PlanetPhysics/Category of Molecular Sets 4

Molecular sets as representations of chemical reactions
A uni-molecular chemical reaction is defined by the natural transformations $$\eta: h^A\longrightarrow h^B,$$ specified in the following commutative diagram: $$ \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}}, $$

with the states of molecular sets $$A_u = a_1, \ldots, a_n$$ and $$B_u = b_1, \ldots b_n$$ being defined as the endomorphism sets $$Hom(A,A)$$ and $$Hom(B,B)$$, respectively. In general, molecular sets $$M_S$$ are defined as finite sets whose elements are molecules; the molecules  are mathematically defined in terms of their molecular observables as specified next. In order to define molecular observables one needs to define first the concept of a molecular class variable or $$m.c.v$$.

A molecular class variables is defined as a family of molecular sets $$[M_S]_{i \in I}$$, with $$I$$ being either an indexing set, or a proper class, that defines the variation range of the $$m.c.v$$. Most physical, chemical or biochemical applications require that $$I$$ is restricted to a finite set, (that is, without any sub-classes). A morphism, or molecular mapping, $$M_t: M_S \to M_S$$ of molecular sets, with $$t \in T$$ being real time values, is defined as a time-dependent mapping or function $$M_S (t)$$ also called a molecular transformation, $$M_t$$.

An $$m.c.v.$$ observable of $$B$$, characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:

$$\gamma : Hom(B,B) \longrightarrow \Re ,$$ where $$\Re$$ is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions: $$ \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},} $$ ~ with $$c: A^*_u \longrightarrow B^*_u$$, and $$A^*_u$$, $$B^*_u$$ being, respectively, specially prepared fields of states of the molecular sets $$A_u$$, and $$B_u$$ within a measurement uncertainty range, $$\Delta$$, which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as $$[A^*, B^*]$$, associated with the observable $$A$$ of molecular set $$A_u$$, and respectively, with the obssevable $$B$$ of molecular set $$B_u$$, in the case of a molecular set $$A_u$$ interacting with molecular set $$B_u$$.

With these concepts and preliminary data one can now define the category of molecular sets and their transformations as follows.

Category of molecular sets and their transformations
The category of molecular sets is defined as the category $$C_M$$ whose objects are molecular sets $$M_S$$ and whose morphisms are molecular transformations $$M_t$$.

This is a mathematical representation of chemical reaction systems in terms of molecular sets that vary with time (or $$msv$$'s), and their transformations as a result of diffusion, collisions, and chemical reactions.

Classification: AMS MSC: 18D35 (category theory; homological algebra :: categories with structure :: Structured objects in a category ) 92B05 (Biology and other natural sciences :: Mathematical biology in general :: General biology and biomathematics) 18E05 (Category theory; homological algebra :: abelian categories :: Preadditive, additive categories) 81-00 (quantum theory :: General reference works )