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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 $$(f, I , n),$$ where $f:X \rightarrow Y $ is a morphism in $\mathcal{D}$, $I\in \phi(X)$, and $n\in \mathbb{N}_{\geq 1}$. If $Y\rightarrow Z$ is given by the triple $(g, J, m)$, then the composition is defined by $$(g, J, m)(f, I, n)= (g\circ f, f^*(J)+mI, mn).$$

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 $\mathbb{F}_{\phi}$.

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 $$(f, I, n, u)$$ 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 $$n\alpha+I=f^*(\beta)+Div(u).$$ 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.

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

--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. My impression is that these `Frobenius-type' maps play a very important role.

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.

--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.

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 'etale $F$-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.

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 {\em algorithmically}. We will return to this as well in a later question.)

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 'etale 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.

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.

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.)

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.