Biology Reference
In-Depth Information
such that for every object X and every family of morphisms
f
j
O
j
→
X
there is a unique morphism fC
→
X such that
f
j
=
f
j
for all j. In the form of a commutative diagram, for every object X in the
category, for every j, there exists a unique morphism f that makes the following
diagram commute for all j:
O
j
f
j
I
j
C
=
O
j
X
f
j
The direct sum is thus a dual concept to the direct product with the arrows
being reversed in the commutative diagrams. In the category of sets,
Set
, the
co-product is the disjoint union C
=
j
O
j
and
j
O
j
→
C is the inclusion.
In the category of abelian groups, the co-product is the group direct sum, often
denoted
C
=
j
O
j
and
j
O
j
→
C are the injections of the jth summand. In the category of groups,
Grp
, the coproduct is the free product of groups. When the objects, from which
the direct product/sum is formed, are abelian, e.g. vector spaces (modules over a
field), or abelian groups (modules over integers) then the direct sum is the same
as the direct product. We return to this question further below in the context of
dynamic pathway modelling.
Next, we define for each O
j
a fixed analytical model, using Eqn 2 as a
template
=
i
a
O
j
ij
O
j
(4)
where
ij
is an observable or state variable on O
j
ij
O
j
→
ij
O
j
q
j
→
ij
q
j
