@BrunoStonek I don't think that's true in general, not even if $QA\to A$ is epi.

@FernandoMuro oh? But $S \to QA \to A$ is the unit of $A$, and $QS\to QA\to A$ is equal to $QS\to S\to A$.

anyway, I don't know if my $QA\to A$ is an epi. Some intuition as to why I shouldn't expect this to be true would also be appreciated

Actually

I think it's true if you use a comonadic cofibrant replacement in an algebraic weak factorization system

yes, pretty sure this is true

not positive though, would have to check to be certain

@BrunoStonek that argument actually seems to need that QA -> A is mono (or maybe you could turn it around and ask that S -> QS is epi). unless QS -> S is an isomorphism this seems really unlikely to hold for a general A, and if you're using the small object argument to get functorial factorizations it seems really likely to fail

Even the alg. small argument?

I've seen it used to produce fantastical objects like "initial contractible objects" with honest universal properties

What is the algebraic small object argument? This is further outside my wheelhouse.

@TylerLawson oh right, thanks for setting me straight on the mono/epi thing. how silly of me :P

@TylerLawson You can construct a pair consisting of a monad and comonad that give an alg. weak factorization system

and cofibrations become coalgebras for the left comonad and trivial fibrations become algebras for the right monad

and so you can do crazy things like find initial and terminal factorizations, it's really neat

Emily Riehl's book "categorical homotopy theory" has a chapter on it, but it is originally due to Garner

Also Emily has a paper where she defines "algebraic model categories" where both AWFS are compatible, it's pretty cool

iirc most/many combinatorial model cats can be algebraized in this way

I mean, if you can find a functorial cofibrant replacement that literally preserves the intial object then that will work; I don't have any intuition for this.

I think maybe you can do this if the model cat is pointed

I haven't studied it in that much detail, but it seems like that could possibly be the case

Also, in the considered case I think there is an easier way

If i'm not mistaken then (it follows from sullivan's conjecture) that in the category of pointed spaces $Map(K,X)$ is contractible whenever $K$ is pi-finite and $X$ is finite dimensional. Are these subcategories "orthogonal" in some stronger more general categorical sense? Or is this really the best statement there is?

@HarryGindi I have to run now, but I'm interested

I'll try to work it out, but no promises =\.

I think you can always rectify Q to some Q' that is still functorial but fixes the initial object, basically

restrict Q to the subcategory minus the initial object, create Q' by defining it to be Q on M-0 and the identity on 0, and just define Q'(u) to be the unique map 0->QA, seems like it works for any modCat

well no, when it's not strictly initial, there are problems, but they seem to be able to be overcome

Suppose that C is a presentably symmetric monoidal oo-category, E a homotopy commutative monoid in C and a: 1 -> E. Surely then the mapping telescope E[1/a] must afford the structure of a homotopy commutative monoid (under E). Where can I read about this?

@TomBachmann when R is also an E_1-ring, this is in section 7.2.3 of Higher Algebra. for the more general result that you want I'm less sure; you can construct a multiplication on the mapping telescope but ensuring that it is associative/commutative/unital seems tricky

I specifically want to deal with something to do with moore spectra, so I cannot assume E_1...

@TylerLawson What's the best way even to define a multiplication? I have been trying to use the day convolution symmetric monoidal structure on Fun(N, C), but I run into having to justify that various things extend up all the higher homotopies ... which is probably true, because how much can possibly go wrong with mapping out of N or N x N, but it seemed like a mess

Somewhat related question. Given an object E in hC and a map a: E -> E in hC, surely this should determine an object E -> E -> E -> ... in Fun(N, C), up to equivalence? Similarly a morphism in Fun(N, C) should not contain more data than homotopies for making all the squares commute. How can I see this rigorously?

@TomBachmann Model your oo-categories as quasicategories. Then your a gives a morphism S^1 -> C, where S^1 = Delta^1/{0,1}, and you want to know how unique the extension to a map NN -> C is, where NN is the nerve of N thought of as a one object category. The space of extensions is contractible, because the inclusion S^1 -> NN is inner anodyne. This is not too hard to prove, building up NN degree by degree by attaching simplices to inner horns.

More generally that argument show that if X is reflexive directed graph (reflexive means each vertex has a distinguished loop destined to play the role of identity morphism), and FX is the free category generated by it, the inclusion X --> NFX is inner anodyne, so that diagrams X --> C have unique extensions to functors NFX --> C.

I think that argument goes back to Dwyer and Kan in one of their simplicial localization papers.

@OmarAntolín-Camarena That's exactly the kind of argument I was looking for, thanks! I guess I will have to finally get more comfortable with the combinatorics of quasi-categories.

Oh, I think what I was remembering from Dwyer and Kan is Proposition 2.9 in Simplicial Localizations of Categories (pdfs.semanticscholar.org/fb5e/…;. The proof is ... not very detailed.

@TomBachmann I think counterexample 12.2 in our paper also applies here. Consider E=Sp^{fin} in the ∞-category of small stable ∞-categories. Then E[1/2] does not admit any binary multiplication with unit E → E[1/2]. If it did, multiplication by 2 on E[1/2] would have a retraction, hence an inverse, which is not possible.

@TomBachmann For constructing objects in Fun(N,C) a useful fact is that the inclusion of the 1-skeleton in the nerve of the poset N is a weak equivalence in the Joyal model structure. This means that giving a sequence of objects E_i and maps E_i → E_{i+1} determines an object of Fun(N,C). Using that ×Δ^1 preserves categorical equivalences (HTT 2.2.5.4), you also get the expected description of morphisms in Fun(N,C).

@MarcHoyois In fact it's even better: the inclusion of the 1-skeleton is inner anodyne

(I think it shouldn't be too hard to do it by hand, but it also follows by the recent theorem that inner anodyne maps are Joyal trivial cofibrations that are a bijection on 0-simplices)

I admit I've never had to consider inner anodyne maps before. Does that have any useful consequences?

You have a bunch of theorems about various versions of "the pushout product of an inner anodyne map with a cofibration is inner anodyne"

I'm generally under the impression there are more of them for inner anodyne maps than trivial cofibrations

For example the fact that the "join pushout product" of an inner anodyne map with a cofibration is inner anodyne is an easy exercise, I don't know if the statement is true for trivial cofibrations, but it cannot be as easy

Oh I see, so in particular it allows you to construct more categorical equivalences. That does sound useful!

Is there a name for the following concept: If $L:C\rightleftarrows D:R$ is an adjoint pair, then it induces an equivalence of full subcategories $C_0\approx D_0$, where $C_0$ is spanned by objects on which the unit is an iso, and $D_0$ is spanned by objects on which the counit is an iso.

Is there a name for the subcategory $C_0=D_0$?

@MarcHoyois @DenisNardin ah that's very helpful!

@CharlesRezk the nLab calls this the fixed point of the adjunction.

That's a good name.

I couldn't find it on nlab

@MarcHoyois Ah I see. I was working out by hand what is required to get a multiplication (commutativity is enough), but apparently more is needed in order for the multiplication to be unital, just as tyler hinted.

@TomBachmann right. so let me try to expand a little

you're taking the colim of the sequence E -> E -> E -> ..., I'll call that diagram I

to get a multiplication you look at the diagram "I tensor I", of products E tensor E with multiplication maps on either factor. since the diagonal I is cofinal, the result E[1/a] ⊗ E[1/a] is the hocolim of (E⊗E -> E⊗E -> ...)

if you reindex the original diagram I to a diagram I' so that it goes twice as fast, then then the multiplication maps E⊗E -> E make a "ladder" diagram (E⊗E-> E⊗E -> ...) => (E -> E -> ...) commute. taking hocolims gives you a map E[1/a]⊗E[1/a] -> E[1/a] extending the multiplication on E. this (a) relies on homotopy commutativity of E and (b) relies on making a choice of homotopy between multiplication-then-multiply-by-a^2 and multiply-each-factor-by-a-then-multiply

for unitality I guess you need to verify that multiplication by a is a self-equivalence of E[1/a], which it seems like Marc said is not necessarily true (I might be misreading)

for associativity you have to verify that your chosen homotopy doesn't screw you up, which I think involves verifying that some hexagonal diagram of homotopies has a filler -- this is probably automatically satisfied for ring spectra that satisfy the analogue of Mac Lane's hexagon axiom

for commutativity there seems to be some analogous pentagonal axiom that's maybe satisfied for things that have E_3 ring structures, or at least the quadratic part of it

[my apologies if i'm repeating things, it sounds like you've already worked some of this out]

ah, i was confused by marc's example because somehow in my head you were asking for E to be in a stable category. stupid lazy brain

@TomBachmann I just noticed by brain played a trick on me. I answered as if you wanted to upgrade the loop shaped diagram of a going from to itself to a whole funtor out of N (as a 1-object category), but instead you wanted E --> E --> E --> ... (with N as a poset). Luckily, the free category thing I said afterwards covers both cases. By the way, the telescope can also be thought of as the left Kan extension from N to Z (as 1-object categories), so even the wrong thing I said wasn't so irrelevant.

@TylerLawson Indeed, in a stable compactly generated ∞-category, a always acts invertibly on the telescope E[1/a]. In a stable presentable ∞-category, a transfinite telescope should work. In my unstable (compactly generated) counterexample, even a transfinite telescope does not work.