# Construction of a Natural Transformation from a Classical to a Quantum 0-Species

## TL;DR

This paper constructs a natural transformation linking classical and quantum 0-species by extending functors into a category of topological linear spaces, highlighting a formal bridge between classical and quantum structures.

## Contribution

It introduces a natural transformation between classical and quantum 0-species within a new categorical framework involving topological linear spaces.

## Key findings

- Established a functorial link between classical and quantum 0-species.
- Extended the category of topological linear spaces to facilitate quantum-classical transition.
- Provided a formal categorical construction connecting classical and quantum dynamical patterns.

## Abstract

A natural transformation $\mathfrak{J}$ between functors valued in the category $\mathfrak{Chdv}_{0}$ is assembled. $\mathfrak{Chdv}_{0}$ is obtained by replacing both the categories $\mathrm{ptls}$ and $\mathrm{ptsa}$ with the category of topological linear spaces in the defining properties of the category $\mathfrak{Chdv}$ introduced in one of our previous papers. By letting a $\mathfrak{dp}$-valued functor be (classical) quantum whenever every its value is a dynamical pattern whose set map takes values in the set of (commutative) noncommutative topological unital $\ast-$algebras, and letting a (classical) quantum $0$-species be a $\mathfrak{Chdv}_{0}$-valued functor factorizing through the canonical functor from $\mathfrak{dp}$ to $\mathfrak{Chdv}_{0}$ into a (classical) quantum $\mathfrak{dp}$-valued functor, we have that the domain and codomain of $\mathfrak{J}$ are a classical and a quantum $0$-species respectively.