Homotopy n-Types

Postnikov tower が \(n\)-stage までしかない空間のホモトピー型, あるいは与えられた空間の \(n\)-stage Postnikov section を homotopy \(n\)-type という。

連結な空間の homotopy \(1\)-type は, その基本群で決まる。 つまり, \(K(\pi ,1)\) 空間である。より一般に, homotopy \(1\)-type は fundamental groupoid で決まる。

このことを出発点に, より高次の homotopy \(n\)-type を, 代数的な構造で記述しようという試みがある。有名なのは Grothendieck の [Gro] だろうか。それ以前に, R. Brown などにより様々な試みがある。 これまでに知られていることについては, Blanc と Paoli の [BP14] の Introduction を見るのが手っ取り早い。

このような対応で, homotopy \(n\)-type に対応する weak \(n\)-groupoid という概念があるという主張を homotopy hypothesis という。

• homotopy hypothesis

小さな \(n\) については, 以下のようなモデルがある。

• crossed module [MW50]
• homotopy double groupoid [Bro+02]
• homotopy bigroupoid [HKK01]
• strict \(2\)-groupoid [MS93]
• double groupoid [CHR12]
• double groupoid with connection [BS76]
• Gray groupoid [Ber99]
• quadratic module [Bau91]

一般の \(n\) については, 次のようなモデルがある。

• \(\mathrm {cat}^n\)-group [Lod82]
• crossed \(n\)-cube [ES87; Por93]
• \(n\)-hyper-crossed complex [CC91]
• Bataninの higher groupoid [Bat98; Cis07]
• \(n\)-hypergroupoid [Gle82]
• Tamsamani の weak \(n\)-groupoid [Tam99; Sim12; Sim]
• Blanc と Paoli の weakly globular pseudo \(n\)-fold groupoid [BP14]

References

[Bat98]

M. A. Batanin. “Monoidal globular categories as a natural environment for the theory of weak \(n\)-categories”. In: Adv. Math. 136.1 (1998), pp. 39–103. url: http://dx.doi.org/10.1006/aima.1998.1724.

[Bau91]

Hans Joachim Baues. Combinatorial homotopy and \(4\)-dimensional complexes. Vol. 2. De Gruyter Expositions in Mathematics. With a preface by Ronald Brown. Walter de Gruyter & Co., Berlin, 1991, pp. xxviii+380. isbn: 3-11-012488-2. url: https://doi.org/10.1515/9783110854480.

[Ber99]

Clemens Berger. “Double loop spaces, braided monoidal categories and algebraic \(3\)-type of space”. In: Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996). Vol. 227. Contemp. Math. Amer. Math. Soc., Providence, RI, 1999, pp. 49–66. url: https://doi.org/10.1090/conm/227/03252.

[BP14]

David Blanc and Simona Paoli. “Segal-type algebraic models of \(n\)-types”. In: Algebr. Geom. Topol. 14.6 (2014), pp. 3419–3491. arXiv: 1204.5101. url: https://doi.org/10.2140/agt.2014.14.3419.

[Bro+02]

Ronald Brown, Keith A. Hardie, Klaus Heiner Kamps, and Timothy Porter. “A homotopy double groupoid of a Hausdorff space”. In: Theory Appl. Categ. 10 (2002), pp. 71–93.

[BS76]

Ronald Brown and Christopher B. Spencer. “Double groupoids and crossed modules”. In: Cahiers Topologie Géom. Différentielle 17.4 (1976), pp. 343–362.

[CC91]

P. Carrasco and A. M. Cegarra. “Group-theoretic algebraic models for homotopy types”. In: J. Pure Appl. Algebra 75.3 (1991), pp. 195–235. url: http://dx.doi.org/10.1016/0022-4049(91)90133-M.

[CHR12]

Antonio Martı́nez Cegarra, Benjamı́n A. Heredia, and Josué Remedios. “Double groupoids and homotopy 2-types”. In: Appl. Categ. Structures 20.4 (2012), pp. 323–378. arXiv: 1003.3820. url: https://doi.org/10.1007/s10485-010-9240-1.

[Cis07]

Denis-Charles Cisinski. “Batanin higher groupoids and homotopy types”. In: Categories in algebra, geometry and mathematical physics. Vol. 431. Contemp. Math. Providence, RI: Amer. Math. Soc., 2007, pp. 171–186. arXiv: math / 0604442. url: http://dx.doi.org/10.1090/conm/431/08272.

[ES87]

Graham Ellis and Richard Steiner. “Higher-dimensional crossed modules and the homotopy groups of \((n+1)\)-ads”. In: J. Pure Appl. Algebra 46.2-3 (1987), pp. 117–136. url: http://dx.doi.org/10.1016/0022-4049(87)90089-2.

[Gle82]

Paul G. Glenn. “Realization of cohomology classes in arbitrary exact categories”. In: J. Pure Appl. Algebra 25.1 (1982), pp. 33–105. url: http://dx.doi.org/10.1016/0022-4049(82)90094-9.

[Gro]

Alexander Grothendieck. Pursuing Stacks. arXiv: 2111.01000.

[HKK01]

K. A. Hardie, K. H. Kamps, and R. W. Kieboom. “A homotopy bigroupoid of a topological space”. In: Appl. Categ. Structures 9.3 (2001), pp. 311–327. url: http://dx.doi.org/10.1023/A:1011270417127.

[Lod82]

Jean-Louis Loday. “Spaces with finitely many nontrivial homotopy groups”. In: J. Pure Appl. Algebra 24.2 (1982), pp. 179–202. url: http://dx.doi.org/10.1016/0022-4049(82)90014-7.

[MS93]

Ieke Moerdijk and Jan-Alve Svensson. “Algebraic classification of equivariant homotopy \(2\)-types. I”. In: J. Pure Appl. Algebra 89.1-2 (1993), pp. 187–216. url: http://dx.doi.org/10.1016/0022-4049(93)90094-A.

[MW50]

Saunders MacLane and J. H. C. Whitehead. “On the \(3\)-type of a complex”. In: Proc. Nat. Acad. Sci. U.S.A. 36 (1950), pp. 41–48. url: https://doi.org/10.1073/pnas.36.1.41.

[Por93]

Timothy Porter. “\(n\)-types of simplicial groups and crossed \(n\)-cubes”. In: Topology 32.1 (1993), pp. 5–24. url: https://doi.org/10.1016/0040-9383(93)90033-R.

[Sim]

Carlos T. Simpson. Homotopy theory of higher categories. arXiv: 1001.4071.

[Sim12]

Carlos Simpson. Homotopy theory of higher categories. Vol. 19. New Mathematical Monographs. Cambridge: Cambridge University Press, 2012, pp. xviii+634. isbn: 978-0-521-51695-2.

[Tam99]

Zouhair Tamsamani. “Sur des notions de \(n\)-catégorie et \(n\)-groupoı̈de non strictes via des ensembles multi-simpliciaux”. In: \(K\)-Theory 16.1 (1999), pp. 51–99. arXiv: alg - geom / 9512006. url: https://doi.org/10.1023/A:1007747915317.