×

Levels in the toposes of simplicial sets and cubical sets. (English) Zbl 1229.55016

In the category of simplicial sets, one has both the notion of an \(n\)-skeleton and an \(n\)-coskeleton. The \(n\)-skeleton, \(sk_nK \), of a simplicial set, \(K\), is everything that is generated by simplices of dimension \(\leq n\) within \(K\). A simplicial set, \(K\), is \(n\)-skeletal if \(sk_nK \cong K\). It is \(n\)-coskeletal if any \(k\)-sphere in \(K\) with \(k>n\), has a unique \(k\)-simplex filling it. The category of simplicial sets is a presheaf topos and these ideas generalise well to more general toposes and for applications in homotopy theory, the cases of cubical sets and reflexive globular sets are of particular interest. Both are presheaf toposes. With reflexive globular sets if \(K\) is \(n\)-skeletal, it is easily seen to be \((n+1)\)-coskeletal. This paper looks at the relationship between ‘skeletal’ and ‘coskeletal’ for both simplicial and cubical sets. Explicitly, it is proved that 6.6mm
(i)
any \(n\)-skeletal cubical set is \(2n\)-coskeletal;
(ii)
any \(n\)-skeletal simplicial set is \((2n-1)\)-coskeletal,
and that these are ‘best possible’.

MSC:

55U10 Simplicial sets and complexes in algebraic topology
18B25 Topoi
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C. Berger, I. Moerdijk, On an extension of the notion of reedy category, 2008. arXiv:0809.3341[math.AT; C. Berger, I. Moerdijk, On an extension of the notion of reedy category, 2008. arXiv:0809.3341[math.AT · Zbl 1244.18017
[2] Connes, A., Cohomologie cyclique et foncteurs \(Ext^n\), C. R. Acad. Sci. Paris Sér. I Math., 296, 23, 953-958 (1983) · Zbl 0534.18009
[3] Dwyer, W. G.; Hopkins, M. J.; Kan, D. M., The homotopy theory of cyclic sets, Trans. Amer. Math. Soc., 291, 1, 281-289 (1985) · Zbl 0594.55020
[4] Gabriel, P.; Zisman, M., Calculus of fractions and homotopy theory, (Ergebnisse der Mathematik und Ihrer Grenzgebiete, Band 35 (1967), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. New York) · Zbl 0186.56802
[5] Grandis, M.; Mauri, L., Cubical sets and their site, Theory Appl. Categ., 11, 8, 185-211 (2003), (electronic) · Zbl 1022.18009
[6] Kan, D. M., Abstract homotopy. I, Proc. Natl. Acad. Sci. USA, 41, 1092-1096 (1955) · Zbl 0065.38601
[7] Kelly, G. M.; Lawvere, F. W., On the complete lattice of essential localizations, actes du Colloque en l’Honneur du Soixantième Anniversaire de René Lavendhomme, Louvain-la-Neuve, 1989. actes du Colloque en l’Honneur du Soixantième Anniversaire de René Lavendhomme, Louvain-la-Neuve, 1989, Bull. Soc. Math. Belg. Sér. A, 41, 2, 289-319 (1989) · Zbl 0686.18005
[8] M.R.C. Kennett, M. Zaks, Analysis of levels in a topos, Preprint, 2002.; M.R.C. Kennett, M. Zaks, Analysis of levels in a topos, Preprint, 2002.
[9] Lawvere, F. W., Unity and identity of opposites in calculus and physics, The European Colloquium of Category Theory, Tours, 1994. The European Colloquium of Category Theory, Tours, 1994, Appl. Categ. Structures, 4, 2-3, 167-174 (1996) · Zbl 0858.18002
[10] Lawvere, F. W., Some thoughts on the future of category theory, (Category Theory (Como 1990). Category Theory (Como 1990), Lecture Notes in Math., vol. 1488 (1991)), 1-13 · Zbl 0779.18001
[11] Lawvere, F. W., Functorial concepts of complexity for finite automata, Theory Appl. Categ., 13, 10, 164-168 (2004), (electronic) · Zbl 1062.18001
[12] F. Lawvere, Open problems in topos theory, in: 88th Peripatetic Seminar on Sheaves and Logic. Available at http://www.cheng.staff.shef.ac.uk/pssl88/lawvere.pdf; F. Lawvere, Open problems in topos theory, in: 88th Peripatetic Seminar on Sheaves and Logic. Available at http://www.cheng.staff.shef.ac.uk/pssl88/lawvere.pdf
[13] Loday, J.-L., Cyclic Homology, (Grundlehren der Mathematischen Wissenschaften. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 301 (1992), Springer-Verlag: Springer-Verlag Berlin), Appendix E by María O. Ronco · Zbl 0719.19002
[14] May, J. P., Simplicial objects in algebraic topology, (Chicago Lectures in Mathematics (1992), University of Chicago Press: University of Chicago Press Chicago, IL), Reprint of the 1967 original · Zbl 0165.26004
[15] M. Roy, The topos of ball complexes, Ph.D. thesis, State University of New York at Buffalo, 1997.; M. Roy, The topos of ball complexes, Ph.D. thesis, State University of New York at Buffalo, 1997.
[16] Street, R., The petit topos of globular sets, Category theory and its applications, Montreal, QC, 1997. Category theory and its applications, Montreal, QC, 1997, J. Pure Appl. Algebra, 154, 1-3, 299-315 (2000) · Zbl 0963.18005
[17] M. Zaks, Does n-skeletal imply \(n + 1\); M. Zaks, Does n-skeletal imply \(n + 1\)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.