ABSTRACT
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A}\to \mathsf{X}$, a `structured cospan' is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. We give a new proof that if $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A}\to \mathbf {Cat}$, a `decorated cospan' is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathbf {Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X}= \smallint F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.
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[1] AlgebraicPetri team, GitHub repository. https://github.com/AlgebraicJulia/AlgebraicPetri.jl.
https://github.com/AlgebraicJulia/AlgebraicPetri.jl
[2] A. Baas, J. Fairbanks, M. Halter, S. Libkind and E. Patterson, An algebraic framework for structured epidemic modeling. https://doi.org/10.48550/arxiv.2203.16345.
https://doi.org/10.48550/arxiv.2203.16345
[3] J. C. Baez and K. Courser, Structured cospans, Theory Appl. Categ. 35 (2020), 1771–1822. https://doi.org/10.48550/arXiv.1911.04630.
https://doi.org/10.48550/arXiv.1911.04630
[4] J. C. Baez, B. Coya and F. Rebro, Props in network theory, Theory Appl. Categ. 33 (2018), 727–783. https://doi.org/10.48550/arXiv.1707.08321.
https://doi.org/10.48550/arXiv.1707.08321
[5] J. C. Baez and J. Erbele, Categories in control, Theory Appl. Categ. 30 (2015), 836–881. https://doi.org/10.48550/arXiv.1405.6881.
https://doi.org/10.48550/arXiv.1405.6881
[6] J. C. Baez and B. Fong, A compositional framework for passive linear networks, Theory Appl. Categ. 33 (2018), 1158–1222. https://doi.org/10.48550/arXiv.1504.05625.
https://doi.org/10.48550/arXiv.1504.05625
[7] J. C. Baez, B. Fong and B. S. Pollard, A compositional framework for Markov processes, Jour. Math. Phys. 57 (2016), 033301. https://doi.org/10.1063/1.4941578. Also available at https://doi.org/10.48550/arXiv.1508.06448.
https://doi.org/10.1063/1.4941578
[8] J. C. Baez and J. Master, Open Petri nets, Math. Struct. Comput. Sci. 30 (2020), 314–341. https://doi.org/10.1017/s0960129520000043. Also available at https://doi.org/10.48550/arXiv.1808.05415.
https://doi.org/10.1017/s0960129520000043
[9] J. C. Baez and B. S. Pollard, A compositional framework for reaction networks, Rev. Math. Phys. 29 (2017), 1750028. https://doi.org/10.1142/S0129055X17500283. Also available at https://doi.org/10.48550/arXiv.1704.02051.
https://doi.org/10.1142/S0129055X17500283
[10] G. Bakirtzis, C. H. Fleming and C. Vasilakopoulou, Categorical semantics of cyber-physical systems theory, ACM Trans. Cyber-Phys. Syst. 5 (2021). https://doi.org/10.1145/3461669. Also available at https://doi.org/10.48550/arXiv.2010.08003.
https://doi.org/10.48550/arXiv.2010.08003
[11] F. Bonchi, P. Sobociński and F. Zanasi, A categorical semantics of signal flow graphs, in CONCUR 2014–Concurrency Theory, eds. P. Baldan and D. Gorla, Lecture Notes in Computer Science 8704, Springer, Berlin, 2014, pp. 435–450. https://doi.org/10.1007/978-3-662-44584-6_30.
https://doi.org/10.1007/978-3-662-44584-6_30
[12] F. Borceux, Handbook of Categorical Algebra, vol. 2, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9780511525865.
https://doi.org/10.1017/CBO9780511525865
[13] M. Bunge and M. Fiore, Unique factorisation lifting functors and categories of linearly-controlled processes, Math. Struct. Comput. Sci. 10 (2000), 137–163. https://doi.org/10.1017/S0960129599003023.
https://doi.org/10.1017/S0960129599003023
[14] D. Cicala and C. Vasilakopoulou, On adjoints and fibrations. In preparation.
[15] K. Courser, A bicategory of decorated cospans, Theory Appl. Categ. 32 (2017), 995–1027. https://doi.org/10.48550/arXiv.1605.08100.
https://doi.org/10.48550/arXiv.1605.08100
[16] K. Courser, Open Systems: a Double Categorical Perspective, Ph.D. thesis, Department of Mathematics, U. C. Riverside, 2020. https://doi.org/10.48550/arXiv.2008.02394.
https://doi.org/10.48550/arXiv.2008.02394
[17] G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS 103 (2006), 8697–8702. https://doi.org/10.1073/pnas.0602767103.
https://doi.org/10.1073/pnas.0602767103
[18] B. Day and R. Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99–157. https://doi.org/10.1006/aima.1997.1649.
https://doi.org/10.1006/aima.1997.1649
[19] B. Fong, Decorated cospans, Theory Appl. Categ. 30 (2015), 1096–1120. https://doi.org/10.48550/arXiv.1502.00872.
https://doi.org/10.48550/arXiv.1502.00872
[20] B. Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, Computer Science Department, University of Oxford, 2016. https://doi.org/10.48550/arXiv.1609.05382.
https://doi.org/10.48550/arXiv.1609.05382
[21] B. Fong, P. Rapisarda and P. Sobocinski, A categorical approach to open and interconnected dynamical systems, in Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), IEEE, New York, 2016, pp. 1–10. https://doi.org/10.1145/2933575.2934556. Also available at https://doi.org/10.48550/arXiv.1510.05076.
https://doi.org/10.1145/2933575.2934556
[22] C. Girault and R. Valk, Petri Nets for Systems Engineering: a Guide to Modeling, Verification, and Applications, Springer, Berlin, 2013. https://doi.org/10.1007/978-3-662-05324-9.
https://doi.org/10.1007/978-3-662-05324-9
[23] R. Gordon, A. J. Power and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 558, 1995. https://doi.org/10.1090/memo/0558.
https://doi.org/10.1090/memo/0558
[24] M. Grandis and R. Paré, Limits in double categories, Cah. Top. Géom. Diff. 40 (1999), 162–220. http://www.numdam.org/item/CTGDC$\underline{ }$1999$\underline{ }$40$\underline{ }$3$\underline{ }$162$\underline{ }$0/.
http://www.numdam.org/item/CTGDC_1999__40_3_162_0/
[25] M. Grandis and R. Paré, Adjoints for double categories, Cah. Top. Géom. Diff. 45 (2004), 193–240. http://www.numdam.org/item/CTGDC$\underline{ }$2004$\underline{ }$45$\underline{ }$3$\underline{ }$193$\underline{ }$0/.
http://www.numdam.org/item/CTGDC_2004__45_3_193_0/
[26] J. Gray, Fibred and cofibred categories, in Proceedings of the Conference on Categorical Algebra: La Jolla 1965, eds. S. Eilenberg et al, Springer, Berlin, 1966, pp. 21–83. https://doi.org/10.1007/978-3-642-99902-4_2.
https://doi.org/10.1007/978-3-642-99902-4_2
[27] P. J. Haas, Stochastic Petri Nets: Modelling, Stability, Simulation, Springer, Berlin, 2002. https://doi.org/10.1007/b97265.
https://doi.org/10.1007/b97265
[28] L. W. Hansen and M. Shulman, Constructing symmetric monoidal bicategories functorially. https://doi.org/10.48550/arXiv.1910.09240.
https://doi.org/10.48550/arXiv.1910.09240
[29] C. Hermida, Some properties of Fib as a fibred 2-category, J. Pure Appl. Alg. 134 (1999), 83–109. https://doi.org/10.1016/S0022-4049(97)00129-1.
https://doi.org/10.1016/S0022-4049(97)00129-1
[30] B. Jacobs, Categorical Logic and Type Theory, Elsevier, Amsterdam, 1999. https://doi.org/10.2307/4212146.
https://doi.org/10.2307/4212146
[31] A. Joyal, M. Nielsen and G. Winskel, Bisimulation from open maps, Inf. Comput. 127 (1996), 164–185. https://doi.org/10.1006/inco.1996.0057.
https://doi.org/10.1006/inco.1996.0057
[32] P. Katis, N. Sabadini and R. F. C. Walters, On the algebra of systems with feedback and boundary, Rendiconti del Circolo Matematico di Palermo Serie II 63 (2000), 123–156. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.51.5758&rep=rep1&type=pdf.
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.51.5758&rep=rep1&type=pdf
[33] G. M. Kelly and R. Street, Review of the elements of 2-categories, in Category Seminar, ed. G. M. Kelly, Lecture Notes in Mathematics 40, Springer, Berlin, 1974, pp. 75–103. https://doi.org/10.1007/BFb0063101.
https://doi.org/10.1007/BFb0063101
[34] I. Koch, Petri nets—a mathematical formalism to analyze chemical reaction networks, Mol. Inform. 29 (2010), 838–843. https://doi.org/10.1002/minf.201000086.
https://doi.org/10.1002/minf.201000086
[35] F. W. Lawvere, State categories and response functors, unpublished manuscript, 1986. https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1986-state-categories-and-response-functors.pdf.
https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1986-state-categories-and-response-functors.pdf
[36] P. McCrudden, Balanced coalgebroids, Theory Appl. Categ. 7 (2000), 71–147. https://www.emis.de/journals/TAC/volumes/7/n6/7-06abs.html.
https://www.emis.de/journals/TAC/volumes/7/n6/7-06abs.html
[37] J. Moeller and C. Vasilakopoulou, Monoidal Grothendieck construction, Theory Appl. Categ. 35 (2020), 1159–1207. https://doi.org/10.48550/arXiv.1809.00727.
https://doi.org/10.48550/arXiv.1809.00727
[38] S. Niefield, Span, cospan, and other double categories, Theory Appl. Categ. 26 (2012), 729–742. https://doi.org/10.48550/arXiv.1201.3789.
https://doi.org/10.48550/arXiv.1201.3789
[39] J. L. Peterson, Petri Net Theory and the Modeling of Systems, Prentice-Hall, New Jersey, 1981. http://dl.icdst.org/pdfs/files3/2bf95f7fde49a09814231bbcbe592526.pdf.
http://dl.icdst.org/pdfs/files3/2bf95f7fde49a09814231bbcbe592526.pdf
[40] B. S. Pollard, Open Markov Processes and Reaction Networks, Ph.D. thesis, Department of Physics, U. C. Riverside, 2017. https://doi.org/10.48550/arXiv.1709.09743.
https://doi.org/10.48550/arXiv.1709.09743
[41] E. Riehl, Category Theory In Context, Dover Publications, New York, 2016. https://math.jhu.edu/ eriehl/context/.
https://math.jhu.edu/~eriehl/context/
[42] P. Schultz, D. Spivak and C. Vasilakopoulou, Dynamical systems and sheaves, Appl. Cat. Struct. 28 (2020), 1–57. https://doi.org/10.1007/s10485-019-09565-x. Also available at https://doi.org/10.48550/arXiv.1609.08086.
https://doi.org/10.48550/arXiv.1609.08086
[43] M. Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008), 650–738. https://doi.org/10.48550/arXiv.0706.1286.
https://doi.org/10.48550/arXiv.0706.1286
[44] M. Shulman, Constructing symmetric monoidal bicategories. https://doi.org/10.48550/arXiv.1004.0993.
https://doi.org/10.48550/arXiv.1004.0993
[45] M. Stay, Compact closed bicategories, Theory Appl. Categ. 31 (2016), 755–798. https://doi.org/10.48550/arXiv.1301.1053.
https://doi.org/10.48550/arXiv.1301.1053
[46] D. Vagner, D. Spivak and E. Lerman, Algebras of open dynamical systems on the operad of wiring diagrams, Theory Appl. Categ. 30 (2015), 1793–1822. https://doi.org/10.48550/arXiv.1408.1598.
https://doi.org/10.48550/arXiv.1408.1598
[47] D. J. Wilkinson, Stochastic Modelling for Systems Biology, Taylor and Francis, New York, 2006. https://doi.org/10.1201/b11812.
https://doi.org/10.1201/b11812
[48] J. C. Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems Magazine 27 (2007), 46–99. https://doi.org/10.1109/MCS.2007.906923.
https://doi.org/10.1109/MCS.2007.906923
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