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Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.

Recently, there has been a flurry of new discussion surrounding Shinichi Mochizuki's interuniversal Teichmueller theory (IUTT). I've been personally quite ashamed about the state of affairs. Regardless of the eventual correctness of the paper in all detail, this is an earnest attempt by an esteemed colleague to present a serious vision of mathematics. Not to have given it proper attention for so long reflects poor manners on my part, to say the least. After an initial attempt to organise a workshop, I've essentially let things slide, assuming things would get sorted out somehow. Clearly, it hasn't happened until now. Some dedicated young people have been reading the papers (for example, I've had a number of illuminating exchanges with Chung Pang Mok recently), but my impression is that they also need some support. Some might ask why I, for example, don't just read the papers carefully myself. Well, as you know, for most people chugging along amid the cares of everyday life, it's quite painful to read long difficult papers without the aid of human discussion and interaction. Mochizuki, by the way, is one of those rare people gifted with the incredible powers of concentration and stamina necessary to sit and do mathematics for long periods in solitude. This was true ever since he was a young student. Perhaps this is a key reason he can't understand why the rest of us are so reluctant. Anyway, I'm moving towards suggesting another workshop in the near future. However, to increase the chance that good mathematicians will participate, it seems sensible to disseminate some more background material on the relevant mathematical structures.

My plan is to do this in a series of questions on MO. How many there will be, I don't have a clear sense of at the moment. Of course, I have myself just a vague general idea of the notions in the IUTT papers, and even what I knew two years ago, I've forgotten. So my plan is to outline something, quite possibly incorrect and certainly incomplete, and then invite others to improve on my exposition. It's my hope that the people who have read the papers much more carefully will contribute and that some new readers will be motivated. For some time, I've been deeply impressed by the energy and integrity of the community surrounding this site. It seems quite appropriate as a venue for unraveling some of the mystery surrounding IUTT.

To read the IUTT papers, a new concept we need to understand well is that of a Frobenioid. To explain why, let me remind you that the main object of study in IUTT is the log-theta lattice associated to an elliptic curve over a number field. The name refers to a collection of oriented paths (Panoramic Overview, Figure 3.1) going between points arranged like a two-dimensional lattice, which represent copies of a $\Theta^{\pm ell}NF$-Hodge theatre. It might be worth emphasising that once you understand this lattice, most of the conceptual background is in place. The extent to which completely new structures are supposed to be endlessly multiplied in these papers has, in my opinion, been greatly exaggerated.

What kind of a thing is a $\Theta^{\pm ell}NF$-Hodge theatre? It is a category, itself glued out of two other categories, a $\Theta^{\pm ell}$-Hodge theatre and a $\Theta NF$-Hodge theatre, but for now, we will deemphasise this particular decomposition. (Once again, in case you're worried, there are no other kinds of Hodge theatres.) But in view of the apparently complicated structure whose details might require some guiding principle to grasp, it is reasonable to ask about the main goal, that is, what exactly the point might be of a $\Theta^{\pm ell}NF$-Hodge theatre. Well, as is stated in a variety of ways by Mochizuki himself, it is supposed to be a categorical model of the spectrum of an algebraic number field $F$, together with some extra structure of a crystalline nature. In some sense, it's the same kind of combinatorial encoding of $F$ as the category of finite étale $F$-algebras, or an `abstract combinatorialization of scheme-theoretic arithmetic geometry.' If you are not used to this point of view, you should try to visualise a category as something like a one-dimensional abstract simplicial complex where even quite complicated objects are reduced to points and their structure encoded in a network of arrows. In the present context, this can be a bit confusing because the Hodge theatres are represented as points connected by paths in the log-theta lattice, but when observed closely, they also resolve into a network.

The 'crystalline nature' of the extra structure is used in a very vague sense. It's just an elliptic curve over the number field, which determines a set of so-called `theta data'. On the one hand, this extra structure is there because we're interested in Szpiro's conjecture (=ABC conjecture). However, a far more essential goal of the theory is to deform the number field in a canonical fashion, in a manner analogous to lifting a curve over a perfect field of positive characteristic to characteristic zero. There as well, one normally doesn't have a canonical lift. However, the $p$-adic Teichmueller theory was developed exactly to deal with this difficulty: If you equip the curve with an ordinary nilpotent indigenous bundle, which exists for a generic curve, then there is a canonical lift. Because these kinds of indigenous bundles are finite in number, one might say that the curve itself has been canonically lifted up to finite indeterminacy. (Even naively, this is much better than the uncountably infinite indeterminacy associated with choosing any lift.) There is by now a long tradition of crystalline philosophy, whereby an object that doesn't itself admit a canonical deformation acquires one through the addition of some natural extra structure. The simplest example of this phenomenon is the structure of a crystal on the universal extension of an abelian variety. The elliptic curve and the attendant theta data are supposed to be exactly this kind of extra structure that allows us to deform the field in a canonical fashion. That is to say, from a certain point of view, it's really the field that's of interest, and the elliptic curve is just auxiliary structure one has to incorporate in order to arrive at a crystal-like situation.

In fact, the log-theta lattice is itself the deformation. At the risk of boring you with repetition, I will restate that a log-theta lattice is made out of copies of a single $\Theta^{\pm ell}NF$-Hodge theatre, which, in turn, is a categorical model of a number field together with a crystalline structure. If you would like to see why a deformation might be made out of the object that it tries to deform, the simplest example to consider is the way in which $\mathbb{Z}_p$ is made out of copies of $\mathbb{F}_p$, especially when considered as the ring of Witt vectors. Perhaps a more elementary remark is that many reasonable deformations in geometry are fibre bundles over the deformation space, so that all fibres are somewhat the same (with varying degrees of rigidity depending on the situation) as the central fibre that we started out with. With the Hodge theatres, Mochizuki's analogy is that they all have the same real-analytic structure, while the holomorphic structure varies.

Before getting to the main point, let me say a word about `glueing'. Now, there may be other notions involved, but one basic one is that of grafting, If $A$ and $B$ are categories equipped with functors $\phi$ and $\psi$ to $C$, then we define the (directed) graft $$A\vdash_C B$$ by taking the union of $Ob(A)$ and $Ob(B)$, then simply defining new morphisms from an object $a$ of $A$ to an object $b$ of $B$ as morphisms $\phi(a)\rightarrow \psi(b)$. This construction is quite simple, and I am very far from understanding the different ways of gluing the categories that occur in IUTT, so more on this later.

For the moment, I wish to concentrate just a little bit on the internal structure of a $\Theta^{\pm ell}NF$-Hodge theatre, leaving aside for now the nature of the paths connecting the copies. A $\Theta^{\pm ell}NF$-Hodge theatre is also glued in various way out of smaller categories, and this brings us to back to our title. The basic building blocks of everything in sight are categories called Frobenioids. Among the prerequisites for studying IUTT, this notion is the really new one. My feeling is that getting a concrete grip on it will already take us a good way towards understanding the whole picture. The other papers on absolute anabelian geometry and so forth are also hard, but still belong to more or less familiar sorts of anabelian geometry, since many people will have heard of the Neukirch-Uchida theorem or Grothendieck's conjectures. (However, the main focus of the absolute anabelian geometry papers is to prove such reconstruction theorems algorithmically. We will return to this as well in a later question.)

The first question in this series is then

What is a Frobenioid?

In the usual spirit of Mathoverflow, I will set up some rudimentary language, give a few examples to show that I'm a serious participant, and wait for contributions from people with more expertise.

I start with a quote from Mochizuki, which we can hope to understand better as more contributions come in:

a Frobenioid may be thought of as a sort of a category-theoretic abstraction of the theory of divisors and line bundles on models of finite separable extensions of a given function field or number field.

The simplest Frobenioid is constructed out of a commutative monoid $M$ (satisfying some natural condition of being divisorial). One can then form a semi-direct product with $\mathbb{N}_{\geq 1}$, the multiplicative monoid of positive integers: $$\mathbb{F}_M:=M\ltimes \mathbb{N}_{\geq 1},$$ which is a non-commutative monoid with composition defined by $$(a,n)(b,m)=(a+nb, nm).$$ For example, $M$ might be a monoid of effective Cartier divisors on a normal variety $B$ (or line bundles equipped with sections). The $\mathbb{N}_{\geq 1}$-action encodes the tensor power operation on line bundles, which is simply absorbed into the structure of $\mathbb{F}_M$. In positive characteristic, the Frobenius map induces the $p$-th tensor power map on line bundles, and this monoid structure enables the construction of a substitute of sorts. Note also that we can regard a monoid as a category with a single object.

The next case starts with a family of monoids. By this, we mean a contravariant functor $$\phi: \mathcal{D}\rightarrow Mon$$ from some category $\mathcal{D}$ to the category of commutative (divisorial) monoids. This $\mathcal{D}$ notation seems to occur rather often and consistently throughout many papers as an indexing category (called a base category in the papers) for some family of monoids. The key example to keep in mind is where $\mathcal{D}$ is a category of finite separable geometrically integral normal covers $X\rightarrow B$ of a normal variety $B$, that is, the usual kind of category of Galois type that occurs in anabelian geometry. In that case, we can take $\phi(X)$ to be a suitable family of effective $\mathbb{Q}$-Cartier divisors on $X$. For example, they might be divisors that lie over a specific monoid of divisors on $B$, say generated by a specific set of prime divisors. (I am going to ignore here the subtleties surrounding pull-backs and types of singularities.)

The earlier semi-direct product construction can now be applied to the functor $\phi$. The objects of $\mathbb{F}_{\phi}$ are just the objects of $\mathcal{D}$ (as an extension of considering a monoid a category with a single object) but a morphism $$ X\rightarrow Y$$ is a triple $$(f, S , n),$$ where $f:X \rightarrow Y $ is a morphism in $\mathcal{D}$, $S\in \phi(X)$, and $n\in \mathbb{N}_{\geq 1}$. If $Y\rightarrow Z$ is given by the triple $(g, T, m)$, then the composition is defined by $$(g, T, m)(f, S, n)= (g\circ f, f^*(S)+mT, mn).$$

Up to here is pretty elementary. But some confusion may arise from the fact that the general Frobenioid involves yet another category $\mathcal{C}$ equipped with a functor $ \mathcal{C}\rightarrow \mathbb{F}_{\phi}$. In fact, $\mathcal{C}$ is in many ways more fundamental than $\mathbb{F}_{\phi}$ and should be thought of as a fiber bundle over the base $\mathbb{F}_{\phi}$, as $\mathbb{F}_{\phi}$ is fibered over $\mathcal{D}$. That is, we have a composition of fibrations

$$\mathcal{C}\rightarrow \mathbb{F}_{\phi}\rightarrow \mathcal{D}.$$

The nature of $\mathcal{C}$ is clarified by the main example, whereby $\mathcal{C}$ is associated to multiplicative groups of rational functions on varieties. The abstract framework for this example, is that of a model Frobenioid, which one constructs out of yet another functor $\psi: \mathcal{D}\rightarrow Ab$ to abelian groups together with a map of functors $$Div:\psi \rightarrow \phi^{gp},$$ where $\phi^{gp}$ denotes the group completion of $\phi$ in an obvious sense. (As the notation suggests, the example to keep in mind is the homomorphism from rational functions to divisors.) Out of this data, we form the category $\mathcal{C}$ whose objects are pairs $$(X, \alpha),$$ with $X$ an object of $\mathcal{D}$ and $\alpha\in \phi(X)^{gp}$. A map from $(X, \alpha)$ to $(Y, \beta)$ is then a quadruple $$(f, S, n, u)$$ where $f: X\rightarrow Y$ is a morphism in $\mathcal{D}$, $I\in \phi(X)$, $n \in \mathbb{N}_{\geq 1}$, $u\in \psi(X)$, and $$n\alpha+S=f^*(\beta)+Div(u).$$ We have merely added the components $\alpha$ to the objects and the components $u$ to the morphisms. There is thus an obvious projection functor to $\mathbb{F}_{\phi}$. The number $n$, by the way, is referred to as the Frobenius degree of the morphism, and seems to be eventually very important.

For the geometric $\phi$ above, we can take $\psi(X)\subset K(X)^*$, a subgroup of the multiplicative group of rational functions on $X$ whose supports are controlled by the divisors in $\phi(X)$. If we view the monoids themselves as being morphisms, this formalism would suggest that $u$ is kind of a 2-morphism, and that $\mathcal{C}$ might better be described as a 2-category in some way. Anyway, I hope you'll agree at this point that the constructions really are not overly exotic.

It's worth working out the morphisms for which all components but one are trivial. We will denote the domain of a morphism by $(X, \alpha)$ and the codomain by $(Y,\beta)$.

--A morphism of the form $(f, 0, 1,1)$, where $f:X\rightarrow Y$ is a morphism of $\mathcal{D}$. For it to map $(X,\alpha)$ to $(Y,\beta)$, we must have $\alpha=f^*\beta$. We might, for example, choose a divisor $\gamma$ on the base $B$,and let $\alpha$ and $\beta$ be pullbacks of $\gamma$. This determines a faithful embedding of $\mathcal{D}$ into $\mathcal{C}$.

--Now consider a morphism of the form $(Id_X, S, 1, 1)$. Of course we must have $Y=X$ and $\alpha+S=\beta$. Thus, this is the `tensor product map' from $(X,\alpha)$ to $(X, \alpha+S)$.

--A morphism of the form $(Id_X, 0, n, 1)$ from $(X,\alpha)$ to $(X,\beta)$ imposes $n\alpha=\beta$. So we have formally adjoined a map from a line bundle to its $n$-th tensor power.

--$(Id_X, 0, 1, u)$ goes from $(X,\alpha)$ to $(X,\beta)$ such that $\alpha=\beta+Div(u)$. This is a map of line bundles in the usual sense.

Obviously, the intention is that as these morphisms intertwine, something interesting will happen.

Hopefully, we will soon see more precise clarifications of my approximate account. But the astute reader will already have noticed something not quite right. Earlier on, it was stated that a Frobenioid is a category, whereas we have indicated that it is a category $\mathcal{C}$ equipped with a functor $\mathcal{C}\rightarrow \mathbb{F}_{\phi}$ for some family of monoids $\phi.$ That some conceptual ambiguity is all right is essentially the main theorem of the first Frobenioid paper: In all natural cases, the category $\mathbb{F}_{\phi}$ and the functor to it are canonically determined by the category $\mathcal{C}$. Thus, it is safe to refer to $\mathcal{C}$ itself as the Frobenioid. Certainly, in the geometric example above, it's clear that $\mathcal{C}$ contains the information for $\mathbb{F}_{\phi}$. But the point of the theorem is that with certain rigid assumptions, this encoding can be detected purely category-theoretically.

There are a number of other theorems of intrinsic interest about categoricity. For example, the base-category $\mathcal{D}$ can be recovered from $\mathcal{C}$. (What conditions are exactly necessary for this, I'm not able to say at the moment.) If $\mathcal{C}_1$ and $\mathcal{C}_2$ are Frobenioids and $E: \mathcal{C}_1\simeq \mathcal{C}_2$, then $E$ must preserve the Frobenius degrees of morphisms.

I think this is all I wish to say for now. Allow me to stress again that I still don't know what a Frobenioid really is and eagerly await corrections and elaborations. There are clearly numerous subtleties and points of emphasis that I am missing. In particular, if someone could give a good account of the main theorem alluded to above, I would be very grateful. I suspect that there are consequences of a rather concrete nature that we can appreciate within the realm of usual arithmetic geometry. This, of course, is the kind of thing that will convince a greater number of people to invest time in understanding the various papers. However, I hope even these superficial paragraphs will provide some indication that the kind of mathematical language developed by Mochizuki is interesting and natural. Indeed, to my untrained mind, the geometric Frobenioids appear very much to be in the spirit of $p$-adic Hodge theory, being subtle composites of structures of étale and De Rham type. Since the earlier Hodge-Arakelov papers had started with the intention of developing a global $p$-adic Hodge theory, maybe this association is not too far from correct.