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created Amitsur complex
In modern generality, an Amitsur complex is associated to any coring with grouplike element, as explained in semifree dga. I suppose that Amitsur has treated mainly noncommutative rings: neither $R$ nor $A$ in Amitsur complex need be commutative.
Are you telling me, or are you warming up for editing the entry?
I am just reminding you in the case you are not aware or forgot of the entry.
On the other hand, two yes, as I will work on this entry relatively soon (within 3-4 weeks) – I should resume my work with Tomasz on noncommutative principal bundles (in fact, special case of cleft Hopf-Galois extensions) over corings; I figured out a remaining obstacle recently and should rework the old ideas in the light of it. I however have a referee report for another paper to answer to and prepare for the Luxembourg conference before that (and 3-4 more things with deadline this or next week).
added at Amitsur complex the proof of the exactness in low degree.
I’m a little confused about some of the notation in this entry. In particular, it claims that the Amitsur complex is the bar resolution associated to a monad. In my experience (and perhaps this is just linguistic confusion), we can associate a bar resolution to a comonad. For instance, for a group G, there is a comonad on G-mod associated to forgetting the G-module structure and then tensoring over Z with Z[G]. The usual bar resolution of a group is the simplicial construction associated to this comonad. In other words, the classical bar construction on G results from taking G as a G-module and constructing the cosimplicial resolution associated to this comonad.
As I’m writing this, I’m starting to think we’re talking about the same thing, just in slightly different ways. For instance, given a comonad S:C–>C, we can construct a bar resolution of an arbitrary element of C. On the other hand, if we have a monad T:D–>D such that C is the category of T-algebras, then the bar construction in nlab terms, which requires a T-algebra to operate, is precisely the simplicial resolution we’ve described above. But this is perhaps a little confusing, since these ideas only really coincide in the case that we’re starting with a monadic functor (I think??).
Hi Jon,
on the one hand, it seems to me that the terminology on the $n$Lab is really standard, on the other hand the entry on “bar construction” could well deserve to be expanded a good bit and might want to have discussion of these points that you mention.
But likely I am missing your point. Maybe you could write down more details of what you are thinking of. (Here or in an $n$Lab entry.)
That’s a good point. I mean, yeah I think it may be true that that description of the bar construction is standard. It also seems to align with Lurie’s notion in chapter 5 of Higher Algebra, so that’s probably good. My main point is that from a comonad, one naturally obtains a simplicial resolution, and from a monad one naturally obtains a cosimplicial resolution. Neither of these take in as data an algebra or coalgebra structure. I’d like to add some things to that nlab entry, especially a bit about a general cobar construction, from a categorical point of view (i.e. less specific than bar and cobar construction). I’ll post something when I do..
At any rate, I believe your reading is correct, that the bar resolution of a monad is (by definition) the simplicial object associated with the comonad on the category of algebras. Actually there is more than one relevant construction here, e.g.,
The induced simplicial object $\Delta^{op} \to [Alg, Alg]$ where the receiver is the endofunctor category on the category of algebras, or
The induced simplicial object $\Delta^{op} \to Alg$ obtained by evaluating the preceding endofunctor-valued simplicial thingy at a given algebra $A$, producing a simplicial algebra (often singling out the case where $A$ is terminal or something similar), or
The simplicial object valued in the base category obtained by applying the forgetful (monadic) functor to the preceding construction; it is this construction which has a contracting homotopy involving decalage valued in the category.
Then of course there is the extremely useful two-sided bar construction, which involves two actions of the monad, one left and one right.
Alright, I made canonical resolution. Please edit or modify as desired. As Todd indicates, there’s an enormous amount one could write on this immensely important idea.
I should mention that a reference that I can’t really put on there is a talk I had with Emily Riehl the other day, where she clarified a number of these more nagging ideas for me.
I can’t tell from what you’ve written whether you’ve absorbed the precise sense in which the bar construction does provide an acyclic simplicial resolution of an object equipped with a $T$-algebra structure. This is for any category $C$ and any monad thereon. I wrote a blog post for the n-Category Café which describes how this works and the sense in which the bar construction is a universal resolution (see bar construction for a link to the blog post).
Sorry Todd. Yeah I think maybe I wasn’t thinking. I had some anxiety about saying it provides an acyclic resolution because I was thinking that I really needed a functor to some additive category to talk about not having higher cohomology groups. But I wasn’t thinking that the extra codegeneracy from the algebra structure is actually a contracting homotopy for the cosimplicial object in a real sense. Is that the right way to think about?
The right way to think about it is essentially that it is the unit of the monad which provides a contracting homotopy. That is, if $T^n A$ is a typical component of the bar construction (this would be in dimension $n-1$, where $T^0 A = A$ is the augmented component in dimension $-1$), then the homotopy takes the form
$u T^n A : T^n A \to T^{n+1} A$where $u: Id \to T$ is the unit, and this can be seen to contract the bar resolution as a simplicial object down to the constant simplicial object at $A$. But it ought to be noted that this homotopy lives not at the level of simplicial algebras $\Delta^{op} \to Alg$ (inasmuch as $u$ isn’t an algebra map of course), but at the level of simplicial objects valued in whatever underlying category the monad acts on. A key word here is decalage.
I’m in the process of writing this stuff up, as I didn’t know how to do diagrams well at the Café when I wrote that blog post.
Jon, thanks for starting canonical resolution!
Todd, looking forward to more of your writeup!
Just one comment re #12, #13: of course whether some map is a resolution depends on the given choice of concept of weak equivalences and, depending on taste and context, also on a choice of concept of (co-)fibration, hence typically on the given choice of model structure on the category of simplicial objects. This might deserve some more discussion.
Okay, I have been adding to bar construction, to flesh out these remarks. I’m not done yet though.
Yes, you’re right Urs (and you too, Jon), and I should not have expressed myself quite so, how shall I say, apodictically in #12. But there is a very robust notion of acyclicity that I am working with here – I introduce a kind of nonce expression ’acyclic structure’ – namely a coalgebra over the decalage functor. It is robust in the sense that if a simplicial object $X: \Delta^{op} \to C$ carries such a coalgebra structure, then so does $G X$ for any functor $G: C \to C'$. For example, even if there is no given model structure on the category $[\Delta^{op}, C]$, under any functor $G: C \to Set$ the simplicial set $G X$ is a decalage coalgebra and is thus automatically acyclic in the standard model structure on simplicial sets (where fibrations are maps satisfying the horn-filler conditions, etc.). So that’s what I should have said earlier, to be clear.
Thanks!
I have made “monoidal functor” into “strong monoidal functor”, to stick with the convention at monoidal functor. Okay?
Absolutely, Urs; thanks.
So I’ve put up more material at bar construction, up to stating the universal property of the bar construction (which I’ve not seen elsewhere in print, except for that blog). At some point I’ll write up the proof (which can also be found here).
May I point out my note The universal simplicial bundle is a simplicial group? Possibly relevant here.
David, please by all means edit in reference to your paper in whatever way you deem best!
I have finished inserting a very detailed proof of the stated universal property of the bar construction. (Too detailed? If so, I can port a chunk of it over to my personal web and link to it there.)
Thanks, impressive!
I would say the proof may easily stay where it is.
Having hit the ’all pages’ tab by mistake, I was surprised to find an entry beginning with ’A’ first when there are non-alphabetic entries to proceed it. It turns out someone set up +Amitsur-Martindale ring of quotient with a ’+’ in front. It also turns out to be empty. Scrap it, or rename it and add something?
Since it’s a thing, no need to scrap it. But I added a pointer to eom, and made some redirects, now at Martindale ring of quotients
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