Towards higher categories by John C. Baez, Michael Shulman (auth.), John C. Baez, J.

By John C. Baez, Michael Shulman (auth.), John C. Baez, J. Peter May (eds.)

The goal of this booklet is to provide heritage in the event you wish to delve into a few greater type concept. it's not a primer on better classification idea itself. It starts with a paper by way of John Baez and Michael Shulman which explores informally, by way of analogy and direct connection, how cohomology and different instruments of algebraic topology are visible during the eyes of n-category theory.

The thought is to provide the various motivations at the back of this topic. There are then survey articles, by means of Julie Bergner and Simona Paoli, approximately (infinity,1) different types and in regards to the algebraic modelling of homotopy n-types. those are parts which are quite good understood, and the place an absolutely built-in concept exists. the main target of the ebook is at the richness to be present in the speculation of bicategories, which supplies the basic start line in the direction of the knowledge of upper specific constructions. an editorial through Stephen Lack provides an intensive, yet casual, advisor to this thought. A paper via Larry Breen at the conception of gerbes indicates how such express buildings seem in differential geometry.

This e-book is devoted to Max Kelly, the founding father of the Australian institution of class thought, and an historic paper by means of Ross highway describes its improvement.

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By John C. Baez, Michael Shulman (auth.), John C. Baez, J. Peter May (eds.)

The goal of this booklet is to provide heritage in the event you wish to delve into a few greater type concept. it's not a primer on better classification idea itself. It starts with a paper by way of John Baez and Michael Shulman which explores informally, by way of analogy and direct connection, how cohomology and different instruments of algebraic topology are visible during the eyes of n-category theory.

The thought is to provide the various motivations at the back of this topic. There are then survey articles, by means of Julie Bergner and Simona Paoli, approximately (infinity,1) different types and in regards to the algebraic modelling of homotopy n-types. those are parts which are quite good understood, and the place an absolutely built-in concept exists. the main target of the ebook is at the richness to be present in the speculation of bicategories, which supplies the basic start line in the direction of the knowledge of upper specific constructions. an editorial through Stephen Lack provides an intensive, yet casual, advisor to this thought. A paper via Larry Breen at the conception of gerbes indicates how such express buildings seem in differential geometry.

This e-book is devoted to Max Kelly, the founding father of the Australian institution of class thought, and an historic paper by means of Ross highway describes its improvement.

Show description

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* 1 Henri Cartan Les travaux de Koszul, I (Lie algebra cohomology)
* 2 Claude Chabauty Le théorème de Minkowski-Hlawka (Minkowski-Hlawka theorem)
* three Claude Chevalley L'hypothèse de Riemann pour les corps de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function)
* four Roger Godement Groupe complexe unimodulaire, I : Les représentations unitaires irréductibles du groupe complexe unimodulaire, d'après Gelfand et Neumark (representation thought of the complicated detailed linear group)
* five Léo Kaloujnine Sur l. a. constitution de p-groupes de Sylow des groupes symétriques finis et de quelques généralisations infinies de ces groupes (Sylow theorems, symmetric teams, countless workforce theory)
* 6. Pierre Samuel l. a. théorie des correspondances birationnelles selon Zariski (birational geometry)
* 7 Jean Braconnier Sur les suites de composition d'un groupe et l. a. journey des groupes d'automorphismes d'un groupe fini, d'après H. Wielandt (finite groups)
* eight Henri Cartan, Les travaux de Koszul, II (see 1)
* nine Claude Chevalley, L'hypothèse de Riemann pour les groupes de fonctions algébriques de caractéristique p, II,, d'après Weil (see 3)
* 10 Luc Gauthier, Théorie des correspondances birationnelles selon Zariski (see 6)
* eleven Laurent Schwartz, Sur un mémoire de Petrowsky : "Über das Cauchysche challenge für ein process linearer partieller Differentialgleichungen im gebiete nichtanalytischen Funktionen" (partial differential equations)
* 12 Henri Cartan, Les travaux de Koszul, III (see 1)
* thirteen Roger Godement, Groupe complexe unimodulaire, II : los angeles transformation de Fourier dans le groupe complexe unimodulaire à deux variables, d'après Gelfand et Neumark (see 4)
* 14 Marc Krasner, Les travaux récents de R. Brauer en théorie des groupes (finite groups)
* 15 Laurent Schwartz, Sur un deuxième mémoire de Petrowsky : "Über das Cauchysche challenge für process von partiellen Differentialgleichungen" (see 11)
* sixteen André Weil Théorèmes fondamentaux de los angeles théorie des fonctions thêta, d'après des mémoires de Poincaré et Frobenius (theta functions)
* 17 André Blanchard, Groupes algébriques et équations différentielles linéaires, d'après E. Kolchin (differential Galois theory)
* 18 Jean Dieudonné, Géométrie des espaces algébriques homogènes, d'après W. L. Chow (algebraic geometry)
* 19 Roger Godement, Sommes keeps d'espaces de Hilbert, I (functional research, direct integrals)
* 20 Charles Pisot, Démonstration élémentaire du théorème des nombres premiers, d'après Selberg et Erdös (prime quantity theorem)
* 21 Georges Reeb, Propriétés des trajectoires de certains systèmes dynamiques (dynamical systems)
* 22 Pierre Samuel, Anneaux locaux ; advent à l. a. géométrie algébrique (local rings)
* 23 Marie-Hélène Schwartz, Compte-rendu de travaux de M. Heins sur diverses majorations de los angeles croissance des fonctions analytiques et sous-harmoniques (complex research, subharmonic functions)
* 24 Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable (connections on fiber bundles)
* 25 Roger Godement, Sommes keeps d'espaces de Hilbert, II (see 19)
* 26 Laurent Schwartz, Sur un mémoire de okay. Kodaira : "Harmonic fields in riemannian manifolds (generalized capability theory)", I (Hodge theory)
* 27 Jean-Pierre Serre, Extensions de groupes localement compacts, d'après Iwasawa et Gleason (locally compact groups)
* 28 René Thom, Les géodésiques dans les variétés à courbure négative, d'après Hopf (geodesics)
* 29 Armand Borel, Groupes localement compacts, d'après Iwasawa et Gleason (see 27)
* 30 Jacques Dixmier, Facteurs : class, size, hint (von Neumann algebras)
* 31 Jean-Louis Koszul, Algèbres de Jordan (Jordan algebras)
* 32 Laurent Schwartz, Sur un mémoire de ok. Kodaira : "Harmonic fields in riemannian manifolds (generalized power theory)", II (see 26)
* 33 Armand Borel, Sous-groupes compacts maximaux des groupes de Lie, d'après Cartan, Iwasawa et Mostow (maximal compact subgroups)
* 34 Henri Cartan, Espaces fibrés analytiques complexes (analytic geometry, fiber bundles)
* 35 Charles Ehresmann, Sur les variétés presque complexes (almost-complex manifolds)
* 36 Samuel Eilenberg, Exposition des théories de Morse et Lusternick-Schnirelmann (Morse idea, Lyusternik-Schnirelmann category)
* 37 Luc Gauthier, Quelques variétés usuelles en géométrie algébrique (algebraic geometry)
* 38 Jean-Louis Koszul, Cohomologie des espaces fibrés différentiables et connexions (Chern-Weil theory)
* 39 Jean Delsarte, Nombre de options des équations polynomiales sur un corps fini, d'après A. Weil (Weil conjectures)
* forty Jacques Dixmier, Anneaux d'opérateurs et représentations des groupes (operator algebras, illustration theory)
* forty-one Roger Godement, Théorie des caractères dans les groupes unimodulaires (unimodular groups)
* forty two Pierre Samuel, Théorie du corps de sessions neighborhood selon G. P. Hochschild (local type box theory)
* forty three Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory)
* forty four Jean-Pierre Serre, Groupes d'homotopie (homotopy groups)
* forty five Armand Borel, Cohomologie des espaces homogènes (cohomology of homogeneous areas of Lie groups)
* forty six Samuel Eilenberg, Foncteurs de modules et leurs satellites, d'après Cartan et Eilenberg (homological algebra)
* forty seven Marc Krasner, Généralisations non-abéliennes de los angeles théorie locale des corps de periods (local fields)
* forty eight Jean Leray, l. a. résolution des problèmes de Cauchy et de Dirichlet au moyen du calcul symbolique et des projections orthogonales et obliques (Dirichlet difficulties and Cauchy difficulties for partial differential equations, symbolic calculus)
* forty nine Pierre Samuel, Sections hyperplanes des variétés normales, d'après A. Seidenberg (algebraic geometry, hyperplane sections, common style)

Additional info for Towards higher categories

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Internalizing n-surjectivity. Tom Fiore: You can define epi and mono categorically and apply them in any category, not just sets. Can you do a similar thing and define analogues of 0-, 1-, and 2-surjectivity using diagrams in any 2-category, etc.? JB: I don’t know. That’s a great question. Eugenia Cheng: You can define a concept of ‘essentially epic’ in any 2-category, by weakening the usual definition of epimorphism. But in Cat, ‘essentially epic’ turns out to mean essentially surjective and full.

Let’s end this section by formulating a hypothesis about the behavior of fibers for n-categories. Generalizing an idea from the first lecture, let’s say that a functor forgets at most k-stuff if it is j-surjective for j > k+1. Hypothesis 17. If a functor between n-categories forgets at most k-stuff, then its fibers are k-categories (which we take to include posetenriched k-categories). The converse is true for n-groupoids and for ncategorical fibrations. We’ve checked this hypothesis above for n = 1 and for n-groupoids (modulo Grothendieck).

The the preimage p−1 (b) is nonempty (since p is essentially surjective), and contractible (since p is full and faithful); thus it is essentially a (−2)-category. These examples, along with the groupoid case we considered in Fact 3, lead us to guess that a functor will forget ‘at most n-stuff’ precisely when its essential preimages are all n-categories. We consider properties to be (−1)-stuff, structure to be 0-stuff, ordinary stuff to be 1-stuff, eka-stuff to be 2-stuff, and so on. However, this guess is not quite right, as we can see by considering some examples that forget structure.

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