Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
Weak $\omega$-category; weak $\omega$-groupoid; weakly invertible cell; equivalence
Weak $\omega$-category; weak $\omega$-groupoid; weakly invertible cell; equivalence
Summary:
We study weakly invertible cells in weak $\omega$-categories in the sense of Batanin–Leinster, adopting the coinductive definition of weak invertibility. We show that weakly invertible cells in a weak $\omega$-category are closed under globular pasting. Using this, we generalise elementary properties of weakly invertible cells known to hold in strict $\omega$-categories to weak $\omega$-categories, and show that every weak $\omega$-category has a largest weak $\omega$-subgroupoid.
References:
[1] Batanin, M. A.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math., Vol. 136, Iss. 1, 39-103, https://doi.org/10.1006/aima.1998.1724, DOI:10.1006/aima.1998.1724 DOI 10.1006/aima.1998.1724
[2] Cheng, Eugenia: An \omega-category with all duals is an \omega-groupoid. Appl. Categ. Structures, Vol. 15, Iss. 4, 439-453, https://doi.org/10.1007/s10485-007-9081-8, DOI:10.1007/s10485-007-9081-8 DOI 10.1007/s10485-007-9081-8 | MR 2350215
[3] clingman, tslil: Towards the theory of proof-relevant categories. PhD thesis, Johns Hopkins University
[4] Cottrell, Thomas, Fujii, Soichiro: Hom weak \omega-categories of a weak \omega-category. Math. Structures Comput. Sci., Vol. 32, Iss. 4, 420-441, https://doi.org/10.1017/S0960129522000111, DOI:10.1017/S0960129522000111 DOI 10.1017/S0960129522000111 | MR 4522579
[5] Fujii, Soichiro, Hoshino, Keisuke, Maehara, Yuki: $\omega$-weak equivalences between weak \omega-categories. In preparation
[6] Garner, Richard: A homotopy-theoretic universal property of Leinster’s operad for weak \omega-categories. Math. Proc. Cambridge Philos. Soc., Vol. 147, Iss. 3, 615-628, https://doi.org/10.1017/S030500410900259X, DOI:10.1017/S030500410900259X DOI 10.1017/S030500410900259X
[7] Garner, Richard: Homomorphisms of higher categories. Adv. Math., Vol. 224, Iss. 6, 2269-2311, https://doi.org/10.1016/j.aim.2010.01.022, DOI:10.1016/j.aim.2010.01.022 DOI 10.1016/j.aim.2010.01.022 | MR 2652207
[8] Lafont, Yves, Métayer, François, Worytkiewicz, Krzysztof: A folk model structure on omega-cat. Adv. Math., Vol. 224, Iss. 3, 1183-1231, https://doi.org/10.1016/j.aim.2010.01.007, DOI:10.1016/j.aim.2010.01.007 DOI 10.1016/j.aim.2010.01.007 | MR 2628809
[9] Leinster, Tom: A survey of definitions of n-category. Theory Appl. Categ., Vol. 10, 1-70 MR 1883478
[10] Leinster, Tom: Higher operads, higher categories. London mathematical society lecture note series, Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511525896, ISBN:0-521-53215-9, DOI:10.1017/CBO9780511525896 DOI 10.1017/CBO9780511525896
[11] Power, A. J.: A 2-categorical pasting theorem. J. Algebra, Vol. 129, Iss. 2, 439-445, https://doi.org/10.1016/0021-8693(90)90229-H, DOI:10.1016/0021-8693(90)90229-H DOI 10.1016/0021-8693(90)90229-H
[12] Street, Ross: The algebra of oriented simplexes. J. Pure Appl. Algebra, Vol. 49, Iss. 3, 283-335, https://doi.org/10.1016/0022-4049(87)90137-X, DOI:10.1016/0022-4049(87)90137-X DOI 10.1016/0022-4049(87)90137-X
[13] Tarski, Alfred: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., Vol. 5, 285-309, http://projecteuclid.org/euclid.pjm/1103044538 DOI 10.2140/pjm.1955.5.285
[14] Verity, Dominic: Enriched categories, internal categories and change of base. Repr. Theory Appl. Categ., Iss. 20, 1-266 MR 2844536
[15] Weber, Mark: Symmetric operads for globular sets. PhD thesis, Macquarie University
[16] Weber, Mark: Generic morphisms, parametric representations and weakly Cartesian monads. Theory Appl. Categ., Vol. 13, No. 14, 191-234 MR 2116333
[17] Wolff, Harvey: V-cat and V-graph. J. Pure Appl. Algebra, Vol. 4, 123-135, https://doi.org/10.1016/0022-4049(74)90018-8, DOI:10.1016/0022-4049(74)90018-8 DOI 10.1016/0022-4049(74)90018-8
Search
Partner of
