Database Reference
In-Depth Information
where, as usual, the square brackets denote the equivalence classes for terms
t
i
).
B
in
DB
sk
returns the homomorphism
f
H
=
U
1
(f )
from the
Σ
R
-algebra
U(A)
into the
Σ
R
-algebra
U(B)
such that for any term
o
i
(
[
t
1
]
,...,
[
t
n
]
)
∈
U(A)
,
o
i
∈
Σ
R
,we
obtain
f
H
(o
i
(
[
t
1
]
,...,
[
t
n
]
))
=
o
i
(f
H
(
[
t
1
]
),...,f
H
(
[
t
n
]
))
∈
U(B)
.
This universal functor
U
, for any simple morphism
f
:
A
−→
•
Its adjoint forgetful functor
F
DB
sk
such that, for any free algebra
U(A)
in
K
, the object
F(U(A))
in
DB
sk
is equal to its carrier-set
A
(each term
o
i
(
:
K
−→
U(A)
is evaluated into a relation that is an element of this
closed object
A
) and for each arrow
U
1
(f )
,
F
1
(U
1
(f ))
[
t
1
]
,...,
[
t
]
n
)
∈
=
f
, i.e.,
FU
=
Id
DB
sk
and
UF
=
Id
K
.
Remark
In this universal algebra framework, we preferred to use the skeletal cat-
egory
DB
sk
(which will be used for the algebraic considerations in what follows),
instead of the complete category
DB
, equivalent to it. Note that if we use the
DB
then the unique universal functor
U
:
DB
→
K
is equal to the previous one, and
the adjoint forgetful functor
F
:
K
→
DB
is defined by
F(U(A))
=
TA
(when
A
TA
), with
F
1
(U(f ))
is a closed object then
A
=
=
Tf
and hence
TA
can be sub-
stituted by
A
because they are isomorphic (
A
TA
in
DB
), and
Tf
:
TA
→
TB
is equivalent to
f
:
A
→
B
(i.e.,
f
≈
Tf
) because they have the same information
fluxes,
f
=
Tf
.
It is immediate from the universal property that the mapping
A
→
U(A)
extends
◦
:
DB
sk
−→
to the endofunctor
F
DB
sk
. This functor carries a
monad structure
(F
◦
U,η,μ)
with
F
◦
U
an equivalent version of
T
but for this skeletal database
category
DB
sk
(while for the equivalent version for
DB
we obtain the monad
T
,
as demonstrated in Proposition
58
in Sect.
8.3
). The natural transformation
η
is
given by the obvious “inclusion” of
A
into
(F
U
]
(each view (relation)
R
in a closed object
A
is an equivalence class of all algebra
terms which produce this view). The natural transformation
η
is the unit of this
adjunction of
U
and
F
, and hence it corresponds to an injective function
◦
U)(A)
=
F(U(A))
:
R
−→ [
R
−→
U(A)
in
Set
, given previously. The interpretation of
μ
is almost equally simple.
An element of
(F
:
A
U)
2
(A)
is an equivalence class of terms built up from elements
◦
U)
2
(A)
of
(F
◦
U)(A)
, so that instead of
t(x
1
,...,x
k
)
, a typical element of
(F
◦
[
t
1
]
[
t
k
]
is given by the equivalence class of a term
t(
,...,
)
. The transformation
μ
is defined by 'flattening-terms' map
. This makes
sense because a substitution of provably equal expressions into the same term results
in provably equal terms.
The monadic properties of the endofunctor
T
=
F
◦
U
will be explored deeper
in Sect.
8.3
, after examination of the algebraic lattice, lfp (locally finite presentable
properties of
DB
category and its enrichments.
[
t(
[
t
1
]
,...,
[
t
k
]
)
]→[
t(t
1
,...,t
k
)
]
=
1
≤
j
≤
m
A
j
,
m
Remark
Consider a generalization to complex objects
A
≥
2.
DB
OP
A bifunctor
E
:
sk
×
K
−→
Set
for any
complex
database instance
A
in
DB
sk
generates a functor
E(A,
_
)
:
K
−→
Set
with a universal element
(U(A),)
,
