Skip to main content
SpringerLink
Log in

A Combinatorial-Topological Shape Category for Polygraphs

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner’s directed complexes, which we use to define a realisation functor from constructible polygraphs to \(\omega \)-categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Al-Agl, F., Brown, R., Steiner, R.: Multiple categories: the equivalence of a globular and a cubical approach. Adv. Math. 170(1), 71–118 (2002). https://doi.org/10.1006/aima.2001.2069

    Article  MathSciNet  MATH  Google Scholar 

  2. Ara, D., Maltsiniotis, G.: Joint et tranches pour les \(\infty \)-catégories strictes. arXiv preprint arXiv:1607.00668 (2016)

  3. Baez, J., Dolan, J.: Higher-dimensional algebra III. n-Categories and the algebra of opetopes. Adv. Math. 135(2), 145–206 (1998). https://doi.org/10.1006/aima.1997.1695

    Article  MathSciNet  MATH  Google Scholar 

  4. Batanin, M.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math. 136(1), 39–103 (1998). https://doi.org/10.1006/aima.1998.1724

    Article  MathSciNet  MATH  Google Scholar 

  5. Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260(1), 159 (1980). https://doi.org/10.2307/1999881

    Article  MathSciNet  MATH  Google Scholar 

  6. Björner, A.: Posets, regular CW complexes and Bruhat order. Eur. J. Comb. 5(1), 7–16 (1984). https://doi.org/10.1016/s0195-6698(84)80012-8

    Article  MathSciNet  MATH  Google Scholar 

  7. Björner, A., Wachs, M.: On lexicographically shellable posets. Trans. Am. Math. Soc. 277(1), 323 (1983). https://doi.org/10.2307/1999359

    Article  MathSciNet  MATH  Google Scholar 

  8. Burroni, A.: Higher-dimensional word problems with applications to equational logic. Theor. Comput. Sci. 115(1), 43–62 (1993). https://doi.org/10.1016/0304-3975(93)90054-W

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockett, J., Koslowski, J., Seely, R.: Morphisms and modules for poly-bicategories. Theory Appl. Categ. 11(2), 15–74 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Cockett, J., Seely, R.: Weakly distributive categories. J. Pure Appl. Algebra 114(2), 133–173 (1997). https://doi.org/10.1016/0022-4049(95)00160-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Crans, S.: Pasting schemes for the monoidal biclosed structure on \(\omega \)-cat. Ph.D. thesis, Utrecht University (1995)

  12. Curien, P.L., Ho Thanh, C., Mimram, S.: Syntactic approaches to opetopes. arXiv preprint arXiv:1903.05848 (2019)

  13. Day, B.: On closed categories of functors. In: MacLane, S., et al. (eds.) Lecture Notes in Mathematics, pp. 1–38. Springer, Berlin (1970). https://doi.org/10.1007/bfb0060438

    Chapter  Google Scholar 

  14. Forest, S.: Unifying notions of pasting diagrams (2019). arXiv preprint arXiv:1903.00282

  15. Goerss, P., Jardine, J.: Simplicial Homotopy Theory. Birkhäuser, Basel (2009). https://doi.org/10.1007/978-3-0346-0189-4

    Book  MATH  Google Scholar 

  16. Guiraud, Y., Malbos, P.: Polygraphs of finite derivation type. Math. Struct. Comput. Sci. 28, 1–47 (2016). https://doi.org/10.1017/s0960129516000220

    Article  MathSciNet  MATH  Google Scholar 

  17. Hachimori, M.: Combinatorics of constructible complexes. Ph.D. thesis, University of Tokyo (2000)

  18. Hachimori, M.: Constructible complexes and recursive division of posets. Theor. Comput. Sci. 235(2), 225–237 (2000). https://doi.org/10.1016/s0304-3975(99)00195-4

    Article  MathSciNet  MATH  Google Scholar 

  19. Hadzihasanovic, A.: The algebra of entanglement and the geometry of composition. Ph.D. thesis, University of Oxford (2017)

  20. Hadzihasanovic, A.: Weak units, universal cells, and coherence via universality for bicategories. Theory Appl. Categ. 34(29), 883–960 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Hadzihasanovic, A.: Representable diagrammatic sets as a model of weak higher categories (2019). arXiv preprint arXiv:1909.07639

  22. Henry, S.: Regular polygraphs and the Simpson conjecture (2018). arXiv:1807.02627

  23. Henry, S.: Non-unital polygraphs form a presheaf category. Higher Structures 3(1) (2019)

  24. Hermida, C., Makkai, M., Power, J.: On weak higher dimensional categories i: part 1. J. Pure Appl. Algebra 154(1–3), 221–246 (2000). https://doi.org/10.1016/s0022-4049(99)00179-6

    Article  MathSciNet  MATH  Google Scholar 

  25. Johnson, M.: The combinatorics of n-categorical pasting. J. Pure Appl. Algebra 62(3), 211–225 (1989). https://doi.org/10.1016/0022-4049(89)90136-9

    Article  MathSciNet  MATH  Google Scholar 

  26. Joyal, A., Street, R.: The geometry of tensor calculus, I. Adv. Math. 88(1), 55–112 (1991). https://doi.org/10.1016/0001-8708(91)90003-p

    Article  MathSciNet  MATH  Google Scholar 

  27. Kapranov, M., Voevodsky, V.: \(\infty \)-groupoids and homotopy types. Cah. Topol. Géom. Différ. Catég. 32(1), 29–46 (1991)

    MathSciNet  MATH  Google Scholar 

  28. Kock, J., Joyal, A., Batanin, M., Mascari, J.C.: Polynomial functors and opetopes. Adv. Math. 224(6), 2690–2737 (2010). https://doi.org/10.1016/j.aim.2010.02.012

    Article  MathSciNet  MATH  Google Scholar 

  29. Kozlov, D.: Combinatorial Algebraic Topology. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-71962-5

    Book  MATH  Google Scholar 

  30. Lafont, Y.: Algebra and geometry of rewriting. Appl. Categ. Struct. 15(4), 415–437 (2007). https://doi.org/10.1007/s10485-007-9083-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Lafont, Y., Métayer, F., Worytkiewicz, K.: A folk model structure on omega-cat. Adv. Math. 224(3), 1183–1231 (2010). https://doi.org/10.1016/j.aim.2010.01.007

    Article  MathSciNet  MATH  Google Scholar 

  32. Leinster, T.: Higher Operads, Higher Categories. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/cbo9780511525896

    Book  MATH  Google Scholar 

  33. Lundell, A., Weingram, S.: The Topology of CW Complexes. Springer, New York (1969). https://doi.org/10.1007/978-1-4684-6254-8

    Book  MATH  Google Scholar 

  34. Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971). https://doi.org/10.1007/978-1-4612-9839-7

    Book  MATH  Google Scholar 

  35. May, J.: A Concise Course in Algebraic Topology. University of Chicago Press, Chicago (1999)

    MATH  Google Scholar 

  36. McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/cbo9780511546686

    Book  MATH  Google Scholar 

  37. Métayer, F.: Resolutions by polygraphs. Theory Appl. Categ. 11(7), 148–184 (2003)

    MathSciNet  MATH  Google Scholar 

  38. Métayer, F.: Cofibrant objects among higher-dimensional categories. Homol. Homotopy Appl. 10(1), 181–203 (2008). https://doi.org/10.4310/hha.2008.v10.n1.a7

    Article  MathSciNet  MATH  Google Scholar 

  39. Power, J.: An n-categorical pasting theorem. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Lecture Notes in Mathematics, pp. 326–358. Springer, Berlin (1991). https://doi.org/10.1007/bfb0084230

    Chapter  Google Scholar 

  40. Quillen, D.: Homotopical Algebra. Springer, Berlin (1967). https://doi.org/10.1007/bfb0097438

    Book  MATH  Google Scholar 

  41. Simpson, C.: Homotopy Theory of Higher Categories. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/cbo9780511978111

    Book  Google Scholar 

  42. Steiner, R.: The algebra of directed complexes. Appl. Categ. Struct. 1(3), 247–284 (1993). https://doi.org/10.1007/bf00873990

    Article  MathSciNet  MATH  Google Scholar 

  43. Steiner, R.: Omega-categories and chain complexes. Homol. Homotopy Appl. 6(1), 175–200 (2004). https://doi.org/10.4310/HHA.2004.v6.n1.a12

    Article  MathSciNet  MATH  Google Scholar 

  44. Street, R.: Limits indexed by category-valued 2-functors. J. Pure Appl. Algebra 8(2), 149–181 (1976). https://doi.org/10.1016/0022-4049(76)90013-X

    Article  MathSciNet  MATH  Google Scholar 

  45. Street, R.: The algebra of oriented simplexes. J. Pure Appl. Algebra 49(3), 283–335 (1987). https://doi.org/10.1016/0022-4049(87)90137-X

    Article  MathSciNet  MATH  Google Scholar 

  46. Street, R.: Parity complexes. Cah. Topol. Géom. Différ. Catég. 32(4), 315–343 (1991)

    MathSciNet  MATH  Google Scholar 

  47. Wachs, M.: Poset topology: tools and applications (2006). arXiv preprint arXiv:math/0602226

  48. Zawadowski, M.: On positive face structures and positive-to-one computads (2007). arXiv preprint arXiv:0708.2658

  49. Zawadowski, M.: Positive opetopes with contractions form a test category (2017). arXiv preprint arXiv:1712.06033

Download references

Acknowledgements

Supported by a JSPS Postdoctoral Research Fellowship and by JSPS KAKENHI Grant Number 17F17810. Many thanks to Joachim Kock and Jamie Vicary for their feedback on the parts which overlap with my thesis, and to Dimitri Ara, Yves Guiraud, Simon Henry, and Georges Maltsiniotis for helpful feedback and suggestions.

Author information

Authors and Affiliations

  1. RIMS, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan

    Amar Hadzihasanovic

Authors
  1. Amar Hadzihasanovic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amar Hadzihasanovic.

Additional information

Communicated by Ross Street.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hadzihasanovic, A. A Combinatorial-Topological Shape Category for Polygraphs. Appl Categor Struct 28, 419–476 (2020). https://doi.org/10.1007/s10485-019-09586-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-019-09586-6

Keywords

Mathematics Subject Classification

Search

Navigation