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Categorical Perspectives on Chemical Reaction Networks

Justin Curry, Mauricio Montes

Abstract

We show that the Schur-complement reduction of a chemical reaction network (CRN) from Hirono et al. is the categorical complement of the stoichiometric arrow in the arrow category $[\mathbf{A}_2,\mathbf{Vect}]$. This identifies the ambient category in which topological reduction of chemical reaction networks is functorial and explains the reduced stoichiometric matrix as a universal diagrammatic construction. We further define a reconstruction functor from a restricted subcategory of $[\mathbf{A}_2, \mathbf{Vect}]$ back to CRNs and prove an adjunction with the stoichiometric functor.

Categorical Perspectives on Chemical Reaction Networks

Abstract

We show that the Schur-complement reduction of a chemical reaction network (CRN) from Hirono et al. is the categorical complement of the stoichiometric arrow in the arrow category . This identifies the ambient category in which topological reduction of chemical reaction networks is functorial and explains the reduced stoichiometric matrix as a universal diagrammatic construction. We further define a reconstruction functor from a restricted subcategory of back to CRNs and prove an adjunction with the stoichiometric functor.

Paper Structure

This paper contains 6 sections, 8 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The Category of CRNs
  3. Categorical Comparisons
  4. Quiver Representations, Stoichiometry and Homology
  5. A Categorical Perspective on Topological Reduction
  6. The Categorical Schur Complement

Key Result

proposition 1

Every CRN $\Gamma$ admits an identity morphism to itself. Composition of CRN morphisms exists and is associative, thus making the collection of all CRNs, written CRN, into a category. Moreover, every $\Gamma\in \mathbf{ob}(\textbf{CRN})$ has a unique map to and a unique map from the trivial CRN $\va

Theorems & Definitions (19)

  • definition 1: Chemical Reaction Network, cf. Def. 8 of Hirono2021
  • definition 2: CRN morphism
  • proposition 1
  • proof
  • definition 3
  • proposition 2
  • proof
  • proposition 3
  • proof
  • corollary 1
  • ...and 9 more