PlanetPhysics/Category of Molecular Sets 4

Molecular sets as representations of chemical reactions

A [[../Molecule/|uni-molecular chemical reaction]] is defined by the [[../VariableCategory2/|natural transformations]] $\eta :h^A⟶h^B,$ specified in 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 [[../Molecule/|states of molecular sets]] $A_u=a_1,\dots ,a_n$ and $B_u=b_1,\dots b_n$ 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 [[../Molecule/|molecules]]; the molecules are mathematically defined in terms of their molecular [[../QuantumSpinNetworkFunctor2/|observables]] as specified next. In order to define molecular observables one needs to define first the [[../PreciseIdea/|concept]] of a [[../Molecule/|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 [[../TrivialGroupoid/|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 [[../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 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.

[[../TrivialGroupoid/|Classification]]: AMS MSC: 18D35 ([[../TrivialGroupoid/|category theory]]; homological algebra :: [[../TrivialGroupoid/|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 :: [[../AbelianCategory2/|abelian categories]] :: Preadditive, [[../DenseSubcategory/|additive categories]]) 81-00 ([[../QuantumOperatorAlgebra5/|quantum theory]] :: General reference [[../Work/|works]] )