Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-16T14:43:26.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Hom weak ω-categories of a weak ω-category

Published online by Cambridge University Press:  12 May 2022

Thomas Cottrell  and
Soichiro Fujii Open the ORCID record for Soichiro Fujii [Opens in a new window]
Thomas Cottrell
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Soichiro Fujii*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
*
*Corresponding author. Email: s.fujii.math@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak $\omega$ -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak $\omega$ -category based on an earlier definition by Batanin and construct, for each weak $\omega$ -category $\mathcal{A}$ , an underlying (weak $\omega$ -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak $\omega$ -category $\mathcal{A}(x,y)$ . We also show that our construction is functorial with respect to weak $\omega$ -functors introduced by Garner.

Keywords

Weak ω-categoryweak ω-groupoidweak ω-functoroperadintensional Martin-Löf type theoryidentity type
Type
Special Issue: The Power Festschrift
Information
Mathematical Structures in Computer Science , Volume 32 , Special Issue 4: The Power Festschrift , April 2022 , pp. 420 - 441
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We gratefully acknowledge the support of Royal Society grant IE160402. The second author is supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. No data were generated in association with this paper.

References

Altenkirch, T. and Rypacek, O. (2012). A syntactical approach to weak $\omega$ -groupoids. In: Computer Science Logic (CSL’12)-26th International Workshop/21st Annual Conference of the EACSL, Schloss Dagstuhl-Leibniz-Zentrum für Informatik.Google Scholar
Ara, D. (2010). Sur les $\infty$ -groupoïdes et une variante $\infty$ -catégorique. Phd thesis, Université Paris Diderot.Google Scholar
Ara, D. and Lucas, M. (2020). The folk model category structure on strict $\omega$ -categories is monoidal. Theory and Applications of Categories 35 (21) 745–808.Google Scholar
Baez, J. and Dolan, J. (1998). Categorification. In: Getzler, E. and Kapranov, M. (eds.) Higher Category Theory, Contemporary Mathematics, vol. 230, Providence, Rhode Island, American Mathematical Society, 1–36.CrossRefGoogle Scholar
Batanin, M. A. (1998). Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics 136 (1) 39–103.CrossRefGoogle Scholar
Bénabou, J. (1967). Introduction to bicategories. In: Reports of the Midwest Category Seminar, Springer, 1–77.CrossRefGoogle Scholar
Benjamin, T. (2020). A Type Theoretic Approach to Weak $\omega$ -Categories and Related Higher Structures. Phd thesis, Institut Polytechnique de Paris.Google Scholar
Benjamin, T., Finster, E. and Mimram, S. (2021). Globular weak $\omega$ -categories as models of a type theory. arXiv:2106.04475.Google Scholar
Bourke, J. and Garner, R. (2016a). Algebraic weak factorisation systems I: Accessible AWFS. Journal of Pure and Applied Algebra 220 (1) 108–147.CrossRefGoogle Scholar
Bourke, J. and Garner, R. (2016b). Algebraic weak factorisation systems II: Categories of weak maps. Journal of Pure and Applied Algebra 220 (1) 148–174.CrossRefGoogle Scholar
Brunerie, G. (2016). On the Homotopy Groups of Spheres in Homotopy Type Theory. Phd thesis, Université Nice Sophia Antipolis.Google Scholar
Cartmell, J. (1986). Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32 209–243.CrossRefGoogle Scholar
Cheng, E. (2007). An $\omega$ -category with all duals is an $\omega$ -groupoid. Applied Categorical Structures 15 (4) 439–453.CrossRefGoogle Scholar
Cheng, E. and Leinster, T. (2019). Weak $\infty$ -categories via terminal coalgebras. Theory and Applications of Categories 34 (34) 1073–1133.Google Scholar
Cottrell, T., Fujii, S. and Power, J. (2018). Higher dimensional categories: Induction on extensivity. Electronic Notes in Theoretical Computer Science 341 73–90. Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXIV).CrossRefGoogle Scholar
Finster, E. and Mimram, S. (2017). A type-theoretical definition of weak $\omega$ -categories. In: 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), IEEE, 1–12.Google Scholar
Garner, R. (2009a). A homotopy-theoretic universal property of Leinster’s operad for weak $\omega$ -categories. Mathematical Proceedings of the Cambridge Philosophical Society 147 (3) 615–628.CrossRefGoogle Scholar
Garner, R. (2009b). Understanding the small object argument. Applied Categorical Structures 17 (3) 247–285.CrossRefGoogle Scholar
Garner, R. (2010). Homomorphisms of higher categories. Advances in Mathematics 224 (6) 2269–2311.CrossRefGoogle Scholar
Gordon, R., Power, J. and Street, R. (1995). Coherence for Tricategories, vol. 558, Providence, Rhode Island, American Mathematical Society.Google Scholar
Grothendieck, A. (1983). Pursuing stacks. unpublished manuscript.Google Scholar
Hofmann, M. and Streicher, T. (1998). The groupoid interpretation of type theory. In: Sambin, G. and Smith, J. (eds.) Twenty-Five Years of Constructive Type Theory, New York, Oxford University Press, 83–111.Google Scholar
Kelly, G. M. (1982). Basic Concepts of Enriched Category Theory , London Mathematical Society Lecture Note Series, vol. 64, Cambridge-New York, Cambridge University Press.Google Scholar
Kelly, G. M. (1992). On clubs and data-type constructors. Applications of Categories in Computer Science 177 163–190.CrossRefGoogle Scholar
Lafont, Y., Métayer, F. and Worytkiewicz, K. (2010). A folk model structure on omega-cat. Advances in Mathematics 224 (3) 1183–1231.CrossRefGoogle Scholar
Leinster, T. (2002). A survey of definitions of n-category. Theory and Applications of Categories 10 (1) 1–70.Google Scholar
Leinster, T. (2004). Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series, vol. 298, Cambridge, Cambridge University Press.Google Scholar
Lumsdaine, P. L. (2010). Weak $\omega$ -categories from intensional type theory. Logical Methods in Computer Science 6 (3) 1–19.Google Scholar
Makkai, M. (1996). Avoiding the axiom of choice in general category theory. Journal of Pure and Applied Algebra 108 (2) 109–173.CrossRefGoogle Scholar
Maltsiniotis, G. (2010). Grothendieck $\infty$ -groupoids, and still another definition of $\infty$ -categories. arXiv:1009.2331.Google Scholar
May, J. P. (1972). The Geometry of Iterated Loop Spaces , Lecture Notes in Mathematics, vol. 271, New York, Springer-Verlag.Google Scholar
Métayer, F. (2003). Resolutions by polygraphs. Theory and Applications of Categories 11 (7) 148–184.Google Scholar
Nordström, B., Petersson, K. and Smith, J. M. (1990). Programming in Martin-Löf’s Type Theory: An Introduction, New York, Oxford University Press.Google Scholar
Pitts, A. M. (2000). Categorical logic. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, Logic and Algebraic Methods, vol. 5, New York, Oxford University Press, 39–128.Google Scholar
Street, R. (1972). The formal theory of monads. Journal of Pure and Applied Algebra 2 (2) 149–168.CrossRefGoogle Scholar
Univalent Foundations Program. (2013). Homotopy Type Theory: Univalent Foundations of Mathematics, 1st edn. https://homotopytypetheory.org/book/.Google Scholar
van den Berg, B. and Garner, R. (2011). Types are weak $\omega$ -groupoids. Proceedings of the London Mathematical Society 102 (2) 370–394.Google Scholar
Warren, M. A. (2011). The strict $\omega$ -groupoid interpretation of type theory. In: Hart, B., Kucera, T. G., Pillay, A., Scott, P. J. and Seely, R. A. G. (eds.) Models, Logics, and Higher-Dimensional Categories, Providence, Rhode Island, American Mathematical Society, 291–340.CrossRefGoogle Scholar
Wolff, H. (1974). $\mathcal{V}$ -cat and $\mathcal{V}$ -graph. Journal of Pure and Applied Algebra 4 (2) 123–135.CrossRefGoogle Scholar