Editing Natural transformation (section)

==Definition==
If <math> F</math> and <math> G</math> are [[functor]]s between the categories <math> C</math> and <math> D </math> (both from <math> C</math> to <math> D</math>), then a '''natural transformation''' <math> \eta</math> from <math> F</math> to <math> G</math> is a family of morphisms that satisfies two requirements.
# The natural transformation must associate, to every object <math> X</math> in <math> C</math>, a [[morphism]] <math>\eta_X :  F(X) \to G(X)</math> between objects of <math> D </math>. The morphism <math> \eta_X</math> is called "the '''component''' of <math> \eta</math> at <math> X </math>" or "the <math> X </math> component of <math>\eta</math>."
# Components must be such that for every morphism <math> f :X  \to  Y</math> in <math> C</math> we have:

:::<math>\eta_Y \circ F(f) = G(f) \circ \eta_X</math>

The last equation can conveniently be expressed by the [[commutative diagram]].

[[File:Natural Transformation between two functors.svg|center|This is the commutative diagram which is part of the definition of a natural transformation between two functors.]]

If both <math> F</math> and <math> G</math> are [[contravariant functor|contravariant]], the vertical arrows in the right diagram are reversed. If <math> \eta</math> is a natural transformation from <math> F</math> to <math> G </math>, we also write <math> \eta : F \to G</math> or <math> \eta : F \Rightarrow G </math>. This is also expressed by saying the family of morphisms <math> \eta_X: F(X) \to G(X)</math> is '''natural''' in <math> X </math>.

If, for every object <math> X</math> in <math> C  </math>, the morphism <math> \eta_X</math> is an [[isomorphism]] in <math> D </math>, then <math> \eta</math> is said to be a '''{{visible anchor|natural isomorphism}}''' (or sometimes '''natural equivalence''' or '''isomorphism of functors'''). This can be intuitively thought of as an isomorphism <math>\eta_X</math> between ''objects'' <math>F(X)</math> and <math>G(X)</math> ''inside'' <math>D</math> having been created, or "generated," by a natural transformation <math>\eta</math> between ''functors'' <math> F </math> and <math> G</math> ''outside'' <math>D</math>, into <math>D</math>. In other words, <math>\eta</math> is a natural isomorphism [[if and only if]] <math> \eta : F \Rightarrow G </math> entails an isomorphic <math>\eta_X: F(X) \to G(X)</math>, for all objects <math>X</math> in <math> C  </math>; the two statements are equivalent. Even more reductionistically, or philosophically, a natural isomorphism occurs when a natural transformation begets its own respective isomorphism (by name that is) and thus the "natural-ness" (or rather, naturality even) of the natural transformation passes from itself over into that very same isomorphism, resulting in a natural isomorphism. Two functors <math> F </math> and <math> G</math> are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from <math> F</math> to <math> G</math> in their category.

An '''infranatural transformation''' <math> \eta : F \Rightarrow G </math> is simply the family of components for all <math> X</math> in <math> C</math>. Thus, a natural transformation is a special case of an infranatural transformation for which <math> \eta_Y \circ F(f)  = G(f) \circ \eta_X</math> for every morphism <math> f : X \to Y </math> in <math> C</math>. The '''naturalizer''' of <math> \eta </math>, <math>\mathbb{nat}(\eta)</math>, is the largest [[subcategory]] <math>C_S \subseteq C</math> (S for subcategory), we will denote as <math>C_{SL}</math> (L for largest), containing all the objects of <math> C</math>, on which <math> \eta</math> restricts to a natural transformation. Alternatively put, <math>\mathbb{nat}(\eta)</math> is the largest <math>C_S \subseteq C</math>, dubbed <math>C_{SL}</math>, such that <math>\, \eta|_{C_{SL}}\ : \ F|_{C_{SL}} \implies G|_{C_{SL}}</math> or <math>\, \eta|_{\mathbb{nat}{(\eta)}}\ : \ F|_{\mathbb{nat}{(\eta)}} \implies G|_{\mathbb{nat}{(\eta)}}</math> for every object <math>X</math> in <math>nat{(\eta)} = C_{SL} \subseteq C</math>.