# 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:
   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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:
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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 )