The n-Category Café

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.

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.

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 $n$.

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.

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.

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).

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 .