Does anyone know an example of a category of algebras over a Lawvere theory which is (1) semiadditive (has finite biproducts/direct sums) but (2) not Mod_R for some ring R.

Just to be clear, by "a category of algebras over a Lawvere theory L", I mean Fun^x(L,Set), the category of product preserving functors from L to Set.

@John: sure, semimodules over a semiring

e.g. commutative monoids

Erm yeah that's what I meant

But since asking I have concluded that there must be a bijection between semiadditive Lawvere theories and semirings

Has this been observed somewhere? I am having trouble with the Lawvere theory literature - it seems pretty chaotic.

@John: I remember @Zhen saying something to me along these lines once... maybe it was in an email

okay, so the statement is that if T is any lawvere theory, then the category of abelian groups internal to T-algebras is equivalent to the category of modules over a ring determined by T

and probably the same is true with abelian groups replaced with commutative monoids and rings replaced with semirins

*rings

but now, if T-algebras is already semiadditive, then the category of commutative monoid objects in T-algebras is just T-algebras again

I don't have a reference, but someone asked the same question last year

@QiaochuYuan do you have a reference for the general statement you made?

it's the Eckmann–Hilton argument

@John: no, but it's not hard to prove. i can copy/paste @Zhen's argument from the email if you want (later, gotta go now)

@JonBeardsley IKR?

Oh man... @TylerLawson and @CraigWesterland... you guys should have a radio show. =P

Let A be a connective E_oo-ring spectrum and B a connective A-algebra. Assume that B is a finite cell object (i.e. can be obtained from A by attaching a finite number of cells). Let I be the homotopy fibre of the morphism of A-modules A -> B. Question: can I recover the A-algebra B from I, by attaching to A 1-cells identifying the generators of pi_0(I) with 0?

I'm guessing no, but I really want it to be true...

what structure do you allow on I?

@TylerLawson I'm just considering it as an A-module (does it have more structure?)

you definitely can't reconstruct it from just the A-module structure.

it has at the least a nonunital A-algebra structure

If you have that I-->A-->B is a sequence of E_\infty rings such that I is the fiber of A-->B in A-modules, can you do it?

look at HZ -> HZ[x]/(x^2 + ax + b)

well, E_\infty... rngs

the fiber is the desuspension of HZ

you can't recover the polynomial coefficients from that

hmm, right

how does the nonunital A-algebra structure arise?

the composite I^I -> A^A -> A -> B is null, so it lifts to the fiber I

@JonBeardsley this is just delightful

and if you work much harder you can get the rest of the multiplicative structure

I see... do you have any idea if that structure would be enough to recover B?

probably not, you also need the map I -> A to be compatible with the multiplications of both and the A-algebra structure

"quotients" can be a gigantic headache

Ah ok. Looks like I'll have to figure something else out. Thanks!

@Adell I should mention Hovey's paper on Smith ideals does give you a framework that can recover the quotient from the structure on the ideal: arxiv.org/abs/1401.2850

ah I'll take a look, I remember Denis also mentioned this a while ago

@SaulGlasman I know rite! :D

Yeah, IIRC Smith ideals basically tell you exactly when something like this does happen?

yeah, but I don't know that it tells you about the commutative algebra structure on the target

Yeah... Smith ideals aren't really E_\infty at all, are they?

I can't remember.

Yeah, looking now. It was purely for the triangulated homotopy category.

Just looking at Hovey's paper now... if I'm reading right, then it seems that the answer is actually yes, if I do also keep track of the morphism of A-modules I -> A, which I am happy to do

Though maybe there is a subtlety about commutativity? It looks like he's always working with associative monoids.

I'm pretty sure that Smith ideals work also for commutative things

After all $\Delta^1$ is a symmetric monoidal category

I wish someone had written this up for infinity-categories

could be your job~

@DenisNardin would do a much better job

Let me put it this way: we have a symmetric monoidal $\infty$-category $\Fun(\Delta^1,C)$ (with the Day convolution symmetric monoidal structure) and a localizing subcategory of those $I\to A$ where $I=0$. Then the map sending $I\to A$ to $A/I$ is just the localization functor

I had that almost already written when you wrote your message @Adeel I swear

:P

(Also localizing subcategory is almost surely the wrong name, I mean the thing a localization functor has as a target)

reflective subcategory?

Possibly. Or maybe coreflective subcategory. I'm not very good at distinguishing left from right

What I mean is that the inclusion has a left adjoint

so reflective indeed

Anyone know of a description, in ordinary categories, of the monoidal structure on the slice category over a monoid which is NOT the one coming from the fibred product?

Is this in Mac Lane's book or anything?

@DenisNardin So, the statement is that for any E_oo-ring spectrum A, the assignment which sends an A-module morphism I -> A to the homotopy cofibre, is an equivalence between Smith ideals and A-algebras?

Hmm, you need a bit more that just "A-module morphism" if I recall correctly the definition of Smith ideal, but yes, I'm pretty sure that's true

@Adeel

Oh I see, I misread something

though in the commutative case, the additional condition seems automatic

The additional condition should be morally that the two possible maps $I\otimes I\to I$ are the same, which doesn't seem automatic to me but I may be wrong

In any case I think that we can easily rewrite Hovey's article in the language of oo-categories (and for an arbitrary oo-operad no less) without changing much

I could give it a try for E_oo-monoids, but you'll have to do arbitrary operads yourself

A more interesting question might be: what condition to put on I -> A to ensure that the map A -> A/I is surjective on connected components

More specifically, for my last question, given a topological monoid $M$ in the monoidal model category $sSet$ with the Quillen model structure, is there a Day convolution structure inducing a strict monoidal structure on $sSet/M$?

I guess this requires a model category theoretic Grothendieck construction.

Ah.... arxiv.org/pdf/1404.1852v3.pdf

Hrmph, that doesn't address the monoidal question.

So, we have a map CP^oo to suspension of MU, restricting to CP^1 giving the unit, and we call it complex orientation. The same map can be interpreted equivariantly, and we shall get a real orientation in C_2 equivariant homotopy. The last thing, when we look at MO, the map RP^oo to MO(1) induces a similar thing in the unoriented cobordism, shall I call it "unoriented orientation"? When I realize that I wrote it down I really got shocked...

@SaulGlasman if you look at the infinity categorical Day convolution product on Fun(X,Top) for X a Kan complex, is this closed? if so, what's the internal hom?

it would seem (??) that the usual presheaf hom would be closed with respect to the cartesian monoidal structure (i.e. taking the product objectwise in the target).