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A mathematical model for cognitive structures and processes underlying innovations

Jürgen Jost, Massimo Warglien

Abstract

Business innovations are often arising from new combinations of pre-existing systems into new ones, coherently assembling features from their sources. We propose an abstract mathematical concept from category theory, the presheaf, to efficiently represent such coherent feature combinations. Moreover, operations on presheaves will allow us to formally describe and analyze business innovations that arise from novel mergings of systems from different domains. Equipped with such tools we provide an example by analyzing a successful case of such type of recombinant innovation, the digital hub concept proposed by Steve Jobs. The example shows how our framework can be used to bring formal rigor while preserving a fundamentally qualitative reasoning style.

A mathematical model for cognitive structures and processes underlying innovations

Abstract

Business innovations are often arising from new combinations of pre-existing systems into new ones, coherently assembling features from their sources. We propose an abstract mathematical concept from category theory, the presheaf, to efficiently represent such coherent feature combinations. Moreover, operations on presheaves will allow us to formally describe and analyze business innovations that arise from novel mergings of systems from different domains. Equipped with such tools we provide an example by analyzing a successful case of such type of recombinant innovation, the digital hub concept proposed by Steve Jobs. The example shows how our framework can be used to bring formal rigor while preserving a fundamentally qualitative reasoning style.

Paper Structure

This paper contains 14 sections, 7 theorems, 43 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Presheaves and the organization of structures
  3. Categories and presheaves
  4. Operations on presheaves and the reorganization and combination of structures
  5. Diagrams
  6. Categories of power sets
  7. Applications to business innovations. The Apple case
  8. Sections
  9. The digital hub
  10. Amalgamation
  11. Transferring structure
  12. Transferring structure and preserving constraints
  13. Discussion
  14. Acknowledgements

Key Result

Lemma 2.1

defines a functor from $\mathbf{C^{op}}$ to $\mathbf{Sets}$.

Figures (6)

  • Figure 1: Strategy canvas of the U.S. wine industry in the late 1990s, from KM
  • Figure 2: The digital hub
  • Figure 3: iMovie: a pictorial representation of the amalgamation of PC and Camcorder
  • Figure 4: The structure underlying Apple's "Digital hub" wheel
  • Figure 5: iMovie: a pictorial representation of the amalgamation of PC and MP3 player
  • ...and 1 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • Definition 2.6
  • Lemma 2.2
  • Definition 2.7
  • Lemma 2.3
  • ...and 12 more