natural transformation

(where is the natural transformation with for any object of ).

All this is natural transformation, but it usually takes seasons before the consequences of these changes become apparent.

The ability to undergo natural transformation among bacterial species is widespread.

For example, the category of small categories is in fact a 2-category, with natural transformations as second degree arrows.

A natural transformation is a relation between two functors.

Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms.

More precisely, it is the following natural transformation of sheaves:

That is, to translate the natural transformation given by currying:

For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine.

The precise formulation of this idea involves the concept of natural transformation.

At the next step, one obtains cobordisms with corners via natural transformations of such 2-functors.

Thus, the mind may use analogies between domains whose internal structures fit according with a natural transformation and reject those that don't.

Every natural transformation is induced by a natural chain map .

These are not allowed in a continuous model, however in this discrete system they arise as natural transformations.

In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism.

This leads to the clarifying concept of natural transformation, a way to "map" one functor to another.

Hence, a natural transformation can be considered to be a "morphism of functors".

Above, the various natural transformations denoted using are parts of the monoidal structure on and .

(as natural transformations ; here denotes the identity transformation from to ).

In a functor category, the morphisms are natural transformations.

Moreover, any element u F(A) defines a natural transformation in this way.

This canonical map is a natural transformation between the identity functor and Gode.

Functors and natural transformations ('naturality') are the key concepts in category theory.

Morphisms in this category are natural transformations between functors.

The coherence conditions for these natural transformations are: