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Stream: deprecated: algebraic geometry

Topic: Morita equivalence +

David Michael Roberts (May 15 2020 at 02:50):

OK, so I might repeat some stuff here that I said on Twitter, updated to take into account something I read after and that simplifies things.

David Michael Roberts (May 15 2020 at 02:52):

I claim that R and S are Morita equivalent if there is a third ring T, injective rng (yes, no i) morphisms f:R -> T <- S:g and projections p,q in T (so qq=q and pp=p) with im(f) = pRp and im(g) = qTq, and such that TpT = T and TqT = T, where TpT is the subring of T generated by things of the form t1.p.t2 etc.

David Michael Roberts (May 15 2020 at 02:53):

In the similar context of C*-algebras, one can even take q = 1-p, and I think this is true here, too.

David Michael Roberts (May 15 2020 at 02:53):

Then it should follow that pTq and qTp are the bimodules that give you the usual Morita equivalence.

John Baez (May 15 2020 at 02:54):

Is that an "if" or an "iff"?

David Michael Roberts (May 15 2020 at 02:54):

if and only if, I mean

David Michael Roberts (May 15 2020 at 02:56):

I have a slow-burning project that aims to write the 2-category of module categories (i.e. of the form RMod, these can be defined axiomatically, I believe) as a bicategorical localisation of the 2-category of rings, homomorphisms and intertwiners.

David Michael Roberts (May 15 2020 at 02:56):

But also with variations, such as for $*$-algebras with only approximate units, $C^*$-algebras and so on. (If I could get all the way to von Neumann algebras I would be very happy)

David Michael Roberts (May 15 2020 at 02:59):

But, I also want to concretely realise these localisations as bicategories of (left) fractions, hence this cospan picture as mentioned above, where now one should write a (suitable) functor $RMod \to SMod$ as a cospan $R \to Q \hookleftarrow S$ where $S \simeq pQp$ ($p$ a projection in $Q$, $QpQ = Q$) and $R\to Q$ is an arbitrary rng map. --- If this is possible

John Baez (May 15 2020 at 02:59):

Nice! Did you see my "conjecture" above? This sort of thing should interact with your program somehow.

David Michael Roberts (May 15 2020 at 03:06):

Yeah, that looks cool. I wouldn't be surprised. I was not thinking of keeping the information about the special object A, though. That might make a difference.

I should have said, my cospan R -> Q <- S consists of rings and rng maps, so there's something "special" going on. This is why I think I can push this stuff away from unital algebras/rings, but not too far, the Morita theory gets really weird in the general case and I think it's not entirely known. There's also an interesting Kleisli category floating around, since one needs maps of the form R -> M(Q) in the approximate unital case, where M is the relative monad that returns the multiplier algebra with a certain canonical topology.

David Michael Roberts (May 15 2020 at 03:07):

Anyway, I'm supposed to be writing an assignment and exam questions. First-year open book home exams for calculus, not pleasant to plan for!

David Michael Roberts (May 15 2020 at 07:10):

Ironically, I married a Breen.

Morgan Rogers (he/him) (May 15 2020 at 10:37):

David Michael Roberts said:

I claim that R and S are Morita equivalent if there is a third ring T, injective rng (yes, no i) morphisms f:R -> T <- S:g and projections p,q in T (so qq=q and pp=p) with im(f) = pRp and im(g) = qTq, and such that TpT = T and TqT = T, where TpT is the subring of T generated by things of the form t1.p.t2 etc.

Is there a reason why you call $p,q$ projections rather than idempotents? I suppose projections are always idempotents, maybe it's just a matter of personal preference.
I know it's shameful self-promotion, but (surprisingly or not) this looks somewhat like the characterisation of Morita-equivalence for monoids, see Corollary 6.6 here; it amounts to extracting an elementary characterisation of conditions for two monoids to have equivalent idempotent-completions. I wasn't the first person to arrive at the elementary condition, but the topos theoretic route I took brought me to a result that I consider more interesting: Theorem 6.5,

Let $\mathfrak{TOP}^∗_{\mathrm{ess}}$ be the 2-category whose objects are Grothendieck toposes having an essential surjective point, whose morphisms are essential geometric morphisms, and whose 2-cells are geometric transformations (natural transformations between the inverse image functors).
Theorem 6.5. The functor $M \mapsto [M^{\mathrm{op}}, \mathbf{Set}]$ is a 2-equivalence from $\mathbf{Mon}^{\mathrm{co}}_{s}$ to $\mathfrak{TOP}^∗_{\mathrm{ess}}$, [where $\mathbf{Mon}_{s}$ is the 2-category of monoids, semigroup homomorphisms (not necessarily preserving the identity), and conjugations are what you might guess them to be].

Morgan Rogers (he/him) (May 15 2020 at 10:44):

David Michael Roberts said:

I should have said, my cospan R -> Q <- S consists of rings and rng maps, so there's something "special" going on. This is why I think I can push this stuff away from unital algebras/rings, but not too far, the Morita theory gets really weird in the general case and I think it's not entirely known.

This might seem like a stupid question, but what actually changes when you pass to the non-unital algebras/rings? Surely for a rng, $RMod \simeq R_1Mod$, where $R_1$ is the ring obtained from $R$ by freely adding a unit? Since the unit is required to act by the identity on all modules, I mean?

Jens Hemelaer (May 15 2020 at 11:11):

Morgan Rogers said:

This might seem like a stupid question, but what actually changes when you pass to the non-unital algebras/rings? Surely for a rng, $RMod \simeq R_1Mod$, where $R_1$ is the ring obtained from $R$ by freely adding a unit? Since the unit is required to act by the identity on all modules, I mean?

Yes, $R-Mod \simeq R_1-Mod$ (similar as in the semigroup/monoid case). So if you only care about the category of modules, then unital rings are more general than non-unital rings!
IIRC, non-unital rings correspond to the unital rings $R$ such that the inclusion of the unit $\mathbb{Z} \subseteq R$ admits a retraction. (These are usually called "augmented" rings).

Jens Hemelaer (May 15 2020 at 11:27):

David Michael Roberts said:

I have a slow-burning project that aims to write the 2-category of module categories (i.e. of the form RMod, these can be defined axiomatically, I believe) as a bicategorical localisation of the 2-category of rings, homomorphisms and intertwiners.

Are you familiar with the work of Julia Ramos González (one my colleagues in Antwerp)? https://arxiv.org/abs/1707.07453
Since rings R are linear sites (with one object) and RMod is the category of presheaves on it, it seems related.
I'm not very fluent in 2-category theory, so the 2-categorical localisation that you are looking for might be a different one than the one she used.

Morgan Rogers (he/him) (May 15 2020 at 12:47):

Rongmin Lu said:

Morgan Rogers said:

This might seem like a stupid question, but what actually changes when you pass to the non-unital algebras/rings?

See the conversation here about non-unitality in the context of C*-algebras.

Aren't these just examples of algebras which are non-unital? Morita-equivalence via modules in the sense above is purely algebraic, so again, why can't we just add a unit for the purposes of Morita-equivalence?

Morgan Rogers (he/him) (May 15 2020 at 12:47):

If you want topological modules, that's a different story, though.

Morgan Rogers (he/him) (May 15 2020 at 13:34):

Ah, a different problem, then. There are many kinds of Morita equivalence. But... at risk of taking us on a tangent, what do we lose by adding the identity as an isolated point, even in that context?

David Michael Roberts (May 15 2020 at 13:47):

In the operator algebra world one wants to keep the relation between commutative $C^*$-algebras and locally compact Hausdorff spaces. The correct maps are not even algebra maps, but maps in a certain Kleisli category, and they certainly don't preserve the unit even when they do arise from algebra maps. The prototypical example for me in the purely algebraic world are groupoid (or, even category) algebras. It's late here, so I can't recall offhand an argument I think I once game to @Rongmin Lu about why free unit addition is no good (something about one-point compactifications and functoriality). In the non-unital algebraic setting, you can get maps that are really of the form R --> M(S), where M denotes the multiplier algebra. Anyway, rambling now. The augmentations would change the picture, I don't know how that would go.

Morgan Rogers (he/him) (May 15 2020 at 14:00):

But I thought the whole point in Morita equivalence problems is that you care about equivalences between categories of actions of things, rather than the things themselves. So the "correct" morphisms are any class which induce functors between the categories of actions and which are sufficiently expressive to capture all equivalences between these categories. This is why we use bimodules instead of homomorphisms in the basic ring case, for example.
So if augmentation doesn't change the category of actions, it makes sense to do it; and if it does, I have more questions :upside_down:

David Michael Roberts (May 15 2020 at 23:10):

The thing is, if I write down something involving cospans of rings, I get to use formal theorems about constructing a model for the localisation of a 2-category. Also, I care about size issues when doing so, and I know this way works without recourse to universes. I also care about geometry, so not just individual rings, or algebras, but bundles of them, or even rings internal to a topos or similar. And also using the same formal localisation theorem for these settings, too.

David Michael Roberts (May 15 2020 at 23:15):

I guess that ultimately the aim: I want people like classical differential geometers and string theorists to understand and be able to use this technology without having to learn about $\infty$-categories.

David Michael Roberts (May 16 2020 at 06:13):

@Rongmin Lu I was responding to this comment:

So the "correct" morphisms are any class which induce functors between the categories of actions and which are sufficiently expressive to capture all equivalences between these categories. This is why we use bimodules instead of homomorphisms in the basic ring case, for example.

I was promoting the use of something other than bimodules that also captures all equivalences, and even (conjecturally) all functors.

Jens Hemelaer (May 16 2020 at 07:57):

For $C^*$-algebras, there is a "trick" to make sure that adding a unit does not give the same category of modules:
You exclude modules $M\neq 0$ of the $C^*$-algebra $A$ such that $m\cdot a = 0$ for all $m \in M,~a \in A$.
This property is automatically satisfied whenever $A$ is unital, so in this case nothing changes.
If you restrict to modules that are 1-dimensional over $\mathbb{C}$, you get the spectrum of the $C^*$ algebra, which is compact if and only if $A$ has a unit.

This "trick" is maybe standard in the theory of $C^*$-algebras? In ring theory this trick is not used, so there adding a unit doesn't change the category of modules.

Morgan Rogers (he/him) (May 16 2020 at 11:33):

See, this is the problem with calling all problems of this form "Morita equivalence". Different people are talking about different categories, which make the answers different. There's a similar problem in Morita equivalence for semigroups: Funk et al. studied some notions of Morita equivalence for inverse semigroups that is distinct from the one in my paper (if we freely add an identity element, it amounts to equivalence between maximal subtoposes). At what point can we start renaming these different equivalences after people in the respective fields instead of after Morita who mostly played with rings?

Reid Barton (May 16 2020 at 11:46):

Well, these terms are different "parts of speech": a Morita equivalence between two k-algebras is a certain kind of bimodule (or equivalence of categories), but a "Tannaka duality between two k-algebras" doesn't make sense (or at least doesn't mean that).

Morgan Rogers (he/him) (May 16 2020 at 12:09):

But these aren't the same thing. I can ignore the topology on a C*-algebra, and ask about its modules as a plain old ring. Then "Morita equivalence" refers to something else; that's the original source of my confusion in this thread, in fact! I would go so far as to argue that using the same name for all instances of this type of problem trivialises the breadth of the category theory involved.

Morgan Rogers (he/him) (May 16 2020 at 12:19):

Rongmin Lu said:

Morgan Rogers said:

I can ignore the topology on a C*-algebra, and ask about its modules as a plain old ring.

Yeah, but then you'll be working in the category of rings.

If you insist, how about Morita equivalence of groups? Without further qualification, I could be talking about the action of groups on sets, on vector spaces, or the modules of the group algebra over a field...

Morgan Rogers (he/him) (May 16 2020 at 12:24):

Look, I'm not against calling this class of problems "Morita equivalences", I'm just saying there should be more distinctive names for the subclasses of the problem that "Morita equivalence of Xs" which is typically ambiguous. Like your distinction between Tannaka duality and Tannaka-Krein duality earlier.

Morgan Rogers (he/him) (May 16 2020 at 12:36):

So we could name these results after whoever actually produces them in the relevant domain, that works for me! It's a little unfair to credit them all to Morita, after all :stuck_out_tongue_wink:

Morgan Rogers (he/him) (May 16 2020 at 12:42):

But as I've already pointed out: I already didn't know what you were talking about re Morita equivalence of $C*$-algebras, and if I already knew enough about C*-algebras to guess which categories of representations were involved, I would also have had plenty of opportunity to be exposed to the conventional name (supposing it existed). Anyway, point taken!

David Michael Roberts (May 17 2020 at 02:16):

"hard Lefschetz" for matroids, "RIemann–Roch" for schemes, "Poincaré duality complex" in algebraic topology, "Pfaffian" in index theory, etc. Even good old "Euclidean" space would be unrecognisable to Euclid.
Not to mention Peano arithmetic. when he was essentially the third person to propose them, and his was the second-order version, when logicians always take the first-order version :-P

Morgan Rogers (he/him) (May 20 2020 at 15:49):

I wrote my first blog post outlining how I think about Morita equivalence. :grinning_face_with_smiling_eyes:

Morgan Rogers (he/him) (May 21 2020 at 09:21):

I wasn't even aware of that terminology! I really hoped it would be more interesting, but there we go. Thanks for reading :heart: