The concept of categories and morphisms were introduced to us today in our Linear Algebra course. I haven't taken abstract algebra yet so this lecture confused me quite a bit. As far as my understanding goes, a category consists of the following:
- A class of objects
- A class of morphisms given any 2 objects from the category
- An identity morphism given an object
- A way to compose 2 morphisms
A few examples that were given by our professor that he claimed would be relevant in the class were the category of vector spaces with linear maps as morphisms and the category of linear maps. But he never explored these concepts in detail and just skimmed over them.
Why must the morphisms of the category of vector spaces have to be linear maps? Why not functions that are non-linear? I have eventually, after trying to prove that they must be linear, think that it is only for the sake of convenience. Is this correct?
And I am quite confused about morphisms of the category of Linear maps (between vector spaces). Does it mean that the morphism takes linear maps as input and outputs an other map in the category?
And in general, do morphisms need only be associative or are there other resitrictions?
I have no prior experience in abstract algebra so please consider me a layman. A few might examples might help.