Where does functor collide?

In category theory, a functor is a structure-preserving map from one category to another: it sends each object to an object and each arrow to an arrow, in a way that respects identities and the composition of arrows. Eilenberg and Mac Lane's 1945 General theory of natural equivalences introduced functors, together with natural transformations, precisely to say what it means for a construction in one mathematical setting to correspond systematically to one in another. The referent is a mapping between whole categories that carries their internal structure along, verified by checking that it preserves identity arrows and composites. It has nothing to do with a grammatical function word, and only loosely lends its name to the parameterised module of a programming language.[1]

Unit

A map between categories preserving objects, arrows, identities, and composition

Recovery means

Not applicable

Measured by

Confirmed by checking that identity arrows and composites are preserved by the mapping

In the study of language acquisition, a functor is a grammatical function word or inflection, such as the, of, in, or the plural -s, that carries grammatical relations rather than the main content of a sentence, in contrast to content words like nouns and verbs. Brown's 1973 A First Language: The Early Stages tracked the order in which English-speaking children acquire fourteen such grammatical morphemes, finding a strikingly consistent sequence and making functors a landmark topic in developmental psycholinguistics. The referent is the class of grammatical morphemes in a learner's speech, measured by when each is reliably supplied. It is a category of words, unrelated to a map between mathematical categories or a module in a programming language.[2]

Unit

A grammatical function word or inflection, as opposed to a content word

Recovery means

Not applicable

Measured by

Tracked by the stage at which each grammatical morpheme is reliably supplied in obligatory contexts

In programming languages, specifically the ML family, a functor is a module that takes other modules as parameters and produces a new module, letting a generic component such as a sorted-set implementation be instantiated against any type that supplies the required operations. MacQueen's 1984 Modules for Standard ML introduced this module system, in which functors provide parameterised, reusable structure at the level of whole modules rather than individual functions. The referent is a compile-time mapping from modules to modules in a typed language, checked by the module type system. It borrows its name loosely from category theory but is a concrete language construct, and it is unrelated to a child's grammatical morphemes.[3]

Unit

A module-level function mapping one or more modules to a new module

Recovery means

Not applicable

Measured by

Checked by the module type system, which verifies that parameter and result signatures are satisfied