Homomorphism
Jarg: homomorphism.
I’m losing steam and getting tired right now, so instead of writing out this page for real, I’m just gonna leave notes for my future self.
Generally speaking, a homomorphism is a transformation of a space.
Generally speaking, a ‘space’ can be many things. Usually they are represented as sets with some extra structure.
Example: vector spaces ()
The “spaces” are the vector spaces.
It is easiest/‘best’ to consider a vector space as “actually consisting” of its bases vectors (for any choise of basis), with the rest of the underlying set being mereley generated from the basis, only existing “virtually”.
Then a homomorphism of spaces is obviously a map of bases, which is exactly what a vector space homomorphism actually is
Example: groups
The homomorphisms are group homomorphisms. I like to think about this as embedding groups “computationally”: a homomorphism witnesses that we can use to model a subgroup of
Oh right most algebraic homomorphisms can be seen as giving a notion of modelling
Example: topological spaces
ummmm I guess I view this as picking out the (physically-)continuous transformations from the space of all transformations
“Homomorphisms preserve structure” — what is meant by structure?
Literally, like, relations between points
If one point in a vector space moves, other points will as well
If one point in a topological space moves, other points will as well (unless the point is isolated)
Metaphor (?): clay
Morphisms are ways to transform things, such as smooshing clay, throwing clay out, baking the clay, adding paint
(This level of precision actually is meaningful. We can’t define morphisms as “sequences of actions on a clay creation” because that’s a free category, which is not an interesting example. We can’t define morphisms as (input, output) pairs because that’s a poset category, which is not an interesting example.)
Exists a morphism between clay horse and clay dinosaur because you can smoosh one into the other. In fact, they are isomorphic
Exists morphisms from unbaked clay into baked clay, but not vice-versa
Exists morphisms from baked clay into baked & painted clay, since you can add paint. Exists vice-versa, too, since you can scrape paint off. But some clay will come with it, so it’s not an isomorphism.