Functorial morphism

natural transformation

An analogue of the concept of a homomorphism of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors themselves play the role of the modules). Suppose that $ F _ {1} $ and $ F _ {2} $ are one-place covariant functors from a category $ \mathfrak K $ into a category $ \mathfrak C $. A functorial morphism $ \phi : F _ {1} \rightarrow F _ {2} $ associates to each object $ A $ of $ \mathfrak K $ a morphism $ \phi _ {A} : F _ {1} ( A) \rightarrow F _ {2} ( A) $, in such a way that for every morphism $ \alpha : A \rightarrow B $ in $ \mathfrak K $ the following diagram is commutative:

$$ \begin{array}{llr} F _ {1} ( A) &\rightarrow ^ { {F _ 1} ( \alpha ) } &F _ {1} ( B) \\ {\phi _ {A} } \downarrow &{} &\downarrow {\phi _ {B} } \\ F _ {2} ( A) & \mathop \rightarrow \limits _ { {F _ {2} ( \alpha ) }} &F _ {2} ( B). \\ \end{array} $$

If $ F _ {1} = F _ {2} $, then, setting $ \phi _ {A} = 1 _ {F _ {1} ( A) } $, one obtains the so-called identity morphism of the functor $ F _ {1} $. If $ \phi : F _ {1} \rightarrow F _ {2} $ and $ \psi : F _ {2} \rightarrow F _ {3} $ are two functorial morphisms, then, setting $ ( \phi \psi ) _ {A} = \phi _ {A} \psi _ {A} $, one obtains the functorial morphism $ \phi \psi : F _ {1} \rightarrow F _ {3} $, called the product of $ \phi $ and $ \psi $. Composition of functorial morphisms is associative. Therefore, for a small category $ \mathfrak K $, all functors from $ \mathfrak K $ into $ \mathfrak C $ and their functorial morphisms form a so-called functor category $ \mathop{\rm Funct} ( \mathfrak K , \mathfrak C ) $, or a category of diagrams with scheme $ \mathfrak K $.

Let $ \phi : F _ {1} \rightarrow F _ {2} : \mathfrak K \rightarrow \mathfrak C $ be a functorial morphism and let $ G: \mathfrak M \rightarrow \mathfrak K $ and $ H: \mathfrak C \rightarrow \mathfrak N $ be two functors. The formulas

$$ \forall B \in \mathop{\rm Ob} \mathfrak M :\ ( G \star \phi ) _ {B} = \phi _ {G ( B) } , $$

$$ \forall A \in \mathop{\rm Ob} \mathfrak R : \ ( \phi \star H) _ {A} = H ( \phi _ {A} ) $$

define functorial morphisms $ G \star \phi : GF _ {1} \rightarrow GF _ {2} $ and $ \phi \star H: F _ {1} H \rightarrow F _ {2} H $, respectively. Then for any functorial morphisms $ \phi : F _ {1} \rightarrow F _ {2} : \mathfrak K \rightarrow \mathfrak C $ and $ \psi : H _ {1} \rightarrow H _ {2} : \mathfrak C \rightarrow \mathfrak N $, the following relationship holds:

$$ ( \phi \star H _ {1} ) ( F _ {2} \star \psi ) = \ ( F _ {1} \star \psi ) ( \phi \star H _ {2} ). $$

A functorial morphism is also called a natural transformation of functors. Functorial morphisms of many-place functors are defined by analogy with functorial morphisms of one-place functors.

Comments

The term "natural transformation" is common.

References

[a1]	B. Mitchell, Theory of categories, Acad. Press (1965)
[a2]	Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer (1998) ISBN 0-387-98403-8