Where the set of maps {A,...,A,M}-->M is {linear orders of {A,...,A,M} in which M is maximal}.
(Note, this is not the way Lurie does it, in particular, he doesn't state the symmetric group actions explicitly)
1.5.1 of this paper arxiv.org/pdf/math/0512576v2.pdf seems to indicate that the set of multimaps {A,...,A,M}-->M should be the same as the set of multimaps {A,...,A}-->A (assuming the domains there have the same number of elements) but this cannot be true?
The symmetric group does not act on the set of maps {A,...,A,M}->M. Rather, it acts on such a map by producing a new map {A,...,M,...,A}->M, where the elements of the n-tuple have been permuted by your permutation. (For a 1-colored operad, you only see a symmetric group action of course.) As for how this action works, it ends up being tautological, since the maps {A,...,M,...,A}->M correspond by definition to the ways to multiply the coordinates in any order, as long as the rightmost one is M.
Well now I am a little confused... It seems like the LMod that Lurie defines in HA 4.2.1.6 is a nonsymmetric operad, because he doesn't allow the kind of permutation of {A,...,A,M} that I just described.
So I think that LMod as defined in 4.2.1.6 is a nonsymmetric operad, but in order to make it symmetric, you have to include as part of the data total orderings on each fiber, as in 4.1.1.1, which would make my original answer valid. Maybe this is a typo?
But at least in the infinity operad story, {A,...,A,M} and {A,...,M,...,A} refer to different objects of the infinity operad. They are equivalent, but in order to specify an equivalence, you need to pick a permutation that permutes one into the other.
What he seems to be describing is a nonsymmetric operad, in which case Fin_* would be replaced by totally ordered finite sets, but I think this can be corrected the same way he built Ass
Well, I mean no it's fine... I just think that it would be helpful to understand 4.1.1.1, this LM object.
Like, all of Lurie's "colored operads" are the same thing as symmetric multicategories, or symmetric colored operads, or whatever. Things for which there is a $\Sigma_n$-action on the sets of multimaps.
And I'm trying to see how to interpret LM in this way.
So, okay... planar LM should have two objects (A,M).... a unique map (A^n,M)-->M and a unique map (A^n)--->A, for all n, and no other maps (except identities)
@JonBeardsley if you're wanting to index over finite ordered sets and order-preserving maps, this is roughly what all that stuff about "approximations" does (doing things for "ordered sets and ordered maps" is enough instead of doing it for "finite based sets" and "maps with a linear ordering on fibers"). for algebras this is 4.1.2.10 and for algebra-module pairs this happens somewhere around ... 4.2.2.8?
this is why sometimes LMod and sometimes Δ-LMod shows up
@TylerLawson right right but that wouldn't work for symmetric stuff, I more mean that Lurie defines colored operads as having maps {X_i}_{i\in I}-->X for every finite set I, but if you look at the construction passing from O to O^\otimes, he only bothers with the ones where I is {1,...,n}
I definitely don't want to work over order preserving maps.
In other words, if you look at the classical literature, the multimaps always go (X_1,...,X_n)-->X, rather than this more general indexing over every finite set.
It's a really dumb point, but it seems unnecessary to use this language at all.
the general framework, because it allows symmetries, works with the latter, and you need the "approximation" stuff to show that the former is equivalent in the nonsymmetric case
But I'm not sure I follow, b/c if you look at, say, the definition of colored operad in this paper of Berger-Moerdijk, arxiv.org/pdf/math/0512576v2.pdf, (on page 3) they index over the natural numbers.
And they add in the symmetric group action as additional data.
So it seems like you could do the same thing (basically) by indexing over the category of sets {1,...,n} and ALL functions between them.
Rather than all finite sets and all functions between them.
I.e. have multimap sets Mul({X_i}_{1<=i<=n},X) for all n.
I've been looking at Evens' thesis, here: dspace.mit.edu/bitstream/handle/1721.1/80453/… There is an intriguing paragraph on page 55. One thing that it says is that "Adams operations provide a way of picking out the actual vector bundles from the virtual vector bundles in K-theory. " How does that work?
@JonBeardsley I think that idea really goes back to the "category of operators" of May and Thomason. Lurie's definition of infinity-operad is an infinity-categorical version of that.
Do we have an analog of J homomorphism for higher K theory or something related to them? When thinking about K(1) case I realize that J is something way more powerful than the Hurewicz map in detecting elements, so if we have such a thing our life might be way easier.
Lurie says in 4.1.3.6 that if you take the subcategory of cofibrant objects of a monoidal model category that it "inherits" a monoidal structure. Is the tensor product of cofibrant objects usually cofibrant again?
@RuneHaugseng do you know what goes wrong with identifying, say, Ass objects in this simplicial operad by actually looking at maps from, like, the Ass colored operad into it?
Ah, right so this was my other question, suppose I want to go {algebras in C}-->{algebras in N(C)}, Lurie seems to only say there's a problem going the other way?
Yeah, I mean, it seems obvious, take an algebra in C, look at the endomorphism operad, End(C), you should have a map Ass-->End(C), giving an algebra the underyling monoidal quasicategory of C
