Today we’re going to finish going over preliminaries and notation from
the Mochizuki masterpiece, Geometry of Frobenioids I: The general theory.
If we are successful at this, we may then be in a position to read the definition of Frobenioid.
Our groups will be Hausdorff topological groups. Subgroups have centralizers.
To a given profinite group is associated a category of finite sets with a action. This category is a Galois category. If every open subgroup of has trivial centralizer, we say that $\Pi$ is slim. This is all we need to know about group theory.
Next we learn about categories. We consider their objects and arrows. We think about one object and one morphism categories. Has anyone ever read Paul Pot’s book, “The one bullet manager?”
We discuss some restrictions of categories to single objects by looking at morphisms all of whose images (or all of whose domains) are a single object. Morphisms of the original category form functors between these restricted categories.
We define what it means for an arrow to be fiberwise surjective. (Basically when it can be put in a commutative diagram with any other arrow having the same target.) A category is of FSM type if every fiberwise surjective morphism is an isomorphism. We define that an endomorphism is a sub-automorphism if there is an automorphism and a map so that applying the automorphism then the map is the same as applying the map then the endomorphism.
Various other notions on categories are discussed. It’s hard to see where this is going. Time for the definition of Frobenioid.