PlanetPhysics/Category of Molecular Sets 2

Molecular Sets and Representations of Chemical Reactions
The uni-molecular chemical reaction is represented by the natural transformations $$\eta: h^A\longrightarrow h^B,$$ as specified by 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 the molecular sets $$A_u = a_1, \ldots, a_n$$ and $$B_u = b_1, \ldots b_n$$ being represented by certain endomorphisms in $$Hom(A,A)$$ and $$Hom(B,B)$$, respectively. In general, molecular sets $$M_S$$ are defined as finite sets whose elements are `molecules' defined in terms of their molecular observables that are specified below. molecular class variables, or $$m.c.v$$'s are defined as families of molecular sets $$[M_S]_{i \in I}$$, with $$I$$ being an indexing set, or class, defining the range of molecular variation of the $$m.c.v$$ ; most applications require that $$I$$ is a proper, finite set, (i.e., without any sub-classes). A morphism $$M_t: M_S \to M_S$$ 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.