Separable frobenius algebras and completely distributive categories

在数学的许多应用中，存在通过函数或其某种推广而彼此相关的感兴趣的对象。为了理解对象，对与对象的某些属性相关联的某些真实的或复量进行观察和测量。用这样的数字进行计算，用这样的变量进行推导，得到方程，方程的数学解扩展了系统的知识。从感兴趣的对象到与选择的属性相关联的选择的量的通道显然是抽象的。一些信息，人们希望只是不相关的信息，丢失了。 有时，对与对象的属性相关联的量执行的计算是可能的，因为计算可以与对象本身一起进行。在这种情况下，在计算之前没有丢弃任何东西，并且抽象较少。对非数字对象的计算，将我们带到了一个非常普遍的计算研究。计算的规则，即使是看起来很熟悉的操作，也可能是不同的。为了增加一般性而失去抽象性是范畴论的一个特征。 这个建议的第一部分涉及的操作和方程的研究中发现的经典研究可分Frobenius代数。我们将应用这些技术的对象比代数中的对象要广泛得多。然而，可分条件和弗罗贝纽斯条件的结合迫使一个对象的内部结构简化，承认这个外部代数结构。 本建议的第二部分研究了方程的推广 a x（B + c）=（a x B）+（a x c） 到一类泛化x的对象操作和一类泛化+的类似对象操作。

In many applications of mathematics, there are objects of interest related to each other by functions or some generalization thereof. To understand the objects, observations and measurements are made of some real or complex quantities associated with some attributes of the objects. Calculations with such numbers and deductions with such variables lead to equations whose mathematical solution expands knowledge of the system. The passage from the objects of interest to a selection of quantities associated with a selection of attributes is manifestly an abstraction. Some information, one hopes only irrelevant information, is lost. Sometimes, the calculations that are performed on quantities associated with attributes of objects are possible because calculations are possible with the objects themselves. In such situations nothing is discarded prior to calculation and there is less abstraction. Calculations with objects that are not numbers takes us to a very generalized study of calculation. The rules for calculation, even with operations that look familiar, may be different. Loss of abstraction in favour of increased generality is a characteristic of Category Theory. The first part of this proposal involves a study of the operations and equations found in the classical study of separable Frobenius algebras. The objects to which we will apply such techniques are considerably more encompassing than those that are found in Algebra. However, the conjunction of the separable and Frobenius conditions forces a simplification of the internal structure of an object admitting this external algebraic structure. The second part of this proposal studies a generalization of the equation a x (b + c) = (a x b) + (a x c) to a large class of object operations that generalize x and a similar class of object operations that generalize +.