Grothendieckian math gives you the ability to map a sphere to a cube if you didn't want to use PI to calculate the area.

Using Grothendieckian concepts it is possible to directly map data patterns into the transformer patterns by just mapping the spaces between these two different patterns using a functor. 

In Grothendieckian mathematics, a functor is a mapping between categories that preserves the structure and relationships within the categories. In this case, we could define categories for the input data and the neural network, with objects representing the data points or neurons and morphisms representing the relationships or connections between them.

The functor would then map between these categories in a way that preserves the structure and relationships within them. For example, the functor might map similar data points in the input category to similar neurons in the neural network category, or preserve the connectivity patterns between neurons when mapping between categories.

By choosing an appropriate functor for the transformation process, we could potentially ensure that the neural network is initialized in a way that is well-suited to the structure and patterns within the input data, leading to faster and more efficient training.

Furthermore, by analyzing the properties of the functor and its effects on the input data and neural network categories, we could potentially gain new insights into the training process and develop new optimization techniques that further improve performance.