# 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:

$\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}},$

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:

$\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},}$

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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 )