Distillations About

“Oh, the morphisms you’ll see!”

17 Oct 2019 ⇐ Notes archive

(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)

There are many prefixes placed in front of the word “morphism”—here’s a glossary of the ones I’ve seen so far:

  • Epimorphism
    • The categorical analog to surjectivity—the “epi” root connotes morphisms mapping “over” the entirety of the codomain. Bartosz covers this well in lecture 2.1 (timestamped) of Part I of his category theory course.
  • Monomorphism
    • Injectivity’s analog and epimorphism’s dual (it blew my mind to realize injectivity and surjectivity, two properties I never thought twice about, are duals). “Mono” in that it generalizes one-to-one functions. Bartosz also mentions them in 2.1.
  • Bimorphism
    • “A morphism that is both an epimorphism and a monomorphism is called a bimorphism.”
    • I don’t quite have the foundation needed to grok when a bimorphism isn’t an isomorphism—maybe because I spend most of my time working in Hask and Swift (Set, in disguise) and Wikipedia mentions “a category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.” On the other hand, and I need to read more into what “split” means in the following, “while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z ⇒ Q is a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism.”
  • Isomorphism
    • Show up all over mathematics. A morphism that “admits a two-sided inverse, meaning that there is another morphism in [the] category [at hand] such that [their forward and backward compositions emit identity arrows on the domain and codomain, respectively].” “Iso” for equal in the sense that if an isomorphism exists, there is an sort of sameness to the two objects.
  • Endomorphism
    • A morphism from an object onto itself that isn’t necessarily an identity arrow. “Endo” for “within” or “inner.” The prefix shed light on why the Point-Free folks named functions in the form (A) -> A, Endo<A>. Looking at that file now, I wonder what the “i” in imap stands for and I may or may not have gotten nerd sniped into checking if imap’s definition shakes out from Func.dimap when dealing with Func<A, A>s and Z == C == B (the B being imap’s generic parameter). Looks like it does?…a few messages later and Peter Tomaselli helped me out! The “i” stands for “invariant,” which reads nicely in that imap’s co- and contravariant parameters kind of cancel out to invariance.
  • Automorphism
    • An isomorphic endomorphism. “Auto” for same or self.
  • Homomorphism
    • The star of algebra, a structure-preserving mapping between two algebraic structures. i.e. a homomorphism f on some structure with a binary operation, say *, will preserve it across the mapping—f(a * b) = f(a) * f(b). I’ll cover the etymological relationship between “hom” and its appearances in category theory—hom-sets and hom-functors—that isn’t quite restricted to sets in the way algebra generally is in a future note.
  • Homeomorphism
    • The one I’m least familiar with—in another life (or maybe later in this one), I want to dig into (algebraic) topology. Seems to be the topologist’s isomorphism (in the category Top).
  • Catamorphism, Anamorphism, and Hylomorphism
    • I’ve only dealt with these three in the functional programming sense. Catamorphisms break down a larger structure into a reduced value (“cata” for “down”), anamorphisms build structure from a smaller set of values (“ana” for “up”), and hylomorphism is an ana- followed by a catamorphism (oddly, “hylo” stands for “matter” or “wood,” wat).
    • I ran into catamorphism the other day when trying to put a name on a function in the form ((Left) -> T) -> ((Right) -> T) -> (Either<Left, Right>) -> T. Turns out folks call this either, analysis, converge, or fold (the last of which was somewhat surprising to me in that the Foldable type class requires a monoidal instance, whereas this transformation doesn’t quite have the same requirement). This function is catamorphic in that it reduces an Either into a T.
    • zip is an example of an anamorphism that builds a Zip2Sequence from two Sequences and by extension, zipWith is a hylomorphism that zips and then reduces down to a summary value by a transformation.
    • Hylomorphisms and imap both seem to be compositions of dual transformations. Wonder if this pattern pops up elsewhere?