View source for Functorial morphism

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421502.png" /> are one-place covariant functors from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421503.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421504.png" />. A functorial morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421505.png" /> associates to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421507.png" /> a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421508.png" />, in such a way that for every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f0421509.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215010.png" /> the following diagram is commutative:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215011.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215012.png" />, then, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215013.png" />, one obtains the so-called identity morphism of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215016.png" /> are two functorial morphisms, then, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215017.png" />, one obtains the functorial morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215018.png" />, called the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215020.png" />. Composition of functorial morphisms is associative. Therefore, for a small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215021.png" />, all functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215023.png" /> and their functorial morphisms form a so-called functor category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215024.png" />, or a category of diagrams with scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215025.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215026.png" /> be a functorial morphism and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215028.png" /> be two functors. The formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215029.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215030.png" /></td> </tr></table>

define functorial morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215032.png" />, respectively. Then for any functorial morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215034.png" />, the following relationship holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042150/f04215035.png" /></td> </tr></table>

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====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,   "Theory of categories" , Acad. Press  (1965)</TD></TR></table>