Functorlike

Functorlike is a term used in mathematics and computer science to describe objects or constructions that behave similarly to functors but do not meet the full formal definition. In category theory, a functor F from a category C to a category D assigns to every object X in C an object F(X) in D and to every morphism f: X → Y a morphism F(f): F(X) → F(Y), preserving identity and composition. A functorlike construction mimics this behavior in one or more respects but may weaken or modify the axioms.

Common relaxations include pseudofunctors, lax functors, and oplax functors. Pseudofunctors preserve composition and identities up to specified isomorphisms, rather than strictly. Lax and oplax functors replace strict preservation by specified 2-cells satisfying coherence conditions. In enriched category theory, one can consider functors between enriched categories that preserve structure up to the enrichment.

In programming languages, many type constructors are described as functors when they provide a map-like operation, typically called fmap or map, that preserves composition and identities in practice. A type constructor that implements the map operation and satisfies the functor laws in all cases is genuinely a Functor; when these laws hold only in restricted contexts or up to observational equivalence, it may be described informally as functorlike.

The term is primarily descriptive and nontechnical in some texts. It signals an expectation of functorial behavior without asserting the full axioms. Related concepts include natural transformations, presheaves, and profunctors, which generalize or relate to functoriality in various directions.