Definition of DB Category - Big Data Integration Theory - page 149

Example 23 Let us consider a number of “conceptualizations” of complex mor-
phisms in Theorem 4 , expressed by the functor T e = (T e ,T e )

: ↓ −→

( DB DB )

DB :
1. The case when f = f 1 + f 2 : A 1 + A 2 → B 1 + B 2 ,g = g 1 + g 2 : C 1 + C 2 → D 1 +
D 2 ,h = h 1 + h 2 : A 1 + A 2 → C 1 + C 2 and k = k 1 + k 2 : B 1 + B 2 → D 1 + D 2 ,
are such that k 1 ◦ f 1 = g 1 ◦ h 1 and k 2 ◦ f 2 = g 2 ◦ h 2 , so that we have the arrow

(h 1 + h 2 ; k 1 + k 2 ) : J(f) → J(g) and hence

: f 1 + f 2 →

T e (h 1 + h 2 ; k 1 + k 2 ) = T e (h 1 ,k 1 ) + T e (h 2 ,k 2 ) g 1 + g 2 .

2. The case when f =[ f 1 ,f 2 ]: A 1 + A 2 → B,g : C → D,h =[ h 1 ,h 2 ]: A 1 +

A 2 → C and k : B → D are such that k ◦ f 1 = g ◦ h 1 and k ◦ f 2 = g ◦ h 2 , so that

we have the arrow ( [ h 1 ,h 2 ]; k) : J(f) → J(g) and hence

T e [ k = T e (h 1 ,k),T e (h 2 ,k) : f 1 + f 2 → h 1 ,h 2 ];

g.

3. The case when f : A → B,g =[ g 1 ,g 2 ]: C 1 + C 2 → D,h = h 1 ,h 2 : A →

C 1 + C 2 and k : B → D are such that k ◦ f = g 1 ◦ h 1 and k ◦ f = g 2 ◦ h 2 , so that

we have the arrow ( h 1 ,h 2 ; k) : J(f) → J(g) and hence

T e k = T e (h 1 ,k),T e (h 2 ,k) : f h 1 ,h 2 ; → g 1 + g 2 .

4. The case when f = f 1 ,f 2 : A → B 1 + B 2 ,g = g 1 ,g 2 : C → D 1 + D 2 ,h :

A → C and k = k 1 + k 2 : B 1 + B 2 → D 1 + D 2 are such that k 1 ◦ f 1 = g 1 ◦ h and

k 2 ◦ f 2 = g 2 ◦ h , so that we have the arrow (h ; k 1 + k 2 ) : J(f) → J(g) and hence

: f T e (h ; k 1 + k 2 ) = T e (h,k 1 ),T e (h,k 2 ) → g 1 + g 2 .

5. The case when f = f 1 ,f 2 : A → B 1 + B 2 ,g : C → D,h = h 1 ,h 2 : A → C

=[

k 1 ,k 2 ]: B 1 + B 2 → D are such that k 1 ◦ f 1 =

◦

h 1 and k 2 ◦ f 2 =

and k g

g ◦ h 2 , so that we have the arrow ( h 1 ,h 2 ;[ k 1 ,k 2 ] ) : J(f) → J(g) and hence

T e ( h 1 ,h 2 ;[ k 1 ,k 2 ] ) =[ T e (h 1 ,k 1 ),T e (h 2 ,k 2 ) ]: f 1 + f 2 → g .
But also more complex cases:
6. The case when f

=[ f 1 ,f 2 ]: A 1 + A 2 → B,g =[ g 1 ,g 2 ]: C 1 + C 2 → D,h =

h 1 + h 2 : A 1 + A 2 → C 1 + C 2 and k : B → D are such that k ◦ f 1 = g 1 ◦ h 1 and

k ◦ f 2 = g 2 ◦ h 2 , so that we have the arrow (h 1 + h 2 ; k) : J(f) → J(g) and hence

T k f 1 ,f 2 ] ◦ T e (h 1 + h 2 ;

=

in g 1 , in g 2 ◦

◦[

(in f 1 + k) in f 2 )

= T e (h 1 ,k),in g 2 ◦

in f 1 ,

◦ ◦

T(k f 1 )

in g 1 ◦

in f 2 ,T e (h 2 ,k) : f 1 + f 2 →

T(k ◦ f 2 ) ◦ g 1 + g 2 .

7. The case when f = f 1 ,f 2 : A 1 + A 2 → B,g =[ g 1 ,g 2 ]: C 1 + C 2 → D,h =
h 1 ,h 2 : A 1 + A 2 → C 1 + C 2 and k =[ k 1 ,k 2 ]: B 1 + B 2 → D 1 + D 2 , so that
we obtain
T e h 1 + k 1 ,k 2 ]

= T e (h 1 ,k 1 ), in g 1 ◦

h 2 ;[

in f 2 ,

T(k 1 ◦ f 1 ) ◦