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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 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 ...
Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.
For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.
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,
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.
is an exact functor, where Ab is the category of abelian groups. An abelian category A {\displaystyle {\mathcal {A}}} is said to have enough projectives if, for every object A {\displaystyle A} of A {\displaystyle {\mathcal {A}}} , there is a projective object P {\displaystyle P} of A {\displaystyle {\mathcal {A}}} and an epimorphism from P to ...
Let C : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let U be the forgetful functor defined above, and let G be any quiver. Then there is a graph homomorphism I : G → U ( C ( G )) and given any category D and any graph homomorphism F : G → U(D) , there is a unique functor F' : C ...