1 upvotes, 2 direct replies (showing 2)
View submission: Ask Anything Wednesday - Engineering, Mathematics, Computer Science
How is category theory different from linear transformation (that I studied in my misspent youth)?
Comment by siddharth64 at 26/06/2024 at 21:18 UTC
1 upvotes, 0 direct replies
It's hard to be precise without being quite technical, but essentially, category theory studies any kind of transformation (formally, morphisms). It could be ordinary functions, or about linear transformations or graph homomophisms or ... Ultimately category theory tries to understand how transformations interact amongst each other rather than what a particular transformation is doing to elements/numbers/vectors/etc.
Comment by F0sh at 26/06/2024 at 23:08 UTC
1 upvotes, 0 direct replies
Linear transformations are introduced as a certain kind of function of vectors of real numbers, and they are generally defined as a certain kind of functions of vectors of any *field*.
Category theory studies many areas of mathematics by looking at objects (for example vector spaces over a certain field) and the transformations of objects in those spaces (for example, linear transformations).
In mathematics you often look at objects and study them by examining their subobjects. For example, there are many sub-vector-spaces of the space of 3-dimensional real vectors. Category theory allows you to generalise this idea to that of a subcategory relation, allowing you to prove theorems in a very general way and then apply them not just to vector spaces, but also to all modules, and maybe also to groups, rings, topological spaces and so on.
Category theory isn't concerned with the precise properties of linear transformations; you would never compute the actual numerical values of a transformation applied to a vector when doing category theory, pretty much.