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Review and Constructive Definitions for Mathematically Engineered Systems as Categorical Interpretation

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Systems Engineering in Context

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Abstract

An analysis was performed to compare the expressibility under category theory for algebraic aspects to systems engineering. The purpose is to replicate the developed aspects under “model-based” specification theorems. A categorical definition of ‘system’ is constructed in the category of categories centrally defined by epimorphism and functor. Then the major theorems developed for model-based engineering are reconstructed and whose proof chains are presented through corresponding deductions from Wymore, A mathematical theory of systems engineering—The elements. New York: Wiley, 1967 and Awodey, Category theory, oxford logic guides. New York: Oxford Press, 2010. The resultant is twofold: showing “parallel” deductive abstractions and interpreted “categorical” primitives as deductive results. Algebraic differences exist in universality, indexing, and adjoints for engineering specifications, yet ‘system of system’ (constructions) are expressed by functors via graphical “diagrams.” Finally, type differences in both intension and extension are explored throughout, and supplementary representations are discussed for incorporating “set” and “structural” formalizations into system (engineering) aspects.

This work was supported both from an internship at MITRE Corporation and from research assistance through the Systems Engineering Research Center (SERC) under contract HQ0034-13-D-0004.

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Notes

  1. 1.

    Conditions 1 and 2 make for “functional-like” relations where the “collection” is (injectively) graphical, with each object as the vertex and the arrow an edge.

  2. 2.

    This can be thought of as “surjective on structure.”

  3. 3.

    Category development stems from areas that deal with continua, namely, in logical typings and algebraic topology.

  4. 4.

    His term for “assemblage” is simply the categorical “collection” without further conditions.

  5. 5.

    R stands for set of real numbers.

  6. 6.

    Similarly, different intervals yet the same “space,” different coordinate integers yet the same “shape” [12].

  7. 7.

    The linguistic usage of categories treats these as “semantics,” “interpretation,” and “syntax” labels, respectfully.

  8. 8.

    Should one identify epimorphisms directly or through coequalizers? Possibly different perspectives align with each of these or others in concepts.

  9. 9.

    As Wymore converges over structure, a homomorphism is defined similarly, but in models as set functions equipped with the operand property.

  10. 10.

    Note, the reals are grouped under “basic” operations in Wymore.

  11. 11.

    Not to mention the validity process about the actual artifacts, processes, and space.

  12. 12.

    A “for all unique (of a class)” quantifier condition.

  13. 13.

    Note: Congruence is usually presented using a monomorphism in (system) scientific classifications.

  14. 14.

    Pull-back statements are used, yet this generalizes to push-outs using coequalizers.

  15. 15.

    There is also a notion of antifoundation, situated order [8], as are Galois and Grothendieck underpinnings.

  16. 16.

    Such extensions do appear to be nonunique, for example, “Kan,” “strong,” and “weak” extensions have all been observed in the categorical literature.

  17. 17.

    Wymore centrally establishes duality in Corollary 4.4 (p. 170).

  18. 18.

    Additionally having σ be from the groups gives a “linear representation” [8].

  19. 19.

    The latter is a shared interest in categorical logic and algebraic geometry.

  20. 20.

    Wymore Def. 5.1, pp. 202–3.

  21. 21.

    Wymore Def. 5.2, p. 211.

  22. 22.

    Note that the normal presentation of the embedding is “flipped” given the focus on the epimorphism/right adjoint.

  23. 23.

    The corresponding lemma (Awoday 8.2, p. 188) and theorem (Awoday 8.5, p. 193) show this is a full, faithful, injective “embedding.”

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Acknowledgment

Special thanks to my advisor Michael Pennock for always maintaining his critical realism and for being constantly available. Thank you to my fellow students for their social and academic support. I acknowledge MITRE, where the initial intuition was developed, for its sponsorship, SERC for investigating its semantic use, and Spencer Breiner and Eswaran Subrahmanian of the NIST working group for helpful discussions and references relevant to applying categories in systems. Finally, as in all things, I express my eternal gratitude to friends and family for their unconditional love and support.

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Authors and Affiliations

  1. Socio-Technical Systems, Stevens Institute of Technology, Hoboken, NJ, USA

    Chris Klesges

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  1. Chris Klesges
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Correspondence to Chris Klesges .

Editor information

Editors and Affiliations

  1. Department of Systems and Information Engineering, University of Virginia, Charlottesville, VA, USA

    Stephen Adams

  2. Department of Systems and Information Engineering, University of Virginia, Charlottesville, VA, USA

    Peter A. Beling

  3. Department of Systems and Information Engineering, University of Virginia, Charlottesville, VA, USA

    James H. Lambert

  4. Department of Systems and Information Engineering, University of Virginia, Charlottesville, VA, USA

    William T. Scherer

  5. Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA, USA

    Cody H. Fleming

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Klesges, C. (2019). Review and Constructive Definitions for Mathematically Engineered Systems as Categorical Interpretation. In: Adams, S., Beling, P., Lambert, J., Scherer, W., Fleming, C. (eds) Systems Engineering in Context. Springer, Cham. https://doi.org/10.1007/978-3-030-00114-8_13

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