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Just added a references to the entry Morita equivalence. Noticed that the entry is in a woeful state.
Can’t edit right now, but hereby I move some ancient query boxes that were sitting there from there to here:
— forwarded query boxes:
+–{: .query} David Roberts: More precisely, a Morita morphism is a span of Lie groupoids such that the 'source leg' has an anafunctor pseudoinverse. Anafunctors are only examples of Morita morphisms, in the sense that open covers $\coprod U_i \to M$ are examples of surjective submersions.
I’m also not sure that this should be called the folk model structure, as I don’t think it exists for groupoids internal to $Diff$. Details of the model structure are in a paper by Everaert, Kieboom and van der Linden, but seem to be tailored towards groupoids internal to categories of algebraic things (e.g semiabelian categories). I think the best one can do is a category of fibrant objects, but that is not something I’ve looked at much.
Toby: For me, an anafunctor involves a surjective submersion rather than an open cover, which is how that got in there. The important thing is to have equivalent hom-categories.
David Roberts: I’m not what I was thinking at the time, but you are pretty much right: the definition of anafunctors depends on a choice of a subcanonical singleton Grothendieck pretopology, so it was remiss of me to demand the use of open covers :) As for the definition of Morita morphism, I now can’t remember if that referred to the span which is an arrow in the localised 2-category or the arrow in the unlocalised 2-category that is sent to an equivalence under localisation. At least for me, the terminology Morita morphism evokes the generalisation from a Morita equivalence (a span of weak equivalences) to a more general morphism in that setting.
Toby: I think that a Morita morphism should be a span, although now I'm not sure that this is what the text says, is it? I should check a reference and then change it.
David Roberts: I’m fairly sure I’ve heard Lie groupoid people (Ping Xu springs immediately to mind) speak about a Morita morphism as being a fully faithful, essentially surjective (in the appropriate sense) internal functor, but I disagree with their usage. If this is indeed the case, we could note the terminological discrepancies =–
+– {: .query}
So is it true that there is a model category structure on algebras such that Morita equivalences of algebras are spans of acyclic fibrations with respect to that structure?
Zoran Škoda: Associative (nonunital) algebras make a semi-abelian category, ins’t it ? So one could then apply the general results of van den Linden published in TAC to get such a result, using regular epimorphism pretopology, it seems to me. It is probably known to the experts in this or another form.
=–
Is there an abstract general account of Morita equivalence? One which would take up Morita equivalence as an aspect of a 3-groupoid (TWF209), as well as Lurie’s claim “The notion of Morita equivalence is most naturally formulated in the language of classifying ∞-topoi” (DAG V).
I see there are also suggestions of Morita equivalence accounting for some string dualities, such as here.
Comment #2 was inspired by a look at Olivia Caramello’s work. It’s got me wondering whether after all there might be some way of thinking of duality as used in mathematics and as used in physics as belonging together. Remember the discussion on the latter, beginning around here.
If physics ’dualities’ could be thought of as a kind of Morita equivalence, it might explain why there isn’t generally a single dual. Just as an associative algebra over a commutative ring can have many Morita equivalents, so a string background can have many T-duals.
As you can only recover the ring, $R$, up to Morita equivalence by Tannakian construction from $R-Mod$ without the underlying functor, presumably you only reconstruct theories up to similar equivalence in Gabriel–Ulmer duality cases.
I guess Caramello’s comment provides a connection here
Two associative rings with unit are Morita-equivalent (in the classical, ring-theoretic, sense) if and only if the algebraic theories axiomatizing the (left) modules over them are Morita-equivalent (in the topos-theoretic sense). In fact, two rings are Morita-equivalent if and only if the cartesian syntactic categories of these theories are equivalent.
Is there something parallel happening between
I guess I should follow up her further comment:
Categorical dualities or equivalences between ’concrete’ categories can often be seen as arising from the process of ’functorializing’ Morita-equivalences which express structural relationships between each pair of objects corresponding to each other under the given duality or equivalence.
I see also there are more people looking at T-duality and Morita, e.g., Szabo’s slides
Open string T-duality as categorical KK-equivalence (refines/generalizes Morita equivalence)
Gabriel–Ulmer duality lets you reconstruct cartesian theories up to equivalence. But that’s because cartesian theories are equivalent if and only if they are Morita equivalent. (Here, I am thinking of cartesian theories as small categories with finite limits, rather than special theories in first-order logic.)
At the webpage for the conference I attended on dualities in physics:
Nic Teh, “‘Duality’ as an inter-theoretical equivalence between Lagrangian and Hamiltonian mechanics”:
“Duality” is often described as a form of inter-theoretical equivalence. In this talk (based on joint work with Dimitris Tsementzis), I will discuss the case of Hamiltonian and Lagrangian mechanics, and whether (in the hyper-regular scenario) one can formulate an equivalence between these theories. We shall see that this is indeed the case, based on a “symplecticization” of the relevant theories, which leads in turn to a notion of “Morita equivalence” (and indeed inter-definability). I will then compare the framework for this intertheoretic equivalence with some more sophisticated dualities in contemporary physics.
Morita equivalence is directly analogous to duality in quantum field theory in that in both cases one is dealing with different presentations of a single structure thus presented.
Where in duality in physics by the discussion which we enterd there we are looking at epimorphisms
$LagrangianData \stackrel{quantization}{\longrightarrow} LagrangianQFT$and find the dualities as the 1-cells in the Cech nerve of this map, for Morita equivalence of algebras this is simply replaced by
$Algebra \stackrel{Mod_{(-)}}{\longrightarrow} LinearCategories$or for morita equivalences of sites (which is what Olivia Caramello studies) this is replaced by
$Sites \stackrel{Sh(-)}{\longrightarrow} Toposes \,.$Presumably there are homotopified versions of the last two, such as
$(\infty,1)-Sites \stackrel{Sh(-)}{\longrightarrow} (\infty,1)-Toposes \,.$I’m still wondering whether in the article I wrote I should have stressed the difference between typical concrete dualities and these ones in physics. If, as Caramello writes,
Categorical dualities or equivalences between ’concrete’ categories can often be seen as arising from the process of ’functorializing’ Morita-equivalences which express structural relationships between each pair of objects corresponding to each other under the given duality or equivalence,
can’t we bundle everything into one simple story of generalized equivalence relations? Is it that classical dualities are a very special kind of Morita equivalence?
Yes, all this applies to $(k,l)$-category theory for any values of $(k,l)$.
Regarding the question: I agree that Morita equivalence is a good analogy for duality in QFT, via the reason in #6. But it seems sufficiently different from the discussion of “abstract and concrete duality”. Of course at the bottom of it you may argue that one is dealing with equivalences of categories, one way or other. But as in #6, I would say that Morita equivalence is about different presentations yielding equivalent presented-categories.
By the way, I am not sure where the above quote from Olivia is taken from, but I see (you will have long seen this yourself) that she has one page specifically on morphisms as sites yielding equivalences on toposes as Morita equivalences here.
Do we see higher equivalences in standard Morita equivalence situations, like the dualities between dualities we were looking for in the physics case?
How about in the situation of Morita equivalence of theories, which Caramello studies? Could this detect, say, the non-trivial auto-equivalence of the theory of projective geometry?
I guess you’re talking about the auto-equivalence that exchanges points and lines in the projective plane. That’s just an ordinary Morita equivalence. To give a very much simplified example, consider the first-order theory with two sorts, no function or relation symbols, and no axioms. Then that theory has a non-trivial auto-equivalence exchanging the two sorts, and that’s certainly a Morita equivalence.
Ah, Ok. So how could higher equivalences appear in this or a related situation? Something like the isomorphism between $(A,B)$-bimodules of TWF209, where the $(A,B)$-bimodules are providing Morita equivalences for $A$ and $B$.
I see that John has a reference there to
Francis Borceux and Enrico Vitale, Azumaya categories, available at http://www.math.ucl.ac.be/AGEL/Azumaya_categories.pdf
These names appear in one of a couple of references I justed added to Morita equivalence:
Hans Porst, Generalized Morita Theories: The power of categorical algebra, (pdf)
Francis Borceux and Enrico Vitale, On the Notion of Bimodel for Functorial Semantics, Appl. Categorical Structures, 2:283–295, 1994 (pdf)
The latter is looking for a notion of ’bimodel’ to play an analogous role to bimodule in the Morita theory of rings. So maybe I’m after equivalences between bimodels.
Higher group versions of Brauer groups appear on the $n$Lab
in the context of higher algebra briefly at infinity-group of units – Relation to Picard- and Brauer infinity-group
in the context of derived algebraic geometry at Brauer group – Relation to derived étale cohomology.
Has anyone looked much at Morita equivalence in the context of doctrines and higher doctrines? I see Yanofsky’s Coherence, Homotopy and 2-Theories casts
Every monoidal category is tensor equivalent to a strict monoidal category
as a form of Morita equivalence.
Re Urs #8,
I agree that Morita equivalence is a good analogy for duality in QFT, via the reason in #6. But it seems sufficiently different from the discussion of “abstract and concrete duality”. Of course at the bottom of it you may argue that one is dealing with equivalences of categories, one way or other. But as in #6, I would say that Morita equivalence is about different presentations yielding equivalent presented-categories.
I see from that Schwede article that in the original 1958 paper, Morita does treat equivalences and dualities together.
Morita treats both contravariant equivalences (which he calls dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’)
For some more detail
A bimodule ${_R} U_S$ , will be called a Morita-duality module if it is a balanced injective cogenerator for both $R$-Mod and Mod-$S$. A left $R$-module $M$ is here called $U^{\circ}$-reflexive if the natural $R$-homomorphism $M \to Hom_S(Hom_R(M, U), U)$ is an isomorphism. When ${}_R U_S$, is a Morita-duality module, $Hom_R(\; , U)$ defines a duality between the $U^{\circ}$-reflexive left $R$-modules and the $U^{\circ}$-reflexive right $S$-modules and these subcategories are closed under submodules and epimorphic images. Conversely, if $\mathcal{C} \subseteq R-Mod$ and $\mathcal{D} \subseteq Mod-S$ are full subcategories closed under submodules and epimorphic images and containing ${_R}R$ and $S_{S}$, respectively, then any duality between $\mathcal{C}$ and $\mathcal{D}$ is naturally equivalent to $Hom( \;, U)$ for some Morita-duality module ${}_R U_S$. (J.M Zelmanowitza and W Jansen, Duality modules and Morita duality, Journal of Algebra, Volume 125, Issue 2, August 1989, Pages 257–277)
Re Urs #8 and David_Corfield #9, your quotes from my website are discussed and expanded in section 8.2 of my text available at the address
http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf ,
which forms the first two chapters of my forthcoming book (for Oxford U.P.). This treatement extends the discussion of Morita-equivalences given in the preprint “The unification of Mathematics via Topos Theory” and contains many other remarks on ways in which Morita-equivalences arise (and on their relationship with dualities or equivalences). For instance, I observe in that context that
“Categorical dualities or equivalences between ‘concrete’ categories can often be seen as arising from the process of ‘functorializing’ Morita-equivalences which express structural relationships between each pair of objects corresponding to each other under the given duality or equivalence. In fact, the theory of geometric morphisms of toposes provides various natural ways of ‘functorializing’ bunches of Morita-equivalences”.
This remark is illustrated for instance by my papers “A topos-theoretic approach to Stone-type dualities” and “Gelfand spectra and Wallman compactifications”.
A few other observations which are discussed in that text are:
“The notion of Morita-equivalences materializes in many situations the intuitive feeling of ‘looking at the same thing in different ways’, meaning, for instance, describing the same structure(s) in different languages or constructing a given object in different ways”.
“Different ways of looking at a given mathematical theory can often be formalized as Morita-equivalences”.
“A geometric theory alone generates an infinite number of Morita-equivalences, via its ‘internal dynamics’”.
It is also shown in that text that many notions of Morita-equivalence arising in Mathematics can be naturally formulated as Morita-equivalences between geometric theories (i.e., as equivalences of classifying toposes). Of course, it will be possible to extend all of this to the realm of higher classifying toposes once the notion of ‘higher geometric theory’ is identified.
Thanks. This discussion and these pointers should be added to the $n$Lab, probably right there at Morita equivalence. I don’t have any time right now, but maybe you or David have the energy?
Welcome to the nForum, Olivia. It’s something of a Pro-am affair, at least when I’m involved. I like to push the professionals to give me big pictures, which they sometimes kindly oblige me with.
Something I came across in my recent searches was
He maintains that Morita theory is essentially a bicategorical affair, which makes sense to me in view of the usual account of Morita-equivalence of rings. He goes on to say
we remind the reader of the bicategorical Yoneda Lemma (5.3) and explain that what is often called Morita theory is a corollary (5.4) of this Yoneda Lemma. We encourage the intuition that bicategorical Morita theory is as elementary as the bicategorical Yoneda Lemma.
Does the account there not fit perfectly with Borceux and Vitale’s On the Notion of Bimodel for Functorial Semantics? Their bimodels (p. 285) seem to me the 1-cell/bimodules of Johnson, mediating between categories of models.
Is that notion of bimodel interpreting a theory in the models of another theory used much? In a situation of duality, is that what the dualizing object (aka schizophenic object) amounts to?
In the context of toposes, one can think of the classifying topos as playing the role of a dualizing object between two Morita-equivalent theories $T_{1}$ and $T_{2}$ classified by it, since the Morita-equivalence provides two different representations for it, one in terms of $T_{1}$ and another in terms of $T_{2}$ (think of the syntactic construction of classifying toposes).
One can also see Morita-equivalence as a weak form of bi-interpretability (see my text for more details).
I think of “Morita equivalence” in general as meaning roughly “an equivalence in some bicategory whose objects also admit some stricter notion of equivalence”. (Often this can mean a double category, but I wouldn’t be dogmatic about that.) Rings have isomorphisms in addition to equivalences in $Mod$; theories have isomorphisms of theories in addition to equivalences of their classifying toposes.
Thanks. I guess I’m left wondering to what extent Morita equivalence is just (a subpart of) a generalised form of higher equivalence relation. That seems to be the gist of Urs #6, and maybe Mike #20.
Then what kind of constraint is imposed by taking the equivalences to be generated by topos equivalence. It seems that even the classical rings and modules case fits here, via the theory of $R$-modules for a ring $R$. Are there any Morita phenomena which cannot be cast in (higher) topos terms?
Is that because essentially one looks to something (co)-presheaves-like to generate the stricter equivalence? Is that why Yoneda is at the heart of things for Johnson (#18)?
But then do the physics cases fit this pattern, e.g.,
I still think that in order to be justified to speak of “Morita equivalence” it must be that we are talking about adding new equivalences between “presentations” induced from the evident equivalences between what they present. If one drops that condition then it becomes unclear to me why one should speak of “Morita equivalence” instead of just of “equivalence”.
What Mike has in #20 retains the idea that there are two kinds of equivalences, but if the stricter version of the equivalences is not that of presentations while the weaker one is that of the objects being thus presented, then I would hesitate to still speak of “Morita equivalence”.
On the other hand, I won’t be surprised if now you say that for every situation as in #20 there is a universal way to realize it as an instance of what I am after. But is there?‘
So with Johnson’s article, it seems to work that the equivalence between categories of models turns out to be a component of a natural equivalence between Homs. E.g.,an equivalence between $R-Mod$ and $S-Mod$ for rings $R$ and $S$ is the $\mathbb{Z}$-component of a natural equivalence in pseudofunctors of bicategories. $[\mathcal{M}, Cat]$, between $\mathcal{M}(R, -)$ and $\mathcal{M}(S, -)$, which through bicategorical Yoneda must arise from a bimodule in $\mathcal{M}(R,S)$.
$\mathcal{M}$ is the bicategory of rings, bimodules and bimodule morphisms.
@Urs #22 well, it probably depends on how wide a notion of “presentation” one is willing to admit…
Re Olivia #19, I’m looking at the introduction of bi-interpretation at the end of section 6.3 of your chapter. I can’t quite see, is there the idea, I mentioned above, of associating to an equivalence of categories of models what gets called a bimodel, i.e., a model of a theory within the models of another theory? One needs a pair of such bimodels, theorem 5.6 of this.
Re David #25: Yes, any bi-interpretation (or, more generally, any Morita-equivalence) between two geometric theories $\mathbb{T}$ and $\mathbb{ S}$ yields two bimodels (in Borceux-Vitale’s sense) ${\mathcal{C}}_{\mathbb{T}} \to \mathbf{Sh}({\mathcal{C}}_{\mathbb{S}}, J_{\mathbb{S}})$ and ${\mathcal{C}}_{\mathbb{S}} \to \mathbf{Sh}({\mathcal{C}}_{\mathbb{T}}, J_{\mathbb{T}})$ satisfying the hypotheses of Theorem 5.6 (where $({\mathcal{C}}_{\mathbb{T}}, J_{\mathbb{T}})$ and $({\mathcal{C}}_{\mathbb{S}}, J_{\mathbb{S}})$ are respectively the geometric syntactic sites of the theories $\mathbb{T}$ and $\mathbb{S}$). These bi-models are given by the compositions of the equivalence of classifying toposes $\mathbf{Sh}({\mathcal{C}}_{\mathbb{T}}, J_{\mathbb{T}})\simeq \mathbf{Sh}({\mathcal{C}}_{\mathbb{S}}, J_{\mathbb{S}})$ induced by the Morita-equivalence with the relevant associated sheaf functors and Yoneda embeddings. Notice that a Morita-equivalence between two geometric theories $\mathbb{T}$ and $\mathbb{S}$ yields a model of $\mathbb{T}$ (resp. of $\mathbb{S}$) inside the classifying topos of $\mathbb{S}$ (resp. of $\mathbb{T}$) rather than a model of $\mathbb{T}$ (resp. of $\mathbb{S}$) inside a category of $\mathbb{S}$-models (resp. of $\mathbb{T}$-models).
Thanks, Olivia.
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