a space $X$, consider a commutative algebra object $V=C(X,\mathbb F_2)$ in $D(\mathbb F_2)$ and we know that Steenrod squarings of a class represented by $\mathbb F_2[i]\to V$ could be recovered by applying $\operatorname{Sym}^2$ to it

which is perfectly done in $D(\mathbb F_2)$.

I wonder in the second interpretation, where the non-linearity appears?

The second question is that, we know that Stiefel-Whitney classes could be constructed from Steenrod squarings of the Thom class. Another way is to study the primary obstruction cocycles. I wonder whether we can describe this relation more intuitively in the modern language of spectra?

@FrankScience My intuition in this is that the SW classes measure the difference in the Steenrod action on $\tilde{H}^*(X)$ and on $\tilde{H}^*(X^V)$ (that remember, are two naturally isomorphic vector spaces over $\mathbb{F}_2$!)

It's not a particularly modern point of view, though.

@FrankScience Actually, thinking about this, I think you are confusing the two interpretations of the Steenrod squares

In one they are maps of spectra $H\mathbb{F}_2\to \Sigma^iH\mathbb{F}_2$, in the other they are elements of $\pi_*\mathrm{Sym}^2(H\mathbb{F}_2)$

How would you do the latter in $D(\mathbb{F}_2)$?

(here I assume you mean $\mathrm{Sym}^2(X)=(X⊗_\mathbb{S}X)_{hC_2}$)

$\operatorname{Sym}^2$ is done in $D(\mathbb F_2)$

What's your definition of $\mathrm{Sym}^2$ in $D(\mathbb{F}_2)$? Or did you mean something different from what I wrote above?

www-math.mit.edu/~lurie/917notes/Lecture2.pdf

The one similar to what you wrote but taken in $D(\mathbb F_2)$

Actually, let me think one second

Oh right I was wrong

Oh right, I'm being silly.

So the problem is the following. You can, being entirely in $D(\mathbb{F}_2)$, construct $\mathbb{F}_2$-linear maps $C^*(X;\mathbb{F}_2)\to C^*(X;\mathbb{F}_2)[i]$. That is not a problem

But to get from this to maps $H\mathbb{F}_2\to H\mathbb{F}_2[i]$ you need to pass to spectra, and that's where the $\mathbb{F}_2$-linearity is lost

In order to apply Yoneda, you need to work in functors out of spaces, not out of $D(\mathbb{F}_2)$

Does this make sense?

I am a bit confused, It seems not to be $C^*(X,\mathbb F_2)\to C^*(X,\mathbb F_2)[i]$, but $H^*$, but maybe I am wrong.

@FrankScience if V in D(F_2) admits a map D_2(V)-->V then you can indeed define a squaring operation from H_(V) to H_*(V), but the definition does not specify this operation as a map V--->V[?] in D(F_2), instead the operation is defined via some procedure: start with a class, apply D_2(-), pick a class in H_*(D_2(-))... etc. This procedure provably does not arise from applying H_ to a map V--->V[?] in D(F_2). However, it does arise from applying \pi_* to a map V--->V[i] of underlying spectra

(i don't know why everything got italicized...)

H_\* instead of H_*. The text between two stars will be italicized.

So essentially, it is due to the fact that $D_2(-)$ in Lurie's text, or $\operatorname{Sym}_{\mathbb F_2}^2(-)$ is not linear in $D(\mathbb F_2)$?

I mean, if I want to try to lift the operation on homologies to chain complexes, I would like to play things like $\DeclareMathOperator{\Map}{Map}\Map(\mathbb F_2,V)\xrightarrow{D_2}\Map(D_2(\mathbb F_2),D_2(V))\to\Map(\mathbb F_2[i],D_2(V))\to\Map(\mathbb F_2[i],V)$. Seemingly the other arrows are linear except the first one: this should be the obstruction to have a lift.

well it's a little misleading because D_2 is also not linear in spectra... however, it turns out that you can rewrite the formula for these power operations using a Tate construction computed in spectra- and the Tate construction is exact in spectra. (cf the later lecture in those same notes where Lurie relates power operations to a Goodwillie derivative)

Thanks for this comment, but I failed to find the discussion about power operations in ocw.mit.edu/courses/mathematics/…

it is lecture 24

the thing I referred to as a Tate construction is denoted E' in that lecture

Thanks

About Stiefel-Whitney classes: In fact, I want to understand how the obstruction theoretic description has something to do with cohomology operations. The proof I see (on Milnor's Characteristic Classes) doesn't give me any info about this

@FrankScience what sort of interpretation are you after?