Abstract
In this chapter we give a non-technical introduction to higher categories. We describe some of the contexts that inspired and motivated their development, explaining the idea of higher categories, and the different classes of higher structures. We discuss one of the most important occurrences of higher categories in algebraic topology, which is the algebraic modelling of homotopy types; in particular, we give an account of the use of internal n-fold structures in modelling path-connected (nā+ā1)-types. This provides an historical development of the notion of weak globularity, which is central to this work.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ara, D.: Higher quasi-categories vs higher Rezk spaces. J. K-Theory 14(3), 701ā749 (2014)
Awodey, S., Warren, M.A.: Homotopy theoretic models of identity types. Math. Proc. Cambridge Philos. Soc. 146(1), 45ā55 (2009)
Baez, J.C., Dolan, J.: Higher dimensional algebra and topological quantum field theory. J. Math. Phys 36, 6073ā6105 (1995)
Baez, J.C., Dolan, J.: Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math. 135, 145ā206 (1998)
Bar, K., Vicary, J.: Data structures for quasistrict higher categories. In: Proceedings on 32th Annual ACM/IEEE Symposium on Logic and Computer Science (2017)
Bar, K., Kissinger, A., Vicary, J.: Globular: an online proof assistant for higher-dimensional rewriting. Logical Methods Comput. Sci. 14(1), paper no. 8 (2018)
Barwick, C.: From operator categories to higher operads. Geom. Topol. 22(4), 1893ā1959 (2018)
Barwick, C., Kan, D.M.: Relative categories: another model for the homotopy theory of homotopy theories. Indag. Math. (N.S.) 23, 42ā68 (2012)
Barwick, C., Kan, D.M.: n-relative categories: a model for the homotopy theory of n-fold homotopy theories. Homology Homotopy Applic. 15(2), 281ā300 (2013)
Barwick, C., Schommer-Pries, C.: On the unicity of the homotopy theory of higher categories. ArXiv:1112.0040v4
Batanin, M.A.: Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. Math. 136(1), 39ā103 (1998)
Batanin, M.A.: Symmetrization of n-operads and compactification of real configuration spaces. Adv. Math. 211(2), 684ā725 (2007)
Batanin, M.A.: The Eckmann-Hilton argument and higher operads. Adv. Math. 217(1), 334ā385 (2008)
Batanin, M., Markl, M.: Operadic categories and duoidal Deligneās conjecture. Adv. Math. 285, 1630ā1687 (2015)
Batanin, M., Weber, M.: Algebras of higher operads as enriched categories. Appl. Categ. Structures 19(1), 93ā135 (2011)
Batanin, M., Cisinski, D.C., Weber, M.: Multitensor lifting and strictly unital higher category theory. Theory Appl. Categ. 28(25), 804ā856 (2013)
Baues, H.J.: Combinatorial homotopy and 4-dimensional complexes. In: De Gruyter Expositions in Mathematics, vol. 2. Walter de Gruyter, Berlin (1991)
Baues, H.J.: The algebra of secondary cohomology operations. In: Progress in Mathematics, vol. 247. BirkhƤuser-Springer, Basel (2006)
BĆ©nabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar. Springer, Berlin (1967)
Bergner, J.: Three models for the homotopy theory of homotopy theories. Topology 46, 397ā436 (2007)
Bergner, J.E.: A survey of (ā, ā1)-categories. towards higher categories. IMA Math. Appl. 152, 69ā83 (2010)
Bergner, J.: Models for (ā, n)-categories and the cobordism hypothesis. In: Proceedings of Symposia in Pure Mathematics, vol. 83. American Mathematical Society, Providence (2011)
Bergner, J.E.: The homotopy theory of (ā, ā1)-categories. In: London Mathematical Society Student Texts, vol. 90. Cambridge University Press, Cambridge (2018)
Bergner, J., Rezk, C.: Comparison of models for (ā, n)-categories 1. Geom. Topol. 17(4), 2163ā2202 (2013)
Blanc, D., Paoli, S.: Segal-type algebraic models of n-types. Algebr. Geom. Topol. 14, 3419ā3491 (2014)
Boardman, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer, Berlin (1973)
Brown, R., Loday, J.L.: Van Kampen theorems for diagrams of spaces. Topology 26(3), 311ā335 (1987)
Brown, R., Spencer, C.B.: Double groupoids and crossed modules. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 17(4), 343ā362 (1976)
Calanque, D., Scheimbauer, C.: A note on the (ā, n)-category of bordisms (2018). Preprint arXiv:1509.08906V2
Carrasco, P.C., Cegarra, A.M.: Group theoretic algebraic models for homotopy types. J. Pure Appl. Alg. 75, 195ā235 (1991)
Cheng, E.: Weak n-categories: opetopic and multitopic foundations. J. Pure Appl. Alg. 186(2), 109ā137 (2004)
Cheng, E.: Comparing operadic theories of n-category. Homology Homotopy Applic. 13(2), 217ā249 (2011)
Chu, H., Haugseng, R., Heuts, G.: Two models for the homotopy theory of ā-operads. J. Topol. 11(4), 856ā872 (2018)
Cisinski, D.C., Moerdijk, I.: Dendroidal sets as models for homotopy operads. J. Topol. 4(2), 257ā299 (2011)
Cisinski, D.C., Moerdijk, I.: Dendroidal sets and simplicial operads. J. Topol. 6(3), 705ā756 (2013)
Coecke, B., Kissinger, A.: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge (2017)
ConduchĆ©, D.: Modules croisĆ©s gĆ©nĆ©ralisĆ©s de longueur 2. J. Pure Appl. Alg. 34(2-3), 155ā178 (1984)
Dawson, R., Pare, R., Pronk, D.: The span construction. Theory Appl. Categ. 24(13), 302ā377 (2010)
Dwyer, W., Kan, D.: Calculating simplicial localizations. J. Pure Appl. Alg. 18, 17ā35 (1980)
Dwyer, W., Kan, D.: Simplicial localizations of categories. J. Pure Appl. Alg. 17(3), 267ā284 (1980)
Dwyer, W., Kan, D., Smith, J.: An obstruction theory for simplicial categories. Proc. Kon. Ned. Akad. Wet. Ind. Math. 89(2), 153ā161 (1986)
Dwyer, W., Kan, D., Smith, J.: Homotopy commutative diagrams and their realizations. J. Pure Appl. Alg. 57, 5ā24 (1989)
Ehresmann, A., Ehresmann, C.: Multiple functors II. The monoidal closed category of multiple categories. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 19(3), 295ā334 (1978)
Ehresmann, A., Ehresmann, C.: Multiple functors IV. Monoidal closed structures on cat n. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 20(1), 59ā104 (1979)
Ellis, G.J., Steiner, R.J.: Higher-dimensional crossed modules and the homotopy groups of (nā+ā1)-ads. J. Pure Appl. Alg. 46, 117ā136 (1987)
Fiore, T.M., Paoli, S.: A Thomason model structure on the category of small n-fold categories. Algebr. Geom. Topol. 10, 1933ā2008 (2010)
Fiore, T.M., Paoli, S., Pronk, D.A.: Model structures on the category of small double categories. Algebr. Geom. Topol. 8, 1855ā1959 (2008)
Garner, R.: Understanding the small object argument. Appl. Categ. Structures 20(2), 103ā141 (2012)
Garner, R., Gurski, N.: The low-dimensional structures formed by tricategories. Math. Proc. Camb. Philos. Soc. 146(3), 551ā589 (2009)
Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Birkhauser, Basel (2009)
Gordon, R., Power, A.J., Street, R.: Coherence for tricategories. Mem. AMS 117(558), 558 (1995)
Grandis, M., ParĆ©, R.: Limits in double categories. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 40(3), 162ā220 (1999)
Grandis, M., ParĆ©, R.: Adjoints for double categories. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 45, 193ā240 (2004)
Grothendieck, A.: Pursuing stacks. Available at http://www.math.jussieu.fr/maltsin/ps.html
Gurski, N.: Coherence in Three-Dimensional Category Theory. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2013)
Hackney, P., Robertson, M., Yau, D.: Infinity Properads and Infinity Wheeled Properads. Lecture Notes in Mathematics, vol. 2147. Springer, Cham (2015)
Heuts, G., Hinich, V., Moerdijk, I.: On the equivalence between Lurieās model and the dendroidal model for infinity-operads. Adv. Math. 302, 869ā1043 (2016)
Hirschowitz, A., Simpson, C.: Descente pour les n-champs. Preprint arXiv:math/9807049
Horel, G.: A model structure on internal categories in simplicial sets. Theory Appl. Categ. 30(20), 704ā750 (2015)
Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Alg. 175(1ā3), 207ā222 (2002)
Kapulkin, C., Lumsdaine, P.L.F.: The simplicial model of univalent foundations (after Voevodsky). Preprint arXiv:1211.2851
Leinster, T.: A survey of definitions of n-category. Theory Appl. Categ. 10(1), 1ā70 (2002)
Leinster, T.: Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004)
Loday, J.L.: Spaces with finitely many nontrivial homotopy groups. J. Pure Appl. Alg. 24, 179ā202 (1982)
Loday, J.L.: La renaissance des opĆ©rades. AstĆ©risque 1994/95, 47ā74 (1996)
Lumsdaine, P.L.: Weak Ļ-categories from intensional type theory. Log. Methods Comput. Sci. 6(3), 3ā24 (2010)
Lurie, J.: On the classification of topological field theories. Curr. Dev. Math. 2008(2009), 129ā280 (2008)
Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)
Lurie, J.: Higher algebra. Available at http://www.math.harvard.edu/lurie/papers/HA.pdf
MacLane, S., Whitehead, J.H.C.: On the 3-type of a complex. Proc. Natl. Acad. Sci. USA 36, 41ā48 (1950)
Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics. In: Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence (2002)
May, J.P.: Simplicial Objects in Algebraic Topology. Chicago Lectures in Mathematics Series. University of Chicago Press, Chicago (1967)
May, J.P.: The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics, vol. 271. Springer, Berlin (1972)
May, J.P.: H ā Ring Spaces and H ā Ring Spectra. Lecture Notes in Mathematics, vol. 577. Springer, Berlin (1977)
Moerdijk, I., Tƶen, B.: Simplicial Methods for Operads and Algebraic Geometry. Advanced Courses in Mathematics. BirkhƤuser/Springer Basel AG (2010)
Moerdijk, I., Weiss, I.: On inner Kan complexes in the category of dendroidal sets. Adv. Math. 221(2), 343ā389 (2009)
Paoli, S.: Weakly globular catn-groups and Tamsamaniās model. Adv. Math. 222, 621ā727 (2009)
Porter, T.: N-types of simplicial groups and crossed n-cubes. Topology 32(1), 5ā24 (1993)
Postnikov, M.M.: Determination of the homology groups of a space by means of the homotopy invariants. Doklady Akad. Nauk SSSR (N.S.) 76, 359ā362 (1951)
Pridham, J.P.: Presenting higher stacks as simplicial schemes. Adv. Math. 238, 184ā245 (2013)
Quillen, D.G.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43. Springer, Berlin (1967)
Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. 353(3), 973ā1007 (2001)
Rezk, C.: A cartesian presentation of weak n-categories. Geom. Topol. 14(1), 521ā571 (2010)
Riehl, E.: Categorical homotopy theory. In: New Mathematical Monographs, vol. 24. University Press, Cambridge (2014)
Riehl, E., Verity, D.: The 2-category theory of quasi-categories. Adv. Math. 280, 549ā642 (2015)
Riehl, E., Verity, D.: Homotopy coherent adjunctions and the formal theory of monads. Adv. Math. 286, 802ā888 (2016)
Schommer-Pries, C.: The classification of two-dimensional extended topological field theories. Ph.D. thesis, University of California, Berkeley (2009)
Schommer-Pries, C., Christopher, J.: Dualizability in low - dimensional higher category theory. In: Contemp. Math. Topology and field theories, pp. 111ā176. American Mathematical Society, Providence (2004)
Segal, G.: Classifying spaces and spectral sequences. Inst. Hautes Ćtudes Sci. Publ. Math. 34(1), 105ā112 (1968)
Simpson, C.: Homotopy types of strict 3-groupoids (1988). Preprint, arXiv:math/9810059V1
Simpson, C.: Homotopy theory of higher categories. In: New Math. Monographs, vol. 19. Cambridge University Press, Cambridge (2012)
Spanier, E.H.: Algebraic Topology. Springer, Berlin (1966)
Stasheff, J.D.: Homotopy associativity of h-spaces. I, II. Trans. Amer. Math. Soc. 108, 275ā292, ibid. 293ā312 (1963)
Street, R.: The algebra of oriented simplexes. J. Pure Appl. Alg. 59(3), 283ā335 (1987)
Sugawara, M.: On the homotopy-commutativity of groups and loop spaces. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33, 257ā269 (1960/1961)
Tamarkin, D.: What do dg-categories form? Compos. Math. 143(5), 1335ā1358 (2007)
Tamsamani, Z.: Sur des notions de n-catĆ©gorie et n-groupoide non-strictes via des ensembles multi-simpliciaux. K-theory 16, 51ā99 (1999)
Thomason, R.W.: Cat as a closed model category. Cah. Top. GĆ©om. DiffĆ©r. CatĆ©g. 21, 305ā324 (1980)
ToĆ«n, B.: Higher and derived stacks: a global overview. In: Algebraic geometryāSeattle 2005. Proceedings of Symposia in Pure Mathematics, Part 1, vol. 80, pp. 435ā487. R. American Mathematical Society, Providence (2009)
ToĆ«n, B.: Derived algebraic geometry. EMS Surv. Math. Sci. 1(2), 153ā240 (2014)
ToĆ«n, B., Vezzosi, G.: Homotopical algebraic geometry. I. Topos theory. Adv. Math. 193(2), 257ā372 (2005)
Toƫn, B., Vezzosi, G.: Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc. 193(902) (2008)
Univalent Foundations Program, T.: Homotopy Type Theory: Univalent Foundations of Mathematics. https//homotopyrtpetheory.org/book: (2013)
van den Berg, B., Garner, R.: Types are weak Ļ-groupoids. Proc. Lond. Math. Soc. 2 102, 370ā394 (2011)
Verity, D.: Weak complicial sets II: nerves of complicial Gray categories. Comtemp. Math 431 (2007)
Verity, D.: Complicial sets characterising the simplicial nerves of strict Ļ-categories. Mem. Amer. Math. Soc. 193, 184 (2008)
Verity, D.: Weak complicial sets I: Basic homotopy theory. Adv. Math 219, 1081ā1149 (2008)
Vicary, J.: Higher quantum theory. Preprint arXiv:1207.4563
Voevodsky, V.: An experimental library of formalized mathematics based on the univalent foundations. Math. Struct. Comput. Sci. 25(5), 1278ā1294 (2015)
Weizhe, Z.: Gluing pseudo functors via n-fold categories. J. Homotopy Relat. Struct. 12, 189ā271 (2017)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
Ā© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Paoli, S. (2019). An Introduction to Higher Categories. In: Simplicial Methods for Higher Categories. Algebra and Applications, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-05674-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-05674-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05673-5
Online ISBN: 978-3-030-05674-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)