General term for map/fmap

Question

The general term for folds is a catamorphism. The general term for unfold is an Anamorphism. Is there an equivalent term for map? I know that a map is a type of fold however restriction is significant. Also it would be useful to have an alternative term to use when the word “map” is ambiguous.

I strongly recommend against the "catamorphism/anamorphism/etc" terminology. "Fold/unfold" is perfectly fine for the "general" case and more likely to be understood. At any rate, if someone doesn't know what "fold/unfold" is (given suitable context), they are unlikely to recognize "catamorophism/anamorphism". — Derek Elkins left SE, Dec 06 '17 at 00:33

score 5 · Accepted Answer · answered Dec 06 '17 at 16:52

5

In addition to being in agreement with Derek (that the ana/cata terminology doesn't bring much to the table) I'll note that the "category theoretic" terminology for the general map function of type $$\mathrm{map}:\forall \alpha\beta. (\alpha \rightarrow\beta)\rightarrow(F\ \alpha\rightarrow F\beta) $$ for a functor $F$ is simply "the functorial action of $F$", and is often written as $F$! For example, on lists, map f l would be written $\mathrm{List}(f)(l)$.

For some reason, computer scientists seem to prefer to have a seperate name for the type constructor and the action on morphisms, but mathematicians do not really consider non-functorial constructors anyways.

answered Dec 06 '17 at 16:52

cody

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Just for clarification the “!” Isnt’ a part of the notation? – Q the Platypus Dec 08 '17 at 03:00
@QthePlatypus No it is not. Good question. – cody Dec 08 '17 at 19:58
A separate symbol for the action of a functor on morphisms is needed rather in formal (including programming) languages than in computer science. The natural mathematical language has ambiguities which are inconvenient in a formal language. – beroal Feb 13 '18 at 16:37

score 1 · Answer 2 · answered Dec 06 '17 at 16:54

I think fmap, at least in the context of Haskell, is already the general term for the thing it does and the thing you're talking about.

The fact that a map function also exists, which is basically an fmap that only works on lists, is mostly a historical accident AFAIK.

Perhaps you're looking for the word Functor, being the thing that has an fmap?

General term for map/fmap

2 Answers2