![]() It is also more accurate to say that the highway bisects each building(s) and each road, taking each morphism and object from 1 category to the morphisms and objects of another.Ī functor is contravariant if it reverses the direction of every morphism, which means that when the functor brings each road to the other category, it does a 180 turn gradually and reverses the direction of the roads. a functor is a just a function bringing 1 category to another, so it is like a highway from 1 category from another. There are a couple more definitions and interactions we must go through. Great! We have a frobenioid! But no, it is not that simple. Since the category we want to analyse most is called a monoid, so we will assume that roads can be one way (noninvertabiliy) and that morphisms don't have to be commutative (I have no idea how to stretch that analogy that far) these are called domains and codomains respectively. And for each morphism/road, there is an object/building(s) where the morphism starts and ends. To get more technical, there are collections of such morphisms and objects that, themselves. objects are the parts of the category that you can think of as buildings, or sets of such, and morphisms are like the roads between those buildings or groups of buildings. In mathematics, a category has 2 parts: objects and morphisms. We will use this definition as our main tool to understanding the other definitions and consequences.įirst, we must know what a category is. To quote directly from IUTT, a hodge theater is “miniature models of conventional scheme theory - given, roughly speaking, by systems of Frobenioids". I will try to build it up in layman's terms. ![]() The first paper is an introduction to the papers, and lists a lot of the notation, and whatnot. example: although it is very rigorous and abstract, all definitions can be found with a decent knowledge of group theory, algebraic topology, and algebraic geometry. ![]()
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