Article
Full entry |
PDF
(1.1 MB)
Feedback
PDF
(1.1 MB)
Feedback
Keywords:
Opetope; Opetopic set; Operad; Polynomial monad; Projective sketch
Opetope; Opetopic set; Operad; Polynomial monad; Projective sketch
Summary:
We define a family of structures called “opetopic algebras”, which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday’s combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a “free pasting diagram” parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve functors for categories and for planar coloured Set-operads. This paper is the first in a series aimed at using opetopic spaces as models for higher algebraic structures.
References:
[1] Adámek, Jiří, Rosický, Jiří: Locally presentable and accessible categories. London mathematical society lecture note series, Cambridge University Press, Cambridge, ISBN:0-521-42261-2, DOI:10.1017/CBO9780511600579 DOI 10.1017/CBO9780511600579
[2] Baez, John C., Dolan, James: Higher-dimensional algebra. III. n-categories and the algebra of opetopes. Advances in Mathematics, Vol. 135, Iss. 2, 145-206, DOI:10.1006/aima.1997.1695 DOI 10.1006/aima.1997.1695
[3] Berger, Clemens, Melliès, Paul-André, Weber, Mark: Monads with arities and their associated theories. Journal of Pure and Applied Algebra, Vol. 216, Iss. 8-9, 2029-2048, https://doi.org/10.1016/j.jpaa.2012.02.039, DOI:10.1016/j.jpaa.2012.02.039 DOI 10.1016/j.jpaa.2012.02.039 | MR 2925893
[4] Cheng, Eugenia: The category of opetopes and the category of opetopic sets. Theory and Applications of Categories, Vol. 11, No. 16, 353-374 DOI 10.70930/tac/v8omllae | MR 2005691
[5] Cheng, Eugenia: Weak n-categories: Comparing opetopic foundations. Journal of Pure and Applied Algebra, Vol. 186, Iss. 3, 219-231, DOI:10.1016/S0022-4049(03)00140-3 DOI 10.1016/S0022-4049(03)00140-3 | MR 2025588
[6] Cheng, Eugenia: Weak n-categories: Opetopic and multitopic foundations. Journal of Pure and Applied Algebra, Vol. 186, Iss. 2, 109-137, DOI:10.1016/S0022-4049(03)00139-7 DOI 10.1016/S0022-4049(03)00139-7 | MR 2025593
[7] Curien, Pierre-Louis, Ho Thanh, Cédric, Mimram, Samuel: A sequent calculus for opetopes. LICS ’19: Proceedings of the 34th annual ACM/IEEE symposium on logic in computer science
[8] Gabriel, Peter, Ulmer, Friedrich: Lokal präsentierbare Kategorien. Lecture notes in mathematics, vol. 221, Springer-Verlag, Berlin-New York
[9] Gambino, Nicola, Kock, Joachim: Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 154, Iss. 1, 153-192, DOI:10.1017/S0305004112000394 DOI 10.1017/S0305004112000394 | MR 3002590
[10] Gepner, David, Haugseng, Rune, Kock, Joachim: \infty-Operads as Analytic Monads. International Mathematics Research Notices, https://doi.org/10.1093/imrn/rnaa332, DOI:10.1093/imrn/rnaa332 DOI 10.1093/imrn/rnaa332 | MR 4466007
[11] Ho Thanh, Cédric: The equivalence between many-to-one polygraphs and opetopic sets. arXiv e-prints
[12] Kock, Joachim: Polynomial functors and trees. International Mathematics Research Notices, Vol. 2011, Iss. 3, 609-673, DOI:10.1093/imrn/rnq068 DOI 10.1093/imrn/rnq068 | MR 2764874
[13] Kock, Joachim, Joyal, André, Batanin, Michael, Mascari, Jean-François: Polynomial functors and opetopes. Advances in Mathematics, Vol. 224, Iss. 6, 2690-2737, DOI:10.1016/j.aim.2010.02.012 DOI 10.1016/j.aim.2010.02.012 | MR 2652220
[14] Leinster, Tom: Higher operads, higher categories. Cambridge University Press, DOI:10.1017/cbo9780511525896 DOI 10.1017/cbo9780511525896 | MR 2094071
[15] Loday, Jean-Louis: Algebras, operads, combinads. Available at https://irma-web1.math.unistra.fr/~loday/
[16] Moerdijk, Ieke, Weiss, Ittay: Dendroidal sets. Algebraic & Geometric Topology, Vol. 7, 1441-1470, DOI:10.2140/agt.2007.7.1441 DOI 10.2140/agt.2007.7.1441 | MR 2366165
[17] Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture notes in mathematics, vol. 269, Springer-Verlag, Berlin-New York
[18] Weber, Mark: Familial 2-functors and parametric right adjoints. Theory and Applications of Categories, Vol. 18, No. 22, 665-732 DOI 10.70930/tac/9l84qqh9 | MR 2369114
Search
Partner of
