Abstract
This is the first part of a series of three strongly related papers in which three equivalent structures are studied:
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internal categories in categories of monoids, defined in terms of pullbacks relative to a chosen class of spans
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crossed modules of monoids relative to this class of spans
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simplicial monoids of so-called Moore length 1 relative to this class of spans.
The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category.
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Aguiar, M.: Internal Categories and Quantum Groups, Ph.D. thesis, Cornell University (1997)
André, M.: Méthode simpliciale en algèbre homologique et algèbre commutative. Springer LNM 32. Springer, Berlin (1967)
Andruskiewitsch, N., Devoto, J.A.: Extensions of Hopf algebras. Algebra i Analiz 7(1), 22–61 (1995). (English version: St. Petersburg Mathematical Journal 7 no. 1 (1995), 17–52)
Baez, J.C., Lauda, A.D.: Higher-dimensional algebra V: 2-groups. Theory Appl. Categ. 12(4), 423–491 (2004)
Bántay, P.: Characters of crossed modules and premodular categories. In: Lepowsky, J., McKay, J., Tuite, M.P. (eds.) Moonshine—The First Quarter Century and Beyond. Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras. London Math. Soc. LNS 372, pp. 1–12, Cambridge University Press (2010)
Baues, H.-J.: Homotopy Type and Homology. Oxford University Press, Oxford (1996)
Böhm, G.: Crossed modules of monoids II. Relative crossed modules, preprint arXiv:1803.04124
Böhm, G.: Crossed modules of monoids III. Simplicial monoids of Moore length 1, preprint arXiv:1803.04622
Breen, L.: Bitorseurs et cohomologie non abèlienne. In: The Grothendieck Festschrift Vol. I, Progr. Math., vol. 86, pp. 401–476, Birkhäuser Boston, Boston, MA (1990)
Breen, L., Messing, W.: Differential geometry of gerbes. Adv. Math. 198(2), 732–846 (2005)
Brown, R., Higgins, P.J., Sivera, R.: Nonabelian algebraic topology: Filtered spaces, crossed complexes, cubical homotopy groupoids. EMS Monographs in Math., vol. 15. European Mathematical Society (2010)
Brown, R., İçen, İ.: Homotopies and automorphisms of crossed modules of groupoids. Appl. Categ. Struct. 11(2), 185–206 (2003)
Brown, R., Spencer, C.B.: \(\cal{G}\)-groupoids, crossed modules and the fundamental groupoid of a topological group. Indag. Math. (Proc.) 79(4), 296–302 (1976)
Conduché, D.: Modules croisés généralisés de longueur 2. In: Friedlander, E.M., Karoubi, M. (eds.) Proceedings of the Luminy conference on algebraic KK-theory (Luminy, 1983). J. Pure Appl. Algebra, vol. 34(2–3), pp. 155–178 (1984)
Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Commun. Math. Phys. 129(2), 393–429 (1990)
Duskin, J.W.: Preliminary remarks on groups, as quoted in [13]: Unpublished notes, Tulane University (1969)
Ellis, G., Steiner, R.: Higher dimensional crossed modules and the homotopy groups of \((n+1)\)-ads. J. Pure Appl. Algebra 46(2–3), 117–136 (1987)
Emir, K.: The Moore Complex of a Simplicial Cocommutative Hopf Algebra, preprint arXiv:1905.09620
Faria Martins, J.: Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy. J. Geom. Phys. 99, 68–110 (2016)
Fernández Vilaboa, J.M., López López, M.P., Villanueva Novoa, E.: \(\text{ Cat }^1\)-Hopf algebras and crossed modules. Commun. Algebra 35(1), 181–191 (2006)
García-Martínez, X., Van der Linden, T.: A note on split extensions of bialgebras. Forum Math. 30(5), 1089–1095 (2018)
Gran, M., Kadjo, G., Vercruysse, J.: A torsion theory in the category of cocommutative Hopf algebras. Appl. Categ. Struct. 24(3), 269–282 (2016)
Gran, M., Sterck, F., Vercruysse, J.: A semi-abelian extension of a theorem by Takeuchi. J. Pure Appl. Algebra 223, 4171–4190 (2019)
Guin-Waléry, D., Loday, J.-L.: Obstructions á l’excision en K-théorie algèbrique. In: Evanston Conference on Algebraic K-theory 1980, Springer LNM 854, pp. 179-216. Springer (1981)
Higgins, P.J.: Groups with multiple operators. Proc. Lond. Math. Soc. 3–6(6), 366–416 (1956)
Janelidze, G.: Internal crossed modules. Georgian Math. J. 10, 99–114 (2003)
Lavendhomme, R., Roisin, J.-R.: Cohomologie non abélienne de structures algébriques. J. Algebra 67, 385–414 (1980)
Loday, J.-L.: Spaces with finitely many nontrivial homotopy groups. J. Pure Appl. Algebra 24(2), 179–202 (1982)
Mac Lane, S.: Homology. Grundlehren der mathematischen Wissenschaften, vol. 114. Springer, Berlin (1967)
Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, Berlin (1978)
Majid, S.: Strict quantum 2-groups, preprint arXiv:1208.6265
Martins-Ferreira, N., Montoli, A., Sobral, M.: Semidirect products and crossed modules in monoids with operations. J. Pure Appl. Algebra 217(2), 334–347 (2013)
Paoli, S.: Internal categorical structures in homotopical algebra. In: Baez, J.C., May, J.P. (eds.) Towards Higher Categories. IMA Volumes in Mathematics and Its Applications, pp. 85–103. Springer (2009)
Patchkoria, A.: Crossed semimodules and Schreier internal categories in the category of monoids. Georgian Math. J. 5(6), 575–581 (1998)
Porter, T.: Extensions, crossed modules and internal categories in categories of groups with operations. Proc. Edinb. Math. Soc. 30, 373–381 (1987)
Porter, T.: \(n\)-types of simplicial groups and crossed n-cubes. Topology 32, 5–24 (1993)
Porter, T.: The Crossed Menagerie: an introduction to crossed gadgetry and cohomology in algebra and topology. Notes initially prepared for the XVI Encuentro Rioplatense de Álgebra y Geometría Algebraica, in Buenos Aires, 12–15 December 2006, extended for an MSc course (Summer 2007) at Ottawa. https://ncatlab.org/timporter/show/crossed+menagerie
Porter, T.: Topological quantum field theories from homotopy n-types. J. Lond. Math. Soc. 58(3), 723–732 (1998)
Porter, T.: Homotopy quantum field theories meets the crossed menagerie: an introduction to HQFTs and their relationship with things simplicial and with lots of crossed gadgetry. Notes prepared for the Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity Lisbon, February, 2011. https://ncatlab.org/timporter/show/HQFTs+meet+the+Crossed+Menagerie
Porter, T., Turarev, V.G.: Formal homotopy quantum field theories, I: formal maps and crossed C-algebras. J. Homotopy Relat. Struct. 3, 113–159 (2008)
Quillen, D.G.: Homotopical Algebra. Springer LNM 53. Springer, Berlin (1967)
Van der Linden, T.: Homology and homotopy in semi-Abelian categories, Ph.D. thesis, Vrije Universiteit Brussel (2006)
Whitehead, J.H.C.: Combinatorial homotopy I. Bull. Am. Math. Soc. 55, 213–245 (1949)
Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Am. Math. Soc. 55, 453–496 (1949)
Yetter, D.N.: Topological quantum field theories associated to finite groups and crossed G-sets. J. Knot Theory Ramif. 1, 1–20 (1992)
Yetter, D.N.: TQFT’s from homotopy 2-types. J. Knot Theory Ramif. 2, 113–123 (1993)
Acknowledgements
Open access funding provided by MTA Wigner Research Centre for Physics (MTA Wigner FK, MTA EK). The author’s interest in the subject was triggered by the excellent workshop ‘Modelling Topological Phases of Matter—TQFT, HQFT, premodular and higher categories, Yetter-Drinfeld and crossed modules in disguise’ in Leeds UK, 5–8 July 2016. It is a pleasure to thank the organizers, Zoltán Kádár, João Faria Martins, Marcos Calçada and Paul Martin for the experience and a generous invitation. Helpful comments and a list of relevant references from the anonymous referee are highly appreciated. Financial support by the Hungarian National Research, Development and Innovation Office—NKFIH (Grant K124138) is gratefully acknowledged.
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George Janelidze.
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Böhm, G. Crossed Modules of Monoids I: Relative Categories. Appl Categor Struct 27, 641–662 (2019). https://doi.org/10.1007/s10485-019-09570-0
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DOI: https://doi.org/10.1007/s10485-019-09570-0