Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product
Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 52 (2011) no. 3, p. 188-208
@article{CTGDC_2011__52_3_188_0,
author = {Brown, Ronald and Street, Ross},
title = {Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product},
journal = {Cahiers de Topologie et G\'eom\'etrie Diff\'erentielle Cat\'egoriques},
publisher = {Andr\'ee CHARLES EHRESMANN},
volume = {52},
number = {3},
year = {2011},
pages = {188-208},
zbl = {1242.18004},
mrnumber = {2866503},
language = {en},
url = {http://www.numdam.org/item/CTGDC_2011__52_3_188_0}
}

Brown, Ronald; Street, Ross. Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 52 (2011) no. 3, pp. 188-208. http://www.numdam.org/item/CTGDC_2011__52_3_188_0/

[AABS02] Al-Agl, F. A., Brown, R. and Steiner, R. 'Multiple categories: the equivalence of a globular and a cubical approach'. Adv. Math. 170 (1) (2002) 71-118. | MR 1929304 | Zbl 1013.18003

[ArMe10] Ara, D. and Metayer, F. 'The Brown-Golasinski model structure on strict $\infty$-groupoids revisited'. Homology, Homotopy Appl. (2011) (to appear). | MR 2803870 | Zbl 1218.18004

[Bou87] Bourn, D. 'The shift functor and the comprehensive factorization for internal groupoids'. Cahiers Topologie Géom. Différentielle Catég. 28 (3) (1987) 197-226. | Numdam | MR 923629 | Zbl 0655.18007

[Bro70] Brown, R. 'Fibrations of groupoids'. J. Algebra 15 (1970) 103-132. | MR 271194 | Zbl 0194.02202

[Bro99] Brown, R. 'Groupoids and crossed objects in algebraic topology'. Homology Homotopy Appl. 1 (1999) 1-78 (electronic). | MR 1691707 | Zbl 0920.55002

[Bro06] Brown, R. Topology and Groupoids. Printed by Booksurge LLC, Charleston, S. Carolina, (2006) (third and retitled edition of book published in 1968). | MR 2273730 | Zbl 1093.55001

[Bro08a] Brown, R. 'Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups'. J. Homotopy Relat. Struct. 3 (1) (2008) 331-342. | MR 2426184 | Zbl 1191.18009

[Bro08b] Brown, R. 'A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space'. Homology, Homotopy Appl. 10 (1) (2008) 327-343. | MR 2399477 | Zbl 1141.18007

[Bro09] Brown, R. 'Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems'. In 'Michiel Hazewinkel (ed.), Handbook of Algebra', Volume 6. Elsevier (2009), 83-124. | Zbl 1214.18014

[BG89] Brown, R. and Golasiński, M. 'A model structure for the homotopy theory of crossed complexes'. Cahiers Topologie Géom. Différentielle Catég. 30 (1) (1989) 61-82. | Numdam | MR 1000831 | Zbl 0679.55016

[BH81] Brown, R. and Higgins, P. J. 'On the algebra of cubes'. J. Pure Appl. Algebra 21 (3) (1981) 233-260. | MR 617135 | Zbl 0468.55007

[BH81a] Brown, R. and Higgins, P. J. 'Colimit theorems for relative homotopy groups'. J. Pure Appl. Algebra 22 (1) (1981) 11-41. | MR 621285 | Zbl 0475.55009

[BH81b] Brown, R. and Higgins, P. J. 'The equivalence of $\infty$-groupoids and crossed complexes'. Cahiers Topologie Géom. Différentielle 22 (4) (1981) 371-386. | Numdam | MR 639048 | Zbl 0487.55007

[BH81c] Brown, R. and Higgins, P. J. 'The equivalence of $\omega$-groupoids and cubical T-complexes'. Cahiers Topologie Géom. Différentielle 22 (4) (1981) 349-370. | Numdam | MR 639047 | Zbl 0486.55011

[BH87] Brown, R. and Higgins, P. J. 'Tensor products and homotopies for $\omega$-groupoids and crossed complexes'. J. Pure Appl. Algebra 47 (1) (1987) 1-33. | MR 906402 | Zbl 0621.55009

[BH91] Brown, R. and Higgins, R. J. 'The classifying space of a crossed complex'. Math. Proc. Cambridge Philos. Soc. 110 (1) (1991) 95-120. | MR 1104605 | Zbl 0732.55007

[BHS11] Brown, R., Higgins, P. J. and Sivera, R. Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. EMS Tracts in Mathematics Vol. 15 (to appear 2011). www.bangor.ac.uk/r.brown/nonab-a-t.html. | MR 2841564 | Zbl 1237.55001

[BM94] Brown, R. and Mucuk, O. 'Covering groups of nonconnected topological groups revisited'. Math. Proc. Cambridge Philos. Soc. 115 (1) (1994) 97-110. | MR 1253285 | Zbl 0831.57001

[BP96] Brown, R. and Porter, T. 'On the Schreier theory of non-abelian extensions: generalisations and computations'. Proc. Roy. Irish Acad. Sect. A 96 (2) (1996) 213-227. | MR 1641218 | Zbl 0885.20018

[BRS99] Brown, R. and Razak Salleh, A. 'Free crossed resolutions of groups and presentations of modules of identities among relations'. LMS J. Comput. Math. 2 (1999) 28-61 (electronic). | MR 1694824 | Zbl 0939.20029

[Cra99] Crans, S. E. 'On Combinatorial Models for Higher Dimensional Homotopies'. PhD. Thesis, Utrecht (1995). http://home. tiscali.nl/secrans/papers/comb.html

[Dak77] Dakin, M. K. Kan complexes and multiple groupoid structures. PhD. Thesis, University of Wales, Bangor (1977). http://ehres.pagesperso-orange.fr/Cahiers/dakinEM32.pdf | MR 766238

[Day70] Day, B. J. Construction of Biclosed Categories (PhD Thesis, University of New South Wales, 1970) http://www.math.mq.edu.au/~street/DayPhD.pdf | Zbl 0213.29702

[Day70a] Day, B. J. On closed categories of functors ; in "Reports of the Midwest Category Seminar IV", Lecture Notes in Mathematics 137 (Springer, Berlin, 1970) 1-38. | MR 272852

[Day72] Day, B. J. 'A reflection theorem for closed categories'. J. Pure Appl. Algebra 2 (1972) 1-11. | MR 296126 | Zbl 0236.18004

[GM03] Grandis, M. and Mauri, L. 'Cubical sets and their site'. Theory Applic. Categories 11 (2003) 185-201. | MR 1988396 | Zbl 1022.18009

[Hig71] Higgins, P. J. Notes on categories and groupoids, Mathematical Studies, Volume 32. Van Nostrand Reinhold Co.; Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195, London (1971). | MR 2118984

[Hig05] Higgins, P. J. 'Thin elements and commutative shells in cubical $\omega$-categories'. Theory Appl. Categ. 14 (2005) No. 4, 60-74 (electronic). | MR 2122826 | Zbl 1064.18005

[How79] Howie, J. 'Pullback functors and crossed complexes'. Cahiers Topologie Géom. Différentielle 20 (3) (1979) 281-296. | Numdam | MR 557084 | Zbl 0429.18007

[Hue80] Huebschmann, J. 'Crossed n-fold extensions of groups and cohomology'. Comment. Math. Helv. 55 (2) (1980) 302-313. | MR 576608 | Zbl 0443.18019

[ML71] Mac Lane, S. Categories for the working mathematician, Graduate Texts in Mathematics, Volume 5. Springer-Verlag, New York (1971). | Zbl 0232.18001

[Pra09] Pratt, V. 'The Yoneda Lemma as a foundational tool for algebra'. Stanford preprint: http://boole.stanford.edu/pub/yon.pdf (2009) 1-18.

[Smi51] Smith, P. A. 'The complex of a group relative to a set of generators. I'. Ann. of Math. (2) 54 (1951) 371-402. | MR 48461 | Zbl 0044.19804

[Ste06] Steiner, R. 'Thin fillers in the cubical nerves of omega-categories'. Theory Appl. Categ. 16 (2006) No. 8, 144-173 (electronic). | MR 2223033 | Zbl 1121.18005

[Str87] Street, R. 'The algebra of oriented simplexes'. J. Pure Appl Algebra 49 (3) (1987) 283-335. | MR 920944 | Zbl 0661.18005

[Str88] Street, R. 'Gray's tensor product of 2-categories.' http://www.math.mq.edu.au/~street/GrayTensor.pdf (Feb, 1988). Handwritten notes.

[SV10] Street, R. and Verity, D. 'The comprehensive factorization and torsors'. Theory Appl. Categ. 23 (2010) 42-75. | MR 2593385 | Zbl 1315.18014

[Ton94] Tonks, A. Theory and applications of crossed complexes. Ph.D. thesis, University of Wales (1994).

[Ton03] Tonks, A. P. 'On the Eilenberg-Zilber theorem for crossed complexes'. J. Pure Appl. Algebra 179 (1-2) (2003) 199-220. | MR 1958384 | Zbl 1038.55012

[Ver08] Verity, D. 'Complicial sets characterising the simplicial nerves of strict $\omega$-categories'. Mem. Amer. Math. Soc. 193 (905) (2008) xvi+184. | MR 2399898 | Zbl 1138.18005

[Whi49] Whitehead, J. H. C. 'Combinatorial homotopy. II'. Bull. Amer. Math. Soc. 55 (1949) 453-496. | MR 30760 | Zbl 0040.38801

[Whi50] Whitehead, J. H. C. 'Simple homotopy types'. Amer. J. Math. 72 (1950) 1-57. | MR 35437 | Zbl 0040.38901