### 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:

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.

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.

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

semanticsfunctor. It has a partial left adjoint, which is called thestructurefunctor (cf. [Dub70], [Str72]). Thepartialhere 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

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?

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?

## Re: Doctrinal and Tannakian Reconstruction

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

Hmm,

And he talks earlier of

Why the scare quotes?