"Natural operation" redirects here. For the natural sum and natural product on ordinals, see Ordinal arithmetic § Natural operations.
This article is about natural transformations in category theory. For the natural competence of bacteria to take up foreign DNA, see Transformation (genetics). For other uses, see Transformation (mathematics) (disambiguation).
Central object of study in category theory
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.
Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.
and 26 Related for: Natural transformation information
In category theory, a branch of mathematics, a naturaltransformation provides a way of transforming one functor into another while respecting the internal...
to D. A naturaltransformation is a relation between two functors. Functors often describe "natural constructions" and naturaltransformations then describe...
another Naturaltransformation between functors in category theory. Unitary transformation, between two Hilbert spaces Geometric transformation, between...
category theory, specifically the theory of 2-categories, a lax naturaltransformation is a kind of morphism between 2-functors. Let C and D be 2-categories...
from its environment in the process called transformation. Competence may be differentiated between natural competence, a genetically specified ability...
horizontal gene transfer. Naturaltransformation in A. calcoaceticus may protect against exposure to DNA-damaging conditions in the natural environment of these...
(see Natural competence). Transformation is common in the prokaryotic world, and thus far 67 species are known to be competent for transformation. One...
bacteria. Natural bacterial transformation is an adaptation for transferring DNA from one individual cell to another. Naturaltransformation, including...
shown impaired colonization and adherence to host's intestines. Naturaltransformation is a common bacterial adaptation for DNA transfer that employs numerous...
However, the genome of V. vulnificus is at least one megabase bigger. Naturaltransformation is a bacterial adaptation for DNA transfer between individual cells...
{\displaystyle f:N\to M} in C {\displaystyle {\mathcal {C}}} to the naturaltransformation Δ ( f ) : Δ ( N ) → Δ ( M ) {\displaystyle \Delta (f):\Delta (N)\to...
monoidal naturaltransformation θ : ( F , m ) → ( G , n ) {\displaystyle \theta :(F,m)\to (G,n)} between those functors is a naturaltransformation θ : F...
how the embedded category, of representable functors and their naturaltransformations, relates to the other objects in the larger functor category. It...
which defines a naturaltransformation S → T {\displaystyle S\to T} . The difference between the two notions is that a naturaltransformation is a particular...
different. Natural competence sums up in three methods where bacteria can acquire DNA from their surroundings: conjugation, transformation, and transduction...
media used in the transformation process can affect the efficiency. For example, certain media supplements can increase the natural competence of cells...
Hepatitis B and C are also the result of natural viral transformation of the host cells. Viral transformation can also be induced for use in medical treatments...
functors F : C → D {\displaystyle F:C\to D} and the morphisms are naturaltransformations η : F → G {\displaystyle \eta :F\to G} between the functors (here...
categories C and D, together with two functors F, G : C → D and a naturaltransformation α : F ⇒ G: There are two kinds of composition in a 2-category (called...
recombination. The ability to undergo naturaltransformation is present in at least 67 bacterial species. Naturaltransformation is common among pathogenic bacterial...
category: the functor category. Morphisms in this category are naturaltransformations between functors. Functors are often defined by universal properties;...
systematically. Natural genetic transformation involves the transfer of DNA from one bacterium to another through the surrounding medium. Transformation is a complex...
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that...
For each object Y, the counit of the exponential adjunction is a naturaltransformation e v Y , Z : Z Y × Y → Z {\displaystyle \mathrm {ev} _{Y,Z}:Z^{Y}\times...
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