Functors often describe " natural constructions " and natural transformations then describe " natural homomorphisms " between two such constructions.

Functors are structure-preserving maps between categories.

Functors were first considered in algebraic topology, where algebraic objects ( like the fundamental group ) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.

Functors like these, which " forget " some structure, are termed forgetful functors.

Functors like these are called representable functors.

Functors are often defined by universal properties ; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.

Functors for which this assumption does not hold are called intensional.

Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories.

Functors between one-object categories correspond to monoid homomorphisms.

Functors that forget the extra sets need not be faithful ; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.

Functors belong to the most important categories in logical grammar ( alongside with basic categories like sentence and individual name ): a functor can be regarded as an " incomplete " expression with argument places to fill in.

Categories, Functors, and Natural Transformations.

In 1942 – 45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology.

Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations ; in order to do that, functors had to be defined, which required categories.

Commutative diagram defining natural transformations

* The functor category D < sup > C </ sup > has as objects the functors from C to D and as morphisms the natural transformations of such functors.

In this context, the standard example is Cat, the 2-category of all ( small ) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense.

Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties.

Morphisms in this category are natural transformations between functors.

Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties.

In studying such transformations it is always necessary to distinguish between the material transformation of the economic conditions of production, which can be determined with the precision of natural science, and the legal, political, religious, artistic or philosophic — in short, ideological forms in which men become conscious of this conflict and fight it out.

Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties.

It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category.

for each object A of C, the natural transformations from h < sup > A </ sup > to F are in one-to-one correspondence with the elements of F ( A ).

In this way, Yoneda's Lemma provides a complete classification of all natural transformations from the functor Hom ( A ,-) to an arbitrary functor F: C → Set.

That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms ( in the reverse direction ) between the associated objects.

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism.

Additionally, every pair of adjoint functors comes equipped with two natural transformations ( generally not isomorphisms ) called the unit and counit.

If and are natural transformations between functors, then we can compose them to get a natural transformation.

If is a natural transformation between functors and is a natural transformation between functors, then the composition of functors allows a composition of natural transformations.

Husserl declares that mental and spiritual reality possess their own reality independent of any physical basis, and that a science of the mind (' Geisteswissenschaft ') must be established on as scientific a foundation as the natural sciences have managed:

Or, in the second case, while natural selection can help animals develop ways of killing or escaping from other species, intrasexual selection drives the selection of attributes that allow alpha males to dominate their own breeding partners and rivals .< ref name = hu > Wikipédia-szerkesztők ( Wikipedia contributors ), ' Nemi szelekció ' (' Sexual Selection '), Wikipédia, 2011. május 17.

A function f on an interval I is uniformly continuous if its natural extension f * in I * has the following property ( see Keisler, Foundations of Infinitesimal Calculus (' 07 ), p. 45 ):

The escutcheon's formal blazon is in Irish, translated here as Argent two piles throughout gules three cinquefoils counterchanged (' On white, two red triangles throughout the shield, three cinquefoils in the reverse colour '); the colours are those of the City of London, and the piles form a W. Along with this was granted a crest, On a wreath of the colours a pied wagtail bearing in its beak ragged robin all proper (' On a red and white wreath, a pied wagtail bearing in its beak ragged robin, all in their natural colours ').

While the street was metalled in 1843 (' metal ' is a New Zealand term for gravel road ), the natural stream still often overflowed its banks, and the area was still swampy.

A noted landmark, the natural arch (' Hole in the Rock ') of Piercy Island, lies two kilometres off the cape.

Hermeneuticians such as Wilhelm Dilthey theorized in detail on the distinction between natural and social science (' Geisteswissenschaft '), whilst neo-Kantian philosophers such as Heinrich Rickert maintained that the social realm, with its abstract meanings and symbolisms, is inconsistent with scientific methods of analysis.

Early German hermeneuticians such as Wilhelm Dilthey pioneered the distinction between natural and social science (' Geisteswissenschaft ').

She writes about the natural world, the trickiness of love, the constant threat of tragedy, as well as very Vancouver subjects such as the late painter Emily Carr (' Here Is A Picture ') and the missing women of the Downtown Eastside (' Return of the Kildeer's ' Liza Jane ').

: Article 2a: ' personal data ' shall mean any information relating to an identified or identifiable natural person (' data subject '); an identifiable person is one who can be identified, directly or indirectly, in particular by reference to an identification number or to one or more factors specific to his physical, physiological, mental, economic, cultural or social identity ;

The lowest natural point is the Bieber ’ s riverbed, and the highest is the aptly named Hoher Berg (' High Mountain ').