PlanetPhysics/Category of Molecular Sets 2

Molecular Sets and Representations of Chemical Reactions

The [[../Molecule/|uni-molecular chemical reaction]] is represented by the [[../VariableCategory2/|natural transformations]] $\eta :h^A⟶h^B,$ as specified by the following [[../Commutativity/|commutative diagram]]:

Failed to parse (unknown function "\def"): {\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 the molecular sets $A_u=a_1,\dots ,a_n$ and $B_u=b_1,\dots b_n$ 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 `[[../Molecule/|molecules]]' defined in terms of their molecular [[../QuantumSpinNetworkFunctor2/|observables]] that are specified below. [[../Molecule/|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 [[../TrivialGroupoid/|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 [[../Bijective/|function]] $M_S(t)$ also called a [[../Molecule/|molecular transformation]], $M_t$.

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

$\gamma :Hom(B,B)⟶\mathrm{\Re },$ where $\mathrm{\Re }$ is the set or [[../CosmologicalConstant/|field]] of real numbers. This mcv-observable is subject to the following [[../TrivialGroupoid/|commutativity]] conditions:

Failed to parse (unknown function "\def"): {\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},} }

~

with $c:A_u^{*}⟶B_u^{*}$, 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, $\mathrm{\Delta }$, which is determined by Heisenberg's uncertainty [[../Bijective/|relation]], or the [[../Commutator/|commutator]] of the observable [[../QuantumOperatorAlgebra4/|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 [[../PreciseIdea/|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 [[../Cod/|category]] $C_M$ whose [[../TrivialGroupoid/|objects]] are molecular sets $M_S$ and whose morphisms are molecular transformations $M_t$.

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