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Applying fmap (+1) to a binary tree of integers increments each integer in the tree by one.. In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values inside a generic type without changing the structure of the generic type.
A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D.
A functor from G to Set is then nothing but a group action of G on a particular set, i.e. a G-set. Likewise, a functor from G to the category of vector spaces, Vect K, is a linear representation of G. In general, a functor G → C can be considered as an "action" of G on an object in the category C. If C is a group, then this action is a group ...
The ML family of functional programming languages uses the term functor to represent a mapping from modules to modules, or from types to types and is a technique for reusing code. Functors used in this manner are analogous to the original mathematical meaning of functor in category theory , or to the use of generic programming in C++, Java or Ada .
A commutative diagram in a category C can be interpreted as a functor from an index category J to C; one calls the functor a diagram. More formally, a commutative diagram is a visualization of a diagram indexed by a poset category. Such a diagram typically includes: a node for every object in the index category,
An anafunctor [note 1] is a notion introduced by Makkai (1996) for ordinary categories that is a generalization of functors. [1] In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor. [2]
The 'unitary executive theory' Driving Trump's strategy is a legal framework championed by conservatives, perhaps most notably by Trump's newly-confirmed director of White House Office of ...
The empty set is the unique initial object in Set, the category of sets.Every one-element set is a terminal object in this category; there are no zero objects.. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in thi