## July 11, 2011

### Doctrinal and Tannakian Reconstruction

#### Posted by David Corfield

Interest in doctrines at the Café goes right back to one of its earliest posts nearly five years ago, and even to one a few days earlier. We’ve begun to record some material at the doctrine page in nLab, but I’m sure there’s more wisdom from the blog to be extracted. Looking over the material again raised a question or two in my mind, which I’d like to pose now.

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the recovery of a theory from the category of models of that theory.

There are a range of similar dualities:

1. Adámek, Lawvere and Rosicky established a similar biequivalence between the 2-category of small Cauchy complete categories with finite products, finite product preserving functors and natural transformations, and the 2-category of multisorted finitary varieties, algebraically exact functors and natural transformations in On the duality between varieties and algebraic theories.

2. Todd tells us here of the duality between the 2-category of Cauchy complete categories on one side, and the 2-category of presheaf toposes and bicontinuous functors on the other.

3. I think I have it right that there’s a biequivalence between the 2-category of finite product categories and the 2-category of categories with all limits and colimits and with functors which preserve limits, filtered colimits, and regular epimorphisms. [EDIT: this needs modifying.]

This talk of recovering something from its category of models put me in mind of Tannakian reconstruction of algebraic entities from their category of modules or comodules or whatever. So I was wondering where there is a way to see doctrinal, Gabriel-Ulmer type, duality and Tannaka duality as instances of something larger. The only reference I have found is Brian Day in Enriched Tannaka reconstruction writing

Our approach provides a synthesis of Tannaka reconstruction and Gabriel-Ulmer duality.

Unfortunately, I find that hard to understand.

John seems to see a link. In TWF200 he writes

fans of the Tannaka-Krein reconstruction theorem for quantum groups will recognize this duality between “theories and their category of models” as just another face of the duality between “algebras and their category of representations” - the classic example being the Fourier transform and inverse Fourier transform!

Daniel Schäppi discusses the semantics-structure adjunction in Tannaka duality for comonoids in cosmoi

The functor which sends a comonad to its category of comodules is traditionally known as the semantics functor. It has a partial left adjoint, which is called the structure functor (cf. [Dub70], [Str72]). The partial here means that the left adjoint of the semantics functor is only defined on a certain subcategory. These names go back to Lawvere (see [Law04, p. 77]). A monad can be viewed as a sort of logical theory, and from this viewpoint the semantics functor sends it to its category of models; the study of the models of a logical theory is generally called its semantics.

So, in the case of a fiber functor, say the functor from the category of representations of a group to their underlying vector spaces, Tannaka reconstruction allows us to recover the group from the enriched endomorphisms of the fiber functor. Now, is it possible that this could be described as in the case of doctrinal duality?

I see the Urs of five years ago wrote

So instead of admitting that I am talking about a linear representation of some group $G$, I could equivalently say that I am talking about a model in $Vect$ of the theory $\Sigma(G)$ with respect to the doctrine $Cat$ of no structure.

If $Rep_{C}(G) = Hom_{Cat}(\Sigma (G), C)$ is the category of representations of $G$ in $C$, is there a $C$ and a doctrine, $\mathcal{D}$, for which $Hom_{\mathcal{D}}(Rep_{C}(G), C)$ is equivalent to the category with single object the fiber functor and its invertible endomorphisms, or $\Sigma{G}$?

Mike said back here that $Cat^{op}$ should be something like

“cocomplete categories equipped with a strongly-generating set of tiny objects, and functors having both left and right adjoints (equivalently, by the adjoint functor theorem, preserving small limits and colimits) whose left adjoint preserves the chosen generators”. That’s obviously similar to “complete atomic Boolean algebras” except that we have to consider the “atoms” as structure, rather than a property. If you make it into a property and consider instead “cocomplete categories for which there exists a strongly-generating set of tiny objects, and functors having both left and right adjoints” then you get the subcategory $Cat_{cc}^{op}$ of Cauchy-complete categories.

Is it possible that this is the doctrine $\mathcal{D}$ I’m after?

A couple more questions then occur

1. Given that the groupoidification program seeks to avoid linearisation of representations via vector spaces in favour of groupoids, how do groupoidification and Tannakian reconstruction fit together?

2. I first met fiber functors when reading about reconstructing the fundamental groupoid of a space from its category (topos) of sheaves. Is there a doctrinal dual to Toposes which would allow for a duality for this reconstruction?

You might then think about collecting doctrines together in a 3-category. In the doctrine page of John’s web, it says

Categorified Gabriel-Ulmer duality is all about recovering the syntactic doctrine from the semantic doctrine,

with a view to taking models of the syntactic doctrine in the 2-category of groupoids. But few details are given there.

I wonder if there is something like the Pontryagin duality for locally compact Hausdorff abelian groups, with its induced duality between properties such as discrete and compact or Lie and finite rank? Is there a duality for doctrines, which restricts to one between limit doctrines and some other kind, or to one between syntactic and semantic doctrines?

Posted at July 11, 2011 11:01 AM UTC

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### Re: Doctrinal and Tannakian Reconstruction

I see Jim Dolan uttering ‘doctrine’ and ‘Tannaka-Krein’ in the same breath, or at least same comment.

Hmm,

the finite-dimensional representations of gl(n) form the free symmetric monoidal finitely cocomplete algebroid on one “n-dimensional” object.

And he talks earlier of

the “doctrine” of symmetric monoidal finitely cocomplete algebroids.

Why the scare quotes?

Posted by: David Corfield on July 11, 2011 12:27 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

David wrote:

Why the scare quotes?

Jim Dolan really hates it when I speak for him. But here I go again:

Maybe the square quotes are there because Jim hadn’t in that comment given his favorite definition of ‘doctrine’, which differs from Lawvere’s original definition of ‘doctrine’ as a monad on the category Cat, and also differs from the later definition of ‘doctrine’ as a pseudomonad on the 2-category Cat. I imagine he was wanting to warn us that the best definition is somewhat up for grabs.

Posted by: John Baez on July 15, 2011 7:10 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So many paths in this post lead to Jim. There are bits and pieces spread over various webs. Besides your doctrines, there are Jim’s Algebraic Geometry, Todd’s Notes on conversations with James Dolan in Todd Trimble, and Alex’s Doctrines in Alex Hoffnung. I see also Jeffrey Morton comments on groupoidification and Tannaka reconstruction.

Posted by: David Corfield on July 15, 2011 10:37 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Geoffrey Morton replies to a question of mine on groupoidification and reconstruction.

Posted by: David Corfield on July 20, 2011 1:51 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Hi David,

by what I said in the other reply, all that is necessary to make bare Tannaka reconstruction work is the Yoneda lemma.

So if you want Tannaka reconstruction for representations in $Span(Grpd)$, what you need is a Yoneda-lemma in $Span(Grpd)$-enriched (higher) category theory.

That shouldn’t be a problem.

Posted by: Urs Schreiber on July 20, 2011 8:50 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I’d forgotten about the categorified universal properties of HDA II. E.g.,

$Rep(U(n))$ is the free connected symmetric 2-$H^{\ast}$-algebra on an even object $x$ of dimension $n$.

Posted by: David Corfield on July 19, 2011 10:13 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Here’s something I’d like to understand. Given a finite limit category $A$, we get a new category $FinLim(A, Set)$. Gabriel-Ulmer duality says (among other things) that we can recover $A$ up to equivalence from $FinLim(A, Set)$: it’s the full subcategory of finitely presentable objects, right?

Now take a finite product category $A$. We get a new category $FinProd(A, Set)$. Can we recover $A$ up to equivalence from $FinProd(A, Set)$?

A closely related question: can we recover a Lawvere theory from its category of models (in $Set$)? Of course we can do so if we’re also given the forgetful functor from the category of models to $Set$, but that’s not what I’m asking.

Or to put it another way: can different Lawvere theories have the same category of models?

Posted by: Tom Leinster on July 11, 2011 3:00 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Can we recover $A$ up to equivalence from $FinProd(A,Set)$?

I’ll wait for the experts, but isn’t that what comes about with my example 3, i.e, if you take morphisms in the 2-category of categories with all limits and colimits and with functors which preserve limits, filtered colimits, and regular epimorphisms? That’s what I gleaned.

There was also some interesting material about fixing a concept and looking at how the theory changes in different doctrines beginning with Todd here.

Posted by: David Corfield on July 11, 2011 3:16 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I have an answer to that. Yes! The Lawvere theory given by powers of a 2-element set (as a subcategory of $Fin$) has the same models as the Lawvere theory of powers of a 3-element set. For they both have the same Cauchy completion (cf. the discussion here).

Of course, finite limit theories escape this fate because finitely complete categories are Cauchy complete.

Posted by: Todd Trimble on July 11, 2011 3:20 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

And that Adamek et al. paper says

If a variety $\mathcal{V}$ is viewed as an abstract category, i.e., without reference to underlying sets and mappings, algebraic theories turn out not to be entirely satisfactory: every non-trivial variety has many (non-equivalent) algebraic theories. For example the variety $Set$ of algebras with no operations is equivalent to $Mod \mathcal{T}$ where $\mathcal{T}$ is the full subcategory of $Set^{op}$ whose objects are all natural numbers. But it is also equivalent to $Mod \mathcal{T}'$ where $\mathcal{T}'$ is the full subcategory of $Set^{op}$ whose objects are just the even numbers.

The paper proposes a “remedy”.

Posted by: David Corfield on July 11, 2011 3:25 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Just to be clear, my “yes!” was in reply to Tom’s final question. It implies a “no!” to the other questions: Tannaka reconstruction doesn’t generally hold for finite product theories.

Posted by: Todd Trimble on July 11, 2011 3:27 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

But it’s right to say, isn’t it Todd, that a finite product theory can be recovered from the category of its models in sets, by taking set-valued functors that preserve all limits, filtered colimits, and regular epimorphisms?

Posted by: David Corfield on July 11, 2011 3:37 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Oh, I think I misread Forssell’s thesis and this only holds for equational (or algebraic) theories, Section 1.2.2.

Posted by: David Corfield on July 11, 2011 3:52 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

David, I’m getting a 404 error when I click on that link. But even for algebraic theories, something seems wrong to me here.

A functor $Alg \to Set$ that preserves limits is representable, a representable that preserves filtered colimits is represented by a finitely presented algebra $A$, and on top of all this, if $\hom(A, -): Alg \to Set$ preserves regular epimorphisms, then $A$ is what I would call a finitely presented projective algebra. The category opposite to finitely presented projective algebras ought to be the Cauchy completion of the category opposite to the category of finitely generated free algebras, i.e., the Cauchy completion of a Lawvere theory. But f.p. projective algebras need not be the same as f.g. free algebras, as I was trying to indicate in my example.

Let me be more concrete about this. Take the algebraic theory of Boolean algebras. I claim that any finite (nonterminal) Boolean algebra $B$, not necessarily a free one (with respect to the usual underlying functor $U: Bool \to Set$), is a projective algebra, i.e., that

$Bool(B, -): Bool \to Set$

preserves regular epimorphisms. To prove this, we only need to show $B$ is a retract of a free finite Boolean algebra, since free Boolean algebras are certainly projective.

In the dual Stone space picture, this is easy to see: it’s the fact that any finite nonempty set $S$ with the discrete topology is a retract of the Stone space of a free Boolean algebra, e.g., the power set $2^S$.

Posted by: Todd Trimble on July 11, 2011 4:46 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I wrote

A functor $Alg \to Set$ that preserves limits is representable

but that’s a bunch of baloney. André Joyal posted a counterexample on the categories mailing list recently, which came to my attention via Mike Shulman in a recent answer on Math Overflow.

I’ve made this assertion not once but several times recently; hopefully it won’t pass my lips again. However, a functor $Alg \to Set$ that preserves limits and is accessible will be representable. (And if $Alg \to Set$ preserves filtered colimits, we do get accessibility of course.)

Posted by: Todd Trimble on July 13, 2011 11:37 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Fixed the link. It’s the material leading up to p. 14. This is just the right kind of muddle where, if I can sort it out in my mind, things become clearer for me.

Posted by: David Corfield on July 11, 2011 5:05 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks for the link, David. I saw what you were referring to on page 14, and it seems to me he’s simply made a mistake. (I don’t see a proof given for the assertion.)

From the standpoint of models, since a finite product theory has the same models as its Cauchy completion, maybe Cauchy complete finite product theories are the ‘right’ thing one ought to consider for this sort of reconstruction theorem. Certainly one can repair his assertion by adding the modifier ‘Cauchy complete’.

Posted by: Todd Trimble on July 12, 2011 10:11 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

But Todd’s example is a pair of algebraic theories (single-sorted, even). So something seems to be amiss.

Posted by: Tom Leinster on July 11, 2011 4:35 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Nice example, Todd! This circle of ideas about Boolean algebras gets more and more interesting.

Of course, finite limit theories escape this fate because finitely complete categories are Cauchy complete.

Right, good point. So I was overlooking an obvious source of examples. Let $A$ be a finite product category that isn’t Cauchy complete. Let $\overline{A}$ be its Cauchy completion, which also has finite products. Then

$FinProd(A, Set) = FinProd(\overline{A}, Set)$

for reasons that momentarily escape me. (I’m sure it’s true, and Todd’s been mentioning it recently, again in connection with Boolean algebras; but there’s got to be a simple conceptual reason that I’m not seeing right now.)

Thus, all we have to do is find a finite product category that isn’t Cauchy complete. One way to do this is to take a nontrivial full subcategory of $Set$ whose collection of objects is closed under finite products. Todd’s examples, the finite sets of cardinality $2^n$ or $3^n$, are of this type.

Posted by: Tom Leinster on July 11, 2011 4:47 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

for reasons that momentarily escape me

I don’t think I’ve given any reasons here at the Café, but one point is:

• If $\mathcal{C}$ has finite products, then so does its Cauchy completion $\widebar{\mathcal{C}}$

because the cartesian product of two retracts of objects $A, B$ of $\mathcal{C}$ is a retract of $A \times B$ (just take the cartesian product of the two retraction pairs).

The second point is that any functor $F: \mathcal{C} \to Set$ has, up to isomorphism, a unique extension to a functor $\widebar{\mathcal{C}} \to Set$, and if $F$ preserves products, so does the extension. This should follow straight from the description of cartesian products in $\widebar{\mathcal{C}}$ given in the paragraph above.

Posted by: Todd Trimble on July 11, 2011 5:29 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks, Todd. Am I right in thinking that the same would work for any kind of limit in place of finite products?

I’m wondering if the situation is something like the following. (I apologize for the sketchiness of what I’m about to say; I’m aware as I type that I haven’t thought this through.) The Cauchy completion of a category is its (free) cocompletion under absolute colimits. Tautologously, finite products commute with absolute colimits. I’m wondering if that can be seen as the reason for which

$FinProd(A, Set) = FinProd(\overline{A}, Set)$

for all finite product categories $A$ (or indeed the same for any other finite product category in place of $Set$).

To test out the validity of this idea, I should try it on some less trivial case. (I say “trivial” because all limits commute with absolute colimits.) For example, if what I have in mind is correct then it should work for finite limits vs. filtered colimits. The filtered cocompletion of a category $A$ is $Flat(A^{op}, Set)$, so if my idea is right then, for starters, $Flat(A^{op}, Set)$ should have finite limits whenever $A$ does. That’s not obvious to me, but perhaps it’s true. I should think about this more carefully.

Posted by: Tom Leinster on July 12, 2011 11:48 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Tom wrote in part

(or indeed the same for any other finite product category in place of $Set$)

You’ll also want the receiving category $C$ in place of $Set$ to be Cauchy complete; otherwise you won’t be able to extend to a functor $\widebar{A} \to C$.

Your general surmise reminds me of distributive laws. If $F$ is a free cocompletion monad of a certain type (e.g., free cocompletion w.r.t. absolute colimits, filtered colimits, etc.) and $G$ is a free completion monad, then it looks like we are considering a distributive law of type

$\theta: G F \to F G$

If $A$ has a $G$-algebra (or pseudo $G$-algebra) structure $\alpha: G A \to A$, then $F A$ acquires a $G$-algebra structure via the composition

$G F A \stackrel{\theta A}{\to} F G A \stackrel{F \alpha}{\to} F A$

Here we could take for instance $F$ to be the filtered colimit cocompletion, $G$ to be the finite limit completion, and I’m imagining that the distributive law $\theta$ here is actually an isomorphism, although I haven’t checked that thoroughly. Similarly if $F$ is the absolute colimit cocompletion.

An example of this where I don’t think $\theta$ is an isomorphism is where $F$ is the cocompletion w.r.t. all small colimits and $G$ is the finite limit completion. (In this case, I’m thinking the 2-monad $F G$ “is morally” the free topos monad. Here I’m thinking toposes and left exact left adjoints as 2-monadic over $Cat$, assuming that naive algebraic intuition pans out correctly w.r.t. size issues.) Even though $\theta$ is not an isomorphism, it’s still true that the free cocompletion is finitely complete. Similarly, I think, taking $G$ to be the small limit completion.

Posted by: Todd Trimble on July 13, 2011 1:52 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Panagis Karazeris wrote to me shortly after I posted my question. That was 15 July, but I was away then for CT11 in Vancouver, so my efficiency plummeted and I’ve only just got round to it.

Anyway, here’s what Panagis told me. I wrote:

if my idea is right then, for starters, $Flat(A^{op}, Set)$ should have finite limits whenever $A$ does. That’s not obvious to me, but perhaps it’s true.

Panagis teased me gently for not remembering the contents of his talk in Barcelona in 2007, when he covered exactly such questions. My one-sentence note from that talk was:

Existence of limits in a cocompletion (w.r.t. some class of colimits) is related to some kind of weak limit property in the original category.

The precise statement that Panagis just told me was:

$Flat(A^{op}, Set)$ has finite limits iff for every finite diagram $D$ in $A$, the category of cones on $D$ is filtered.

A category with a terminal object is certainly filtered, so if $A$ has finite limits then $Flat(A^{op}, Set)$ does indeed have finite limits, as I’d hoped.

The result that Panagis mentioned is here:

Panagis Karazeris, Jiří Velebil, Representability relative to a doctrine, Cahiers de Topologie et Géometrie Différentielle Catégoriques 50 (2009), 3–22.

You can also read some talk slides on the sifted case (i.e. finite products rather than finite limits).

Posted by: Tom Leinster on August 1, 2011 3:33 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

The precise statement that Panagis just told me was:

Best to record such facts on the $n$Lab, where they can be found.

Posted by: Urs Schreiber on August 1, 2011 12:57 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Posted by: Tom Leinster on August 1, 2011 1:24 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Incidentally, I think these questions about calculating the Cauchy completion of finite product theories, even of algebraic theories, are quite interesting and often pretty non-trivial.

I think I managed to convince myself once that the Lawvere theory of groups is Cauchy complete (and therefore has a Tannaka reconstruction along the lines David gave). You can see in this Math Overflow answer a clever and concerted response that implicitly shows (if I’m not mistaken) that the Lawvere theory of commutative rings is also Cauchy complete.

It would be nice to have more examples and counterexamples! Computing the class of idempotent algebra maps is all by itself an interesting and often nontrivial problem.

Posted by: Todd Trimble on July 11, 2011 5:44 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Oh dear, my memory here was very wrong. The Lawvere theory of commutative rings is not Cauchy complete. See the very first page of this article, particularly the reference to D.L. Costa’s paper.

Posted by: Todd Trimble on July 11, 2011 5:55 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So, I’d still like to know about my central question, which in terms of groups could be posed as

Is there a common framework which includes the (doctrinal) reconstruction of the theory of groups from its category of models and the (Tannakian) reconstruction of a specific group from its category of representations?

Looking at Isbell duality I guess the answer is ‘Yes’. Is there any duality that isn’t covered by it?

Posted by: David Corfield on July 13, 2011 12:07 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I can offer some very tautological ways in which the answer is ‘yes’.

In Tannaka reconstruction, we have not just a category of representations but also an underlying or forgetful functor to work with. In the most tautological formulations, we work with the category of all representations and its underlying-set functor

$U: Set^G \to Set.$

This is representable: $U = Set^G(G, -)$. A Yoneda lemma argument gives that the monoid of endomorphisms $Endo(U)$ is isomorphic to the monoid of $G$-set maps $G \to G$, and this monoid is isomorphic to $G$ as a group. In this way we recover $G$ from $U$.

For the category of models of a Lawvere theory $T$, we again have a forgetful functor

$U: Mod(T) \to Set$

which is again representable by $F[1]$, the free $T$-algebra on one generator. This time we don’t form the endomorphism monoid, but an “endomorphism theory” on $U$ whose $n$-ary operations are

$Endo(U)[n] = \hom(U^n, U)$

and here too $T$ is isomorphic (as a Lawvere theory) to $Endo(U)$. Again, this is by a Yoneda lemma argument (e.g., $U^n$ is represented by the free algebra $F[n]$).

There has to be a very general sort of doctrinal story here. Thinking of the objects of a doctrine (in Jim’s sense) as ‘theories’, in one case the ‘theories’ are groups or monoids, in another they are Lawvere theories. In these cases the theory is recovered from an “endomorphism theory” of an underlying functor on its category of models.

It might be thought these cases are slight cheats, because we’ve been concentrating on “single-sorted theories”, where we deal with just one underlying functor $U$ which plays the role of a “universal sort” which allows us a toehold to get at the theory. But there’s a trick which allows us to deal with say multi-sorted Lawvere theories, where we can package multiple universal sorts $U_s: Mod(T) \to Set$ (ranging over a set of sorts $S$) as a single functor $U: Mod(T) \to Set/S$. (One role of $Set/S$ here is that it is the free category with small products generated by a discrete category $S$ – it’s a good environment for not just finitary algebraic theories, but various infinitary theories.)

I have to leave soon, but I’d like to come back to Gabriel-Ulmer duality (and beyond).

Posted by: Todd Trimble on July 15, 2011 2:57 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Thanks for this Todd. That’s helpful.

Your ‘and beyond’ sounds intriguing. Presumably we ought to be looking for 2-theories and their models. Like you showed us that the group concept resides in different doctrines, I take it that there will be a Lex concept which will reside as a theory in different 2-doctrines, models from each into $Cat$ giving $Lex$.

Posted by: David Corfield on July 15, 2011 3:56 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Note: if we ever do follow up categorified Gabriel-Ulmer duality, check out Makkai’s talk 2-dimensional Gabriel-Ulmer duality.

Posted by: David Corfield on July 19, 2011 1:51 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Just to amplify on what Todd just said about how Tannaka reconstruction is essentially just the Yoneda lemma argument:

the details of the Yoneda-aspect are at the beginning of the entry Tannaka reconstruction . We once wrote this out when I was thinking about $(\infty,1)$-Tannaka duality. Simple and useful as the argument is, I am not aware that it is stated explicitly anywhere except in this $n$Lab entry. (If you know a reference, please let me know.)

But notice that most of what is called Tannaka duality in the literature does contain a second step, which is not just general abstract:

There is a general-abstract and a concrete aspect to Tannaka duality. The general abstract one says that an algebra/group $A$ is reconstructible from the fiber functor on the category of all its modules/representations. This is a pure general abstract Yoneda argument. The concrete one says that in nice cases it is reconstructible from the category of dualizable (finite dimensional) modules, even if it is itself not finite dimensional.

More precisely…

(see the entry for the details).

But this means, conversely, that if we are content with looking at a kind of Tannaka duality for the collection of all representations, hence just at the Yoneda aspect of it, the argument is of vast generality. In particular it immediately generalizes to ∞-algebras over algebraic ∞-theories. At Tannaka duality in higher category theory two applications are mentioned: Tannaka duality for $(\infty,1)$-permutation representations, and their application to Galois theory in cohesive ∞-toposes .

Posted by: Urs Schreiber on July 19, 2011 8:41 PM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

I first learned that Tannaka duality reduces to abstract nonsense when you look at all modules/representations from Daniel Schäppi’s paper Tannaka duality for comonoids in cosmoi, which is linked from the nLab page. He phrases it in different terminology (the comonadicity theorem for cocontinuous comonads), but all abstract nonsense is interconvertible to any other abstract nonsense, right?

Posted by: Mike Shulman on July 20, 2011 3:13 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

…all abstract nonsense is interconvertible to any other abstract nonsense.

An interesting thesis, in accord with Mac Lane’s

The notion of Kan extensions subsumes all the other fundamental concepts of category theory.

Posted by: David Corfield on July 20, 2011 10:00 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

right?

Right, I should have mentioned this. In fact, I was scanning the references of the entry yesterday trying to remember which one that was, since we had talked about it at length back then. But then I couldn’t find it before I ran out of time.

It still seems to me that the Yoneda argument is noteworthy for its sheer simplicity. As far as I remember, Schäppi’s presentation does not quite convey the striking fact that one can teach half of Tannaka duality as a 1-line excercise proof right after introducing the Yoneda lemma by simply saying: “hint: apply it four times in a row”.

I think this is what I am wondering if nobody noticed it in the literature before.

Posted by: Urs Schreiber on July 20, 2011 10:38 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

Re Tannaka duality for $(\infty,1)$-categories - James Wallbridge (a student of Toen, Murray and Varghese) did this in his thesis, which he recently defended. AFAIK he doesn’t have an online presence.

Posted by: David Roberts on July 20, 2011 7:51 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

James Wallbridge (a student of Toën, Murray and Varghese) did this in his thesis

I didn’t know this. Hopefully I’ll be able to get hold of this thesis.

AFAIK he doesn’t have an online presence.

Hm, I see a homepage of a James Wallbridge in Adelaide here. While this mentions a seminar titled Higher stacks and homotopy theory II: the motivic context it also speaks of a PhD thesis on Noncommutative geometry in string and M theory . Is that him, or not? Did he write two theses?

Indeed, here I see an announcement of a talk by him on Yoneda in homotopical algebraic geometry .

Posted by: Urs Schreiber on July 20, 2011 10:28 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

No, not two theses; the NC-geometry title was a provisional one from when he started his candidature, the higher category one, “Higher Tannaka duality”, is the real one. The talk about higher stacks was also him. I have handwritten notes, but there was a lot of background in the talk, so no new stuff. He gave at the end as an example the Morel-Voevodsky stable motivic (oo,1)-category.

Posted by: David Roberts on July 20, 2011 10:50 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

the NC-geometry title was a provisional one from when he started his candidature, the higher category one, “Higher Tannaka duality”, is the real one.

I see, thanks. Okay, so I have recorded what we have so far here. Let’s update this as soon as we know more.

Posted by: Urs Schreiber on July 20, 2011 10:59 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

So what was in Toën’s MSRI talk Higher Tannaka duality?

Posted by: David Corfield on July 20, 2011 11:55 AM | Permalink | Reply to this

### Re: Doctrinal and Tannakian Reconstruction

There is a general-abstract and a concrete aspect to Tannaka duality. The general abstract one says that an algebra/group $A$ is reconstructible from the fiber functor on the category of all its modules/representations.

Does this suggest that there are (at least) two concrete aspects:

1. Reconstruction from finite dimensional modules.

2. Reconstruction from all modules but without a specified fiber functor.

Posted by: David Corfield on July 20, 2011 8:30 AM | Permalink | Reply to this