Functorial Semantics for Database Schema Mappings - Big Data Integration Theory - page 179

For the set of tgds mappings in M , the construction of this graph G is the same

I

as that presented in Definition 14 , with the mapping edge
M SG = TgdsToSOtgd( ) : S → G

M .

The integrity-constraint mapping
GA = EgdsToSOtgd Σ egd

G ∧

TgdsToConSOtgd Σ tgd

G : G → A ,

Σ egd Σ tgd

G

if Σ G =

G ∪

is a nonempty set of integrity constraints over the global

G

schema , is not explicitly defined as an edge in this graph G I , but it will be

introduced in the form of a mapping-operad arrow in the sketch category of this
system (in Proposition 15 ). Let Sch (

I ) be the category generated by this graph G I

(in Proposition 16 ). We can now define a mapping functor from the schema-level
sketch category into the instance-level category DB :

Proposition 18 Let Sch ( I ) be the sketch category of a given data integration

system I = G , S , M , generated from a graph G I in Definition 30 and Propo-

sition 16 , with the mapping arrow obtained from edge M SG ={ TgdsToSOtgd( M ) }

∈ G ,

I

M SG = MakeOperads( M SG ) : S → G ,

Σ egd Σ tgd

G

with the identity arrows for S , G and A and , if Σ T =

G ∪

is a nonempty

set of integrity constraints over the global schema G , with the integrity-constraint

arrow T GG = MakeOperads( { Φ } ) : G → A , where :

1. If Σ egd
G

and Σ tgd
G

are both nonempty then Φ is equal to

EgdsToSOtgd Σ egd

G ∧

TgdsToConSOtgd Σ tgd

G ;

2. If Σ egd
G

is empty and Σ tgd
G

is nonempty then Φ is TgdsToConSOtgd(Σ tgd

G ) ;

3. If Σ egd
G

is nonempty and Σ tgd
G

is empty then Φ is equal to EgdsToSOtgd(Σ egd
G

) .

If for this data integration system I = G , S , M , for a given instance D of the data

source schema S , there exists the universal ( canonical ) instance G = can( I ,D) of

the global schema legal w . r . t . D , then there exists the mapping-interpretation R-

algebra α and the functor ( categorial Lawvere's model ) α ∗ :

G

Sch ( I ) −→ DB such

that α ∗ ( D and α ∗ (

S ) = G ) = G = can( I ,D) .

Proof Directly from Propositions 16 , 17 , Definition 30 and the properties of DB ,
each morphism in DB represents a denotational semantics for a well defined ex-
change problem between two database instances, so we can define a functor for
such an exchange problem. In fact, if there is a canonical solution G = can( I ,D)

α ∗ ( of the global schema G , that is legal w.r.t. the source data instance D = S ) then,

by assuming that α ∗ ( G ) = can( I ,D) , all integrity constraints in Σ G are satisfied