notion of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category. A functor F : C →...

representation of a functor F to Set is an initial object in the category of elements of F. The notion of final functor (respectively, initial functor) is a generalization...

that the functor C(1,-): C → Set is faithful. For example the category Grph of graphs and their associated homomorphisms is a topos whose final object 1...

topological functor is one which has similar properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits...

be the diagonal functor from Top to the functor category TopJ (this functor sends each space X {\displaystyle X} to the constant functor to X {\displaystyle...

The third condition is equivalent to the requirement that the functor – ×Y (i.e. the functor from C to C that maps objects X to X ×Y and morphisms φ to φ × idY)...

respect to some universe) and the morphisms functors. Fct(C, D), the functor category: the category of functors from a category C to a category D. Set, the...

ind-completed category, denoted Ind(C), are known as direct systems, they are functors from a small filtered category I to C. The dual concept is the pro-completion...

of map instead: (unit ∘ φ) x ↔ ((map φ) ∘ unit) x ↔ x The final leap from applicative functor to monad comes with the second transformation, the join function...

Then the following are equivalent: The constant functor with value ω {\displaystyle \omega } is a final object in the category τ ( Hom _ ( X , C ) ) {\displaystyle...

the carrier of the final coalgebra of 1 + (−) is known as the set of conatural numbers. For a second example, consider the same functor 1 + N × (−) as before...

examples of sites with no final object. There are two natural types of functors between sites. They are given by functors that are compatible with the...

the same as a covariant functor I → C {\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}} . The colimit of this functor is the same as the direct...

must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints. When considering functors between R-linear...

M-types) are (isomorphic to) initial algebras (resp. final coalgebras) of such functors. Polynomial functors have been studied in the more general setting of...

to a functor F {\displaystyle F} , with specific properties as defined below. For both algebras and coalgebras,[clarification needed] a functor is a convenient...

category theory. It follows that adjoint functors induce homotopy equivalences. In particular, if C has an initial or final object, its nerve is contractible...

a map so that every lift exists that is a final object among all lifts. For example, the forgetful functor QCoh → Sch {\displaystyle {\textrm {QCoh}}\to...

obtained by placing the final topology on the corresponding colimits in Set. Unlike many algebraic categories, the forgetful functor U : Top → Set does not...

joining X to a point. The map X ↦ C X {\displaystyle X\mapsto CX} induces a functor C : T o p → T o p {\displaystyle C\colon \mathbf {Top} \to \mathbf {Top}...

should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation). In the untyped lambda calculus, all...

construct-memoized-functor(factorial) The above example assumes that the function factorial has already been defined before the call to construct-memoized-functor is...

functors fmap :: (a -> b) -> (f a -> f b) -- action of functor on morphisms instance Functor (MaybeProd a) where -- turn MaybeProd a into a functor,...

theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the...

contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor, and hence...

free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing...

deformation theory introduced by Schlessinger (1968) that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem...

Standard Library. It provides four components called algorithms, containers, functors, and iterators. The STL provides a set of common classes for C++, such...

and inverse axioms) a functor from schemes over S to the category of groups, such that composition with the forgetful functor to sets is equivalent to...

Michael H.; Larsen, Michael; Wang, Zhenghan (1 June 2002). "A Modular Functor Which is Universal for Quantum Computation". Communications in Mathematical...