Morphisms

This post is a brief, nonexhaustive, nonrigorous listing of morphisms I’ve seen come up in the wild, including homo-, iso-, endo-, auto-, para-, ana-, cata-, hylo-…. There are too many prefixes to keep track of, which is why this post exists.

These kinds of transformations have become more important in my mind since starting my little forays into category theory and other branches of math, so having a place where I can keep them all straight might be helpful in the future.

Homomorphism

Sort of the granddaddy of them all. Preserves operations of structures while transforming between them. Confusingly also just called a “morphism” in some places.

Isomorphism

Bijective homomorphism. If we can transform in one direction, we can also do the inverse transformation in the other direction, preserving structure all the while.

Endomorphism

The source and the target are the same thing (or, to be more “mathy”, the domain equals the codomain). “Endo” kind of means “within”, and this stays within the domain it already knows.

Automorphism

Endo- and iso- together. Therefore, an invertible way to map some structure to itself and back again. This one I don’t have a great mnemonic for beyond “endo plus iso”.

Monomorphism

Injective. Remember that “mono” means “one”, and “injective” means “one-to-one”.

Epimorphism

Surjective. It’s an “epic morphism” because it maps onto every part of its image (i.e. it’s epic because it covers everything.)

Anamorphism

An unfolding which builds up a set of results. Think of “anabolic steroids”, which build up muscle.

Catamorphism

Opposite of anamorphism, so folds a set of results down to one result. If you know the words “anabolism” and “catabolism” then you’re set for these prefixes.

Hylomorphism

Anamorphism along with catamorphism. Builds up a set of results and then folds it back down to one result. 100% reason to remember the name.

Apomorphism

An anamorphism but takes extra stuff (the original information you had) along for the ride. Maybe remember that apo- and ana- look kinda similar?

Paramorphism

Dual of apomorphism. Like a catamorphism, but takes your original thing along for the ride.


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