I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle of ideas, regardless of the eventual status of the main result of interest. That is to say, the current sense of urgency to understand something seems generally a good thing. So I thought I'd give the flimsiest bit of introduction imaginable at this stage. On the other hand, as with many of my answers, there's the danger I'm just regurgitating common knowlege in a long-winded fashion, in which case, I apologize.
For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is
to decipher just a little bit those few paragraphs.
The beginning of the investigation is indeed the function field case (over $\mathbb{C}$, for simplicity), where one is given a family
$$f:E \rightarrow B$$
of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial.
There is an exact sequence
$$0\rightarrow \omega_E \rightarrow H^1_{DR}(E) \rightarrow H^1(O_E)\rightarrow0,$$
which is moved by the logarithmic Gauss-Manin connection of the family.
(I hope I will be forgiven for using standard and non-optimal notation
without explanation in this note.) That is to say, if $S\subset B$ is the finite set of images of the bad fibers, there is a log connection
$$H^1_{DR}(E) \rightarrow H^1_{DR}(E) \otimes \Omega_B(S),$$
which * does not preserve* $\omega_E$. This fact is crucial, since it leads to an
$O_B$-linear Kodaira-Spencer map $$KS:\omega \rightarrow H^1(O_E)\otimes \Omega_B(S),$$ and thence
to a non-trivial map
$$\omega_E^2\rightarrow \Omega_B(S).$$
From this, one easily deduces Szpiro's inequality:
$$\deg (\omega_E) \leq (1/2)( 2g_B-2+|S|).$$
At the most simple-minded level, one could say that Mochizuki's programme has been concerned with
replicating this argument over a number field $F$. Since it has to do with differentiation on $B$, which eventually turns into $O_F$, some philosophical connection to $\mathbb{F}_1$-theory
begins to appear. I will carry on using the same notation as above, except now $B=Spec(O_F)$.
A large part of HAT is exactly concerned with the set-up necessary to implement this idea, where, roughly speaking, the Galois action has to play the role of the GM connection.
Obviously, $G_F$ doesn't act on $H^1_{DR}(E)$. But it does act on $H^1_{et}(\bar{E})$ with
various coefficients. The comparison between these two structures is the subject
of $p$-adic Hodge theory, which sadly works only over local fields rather than a global one. But Mochizuki noted long ago that something like $p$-adic Hodge theory should be a key ingredient at some level because over $\mathbb{C}$, the comparison isomorphism
$$H^1_{DR}(E)\simeq H^1(E(\mathbb{C}), \mathbb{Z})\otimes_{\mathbb{Z}} O_B$$
allows us to completely recover the GM connection, by the condition that
the topological cohomology generates the flat sections.
In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dag}\rightarrow E,$$
(the moduli space of line bundles with flat connection on $E$)
whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the \'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form:
$$\Xi: A_{DR}=\Gamma(E^{\dag}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$
(I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by
a calligraphic $H$. I can't seem to write calligraphic characters here.)
Here, $L$ is a suitably chosen torsion line bundle on $E$, which can then be pulled back
to $E^{\dag}$. The inequality refers to the polynomial degree in the fiber direction of
$E^{\dag} \rightarrow E$. The isomorphism is effected via evaluation of sections at
$$E^{\dag}[\ell]\simeq E[\ell].$$
Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete \'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.
Now for some corrections: As stated, the isomorphism is not true, and must be modified at the places of bad reduction, the places dividing $\ell$, and the infinite places.
This correction takes up a substantial portion of the HAT paper. That is, the isomorphism is generically true over $B$, but to make it true everywhere, the integral structures must be modified in subtle and highly interesting ways, while one must consider also a comparison of metrics, since these will obviously figure in an arithmetic analogue of Szpiro's conjecture. The correction at the finite bad places can be interpreted via coordinates near infinity on the moduli stack of elliptic curves as the subtle phenomenon that Mochizuki refers to as 'Gaussian poles' (in the coordinate $q$). Since this is a superficial introduction, suffice it to say for now that these Gaussian poles end up being a major obstruction in this portion of Mochizuki's theory.
In spite of this, it is worthwhile giving at least a small flavor of Mochizuki's Galois-theoretic KS map. The point is that $A_{DR}$ has a Hodge filtration defined by
$F^rA_{DR}= \Gamma(E^{\dag}, L)^{ < r} $
(the direction is unconventional), and
this is moved around by the Galois action induced
by the comparison isomorphism. So one gets thereby a map
$$G_F\rightarrow Fil (A_{DR})$$
into some space of filtrations on $A_{DR}$.
This is, in essence, the Galois-theoretic KS map. That, is if we consider the equivalence over $\mathbb{C}$ of $\pi_1$-actions
and connections, the usual KS map measures the extent to which the GM connection moves around the Hodge filtration. Here, we are measuring the same kind of motion for the $G_F$-action.
This is already very nice, but now comes a very important variant, essential for understanding the motivation behind the IUTT papers. In the paper GTKS, Mochizuki modified this map, producing instead a 'Lagrangian' version. That is, he assumed the existence of a Lagrangian Galois-stable subspace $G^{\mu}\subset E[l]$ giving rise to another isomorphism
$$\Xi^{Lag}:A_{DR}^{H}\simeq L\otimes O_{G^{\mu}},$$
where $H$ is a Lagrangian complement to $G^{\mu}$, which I believe does not itself need to
be Galois stable. $H$ is acting on the space of sections, again via Mumford's theory.
This can be used to get another KS morphism to filtrations on $A_{DR}^{H}$. But the key point is that
$\Xi^{Lag}$, in contrast to $\Xi$ is free of the Gausssian poles
via an argument I can't quite remember (If I ever knew).
At this point, it might be reasonable to see if $\Xi^{Lag}$ contributes towards a version
of Szpiro's inequality (after much work and interpretation), except for one small problem. A subspace like $G^{\mu}$ has no
reason to exist in general.
This is why GTKS is mostly about the universal elliptic curve over a formal completion near $\infty$ on the moduli stack of elliptic curves, where such a space does exists.
What Mochizuki explains on IUTT page 10 is exactly that
the scheme-theoretic motivation for IUG was to enable the move to a single elliptic curve over $B=Spec(O_F)$, via the intermediate case of an elliptic curve 'in general position'.
To repeat:
A good 'nonsingular' theory of the KS map over number fields requires a global Galois
invariant Lagrangian subspace $G^{\mu}\subset E[l]$.
One naive thought might just be to change base to the field generated by the $\ell$-torsion, except one would then lose the Galois action one was hoping to use. (Remember that Szpiro's inequality is supposed to come from moving the Hodge filtration inside De Rham cohomology.) On the other hand, such a subspace does often exist locally, for example, at a place of bad reduction. So one might ask if there is a way to globally extend such local subspaces.
It seems to me that this is one of the key things going on in the IUTT papers I-IV.
As he say in loc. cit. he works with various categories of collections of local objects that simulate global objects. It is crucial in this process that many of the usual
scheme-theoretic objects, local or global, are encoded as suitable categories with a rich and precise combinatorial structure.
The details here get very complicated, the encoding of a scheme into
an associated Galois category of finite \'etale covers being merely
the trivial case. For example, when one would like to encode the
Archimedean data coming from an arithmetic scheme (which again, will clearly be
necessary for Szpiro's conjecture), the attempt to come up with a category of
about the same order of complexity as a Galois category gives rise to the
notion of a Frobenioid. Since these play quite a central role in Mochizuki's theory,
I will quote briefly from his first Frobenioid paper:
'Frobenioids provide a single framework [cf. the notion of a "Galois category";
the role of monoids in log geometry] that allows one to capture the essential aspects of
both the Galois and the divisor theory of number fields, on the one hand, and function
fields, on the other, in such a way that one may continue to work with, for instance,
global degrees of arithmetic line bundles on a number field, but which also exhibits the new
phenomenon [not present in the classical theory of number fields] of a "Frobenius
endomorphism" of the Frobenioid associated to a number field.'
I believe the Frobenioid associated to a number field is something close to the
finite \'etale covers of $Spec(O_F)$ (equipped with some log structure) together with metrized line bundles on them, although it's
probably more complicated. The Frobenious endomorphism for a prime $p$ is then something like
the functor that just raises line bundles to the $p$-th power.
This is a functor that would come from a map of schemes if we were
working in characteristic $p$, but obviously not in characteristic zero.
But this is part of the reason to start encoding in categories:
We get more morphisms and equivalences.
Some of you will notice at this point the analogy to
developments in algebraic geometry where varieties are encoded in categories,
such as the derived category of coherent sheaves. There as well, one has reconstruction
theorems of the Orlov type, as well as the phenomenon of non-geometric morphisms
of the categories (say actions of braid groups). Non-geometric morphisms
appear to be very important in Mochizuki's theory, such as the Frobenius above,
which allows us to simulate characteristic $p$ geometry in characteristic
zero. Another important illustrative example is a
non-geometric isomorphism between Galois groups of local fields (which can't exist
for global fields because of the Neukirch-Uchida theorem).
In fact, I think Mochizuki was rather fond of Ihara's comment that the positive
proof of the anabelian conjecture was somewhat of a disappointment, since
it destroys the possibility that encoding curves into their fundamental
groups will give rise to a richer category. Anyways, I believe the importance
of non-geometric maps of categories encoding rather conventional objects
is that
they allow us to glue together several standard
categories in nonstandard ways.
Obviously, to play this game well,
some things need to be encoded in rigid ways, while others should
have more flexible encodings.
For a very simple example that gives just a bare glimpse of the general theory, you might consider a category of
pairs $$(G,F),$$ where $G$ is a profinite topological group
of a certain type and $F$ is a filtration on $G$.
It's possible to write down explicit conditions that ensure that
$G$ is the Galois group of a local field and $F$ is its ramification filtration
in the upper numbering (actually, now I think about it, I'm not sure about 'explicit conditions' for the filtration part, but anyways). Furthermore, it is a theorem of Mochizuki
and Abrashkin that the functor that takes a local field to the corresponding
pair is fully faithful. So now, you can consider triples
$$(G,F_1, F_2),$$
where $G$ is a group and the $F_i$ are two filtrations of the right type.
If $F_1=F_2$, then this 'is' just a local field. But now you can have
objects with $F_1\neq F_2$, that correspond to strange amalgams of
two local fields.
As another example, one might take
a usual global object, such as $$ (E, O_F, E[l], V)$$ (where $V$
denotes a collection of valuations of $F(E[l])$ that restrict bijectively to
the valuations $V_0$ of $F$), and associate to it a collection of local categories
indexed by $V_0$ (something like Frobenioids corresponding to the $E_v$ for $v\in V_0$). One can then try to glue them together
in non-standard ways along sub-categories, after performing a number of non-standard transformations. My rough impression at the moment is that
the 'Hodge theatres' arise in this fashion. [This is undoubtedly a gross oversimplification, which I will correct
in later amendments.] You might further imagine that some
construction of this sort will eventually retain the data necessary to get the height of
$E$, but also have data corresponding to the $G^{\mu}$, necessary for the Lagrangian KS map.
In any case, I hope you can appreciate that a good deal of 'dismantling' and 'reconstructing' will be necessary.
I can't emphasize enough times that much of what I write is based on
faulty memory and guesswork. At best, it is superficial, while at worst,
it is (not even) wrong.
I have not yet done anything with the current papers than give them a rough glance.
If I figure out more in the coming weeks, I will make corrections.
But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.
Allow me to make one remark about set theory, about which I know next to nothing.
Even with more straightforward papers in arithmetic geometry, the question sometimes arises about Grothendieck's universe axiom, mostly because universes appear to be used in SGA4. Usually, number-theorists (like me) neither understand, nor care about such foundational matters, and questions about them are normally
met with a shrug. The conventional wisdom of course is that any of the usual
theorems and proofs involving Grothendieck cohomology theories or topoi do
not actually rely on the existence of universes, except general laziness allows us
to insert some reference that eventually follows a trail back to SGA4.
However, this doesn't seem to be the case with
Mochizuki's paper. That is, universes and
interactions between them seem to be important actors rather than conveniences.
How this is really brought about, and whether more than the universe axiom is necessary for the arguments, I really don't understand enough yet to say.
In any case, for a number-theorist or an algebraic geometer, I would guess it's still prudent to acquire a reasonable feel for the
'usual' background and motivation (that is, HAT, GTKS, and anabelian things) before worrying too much about deeper issues of set theory.
