# Zoran Skoda hradec orbifolds 2018

Supplementary information to the course on quotients, orbifolds and stacks at the International School on Geometry and Topology in Hradec Králové, May 2018 (still under construction). Web page of the school is here. Lectures were rather informal as an overview of a wide area was envisoned and we had to start from basics (there were few presentation errors which will be noted below at a later point).

The basics of category theory can be found at very many places, e.g. in a quite pedagogical book of Leinster,

• Tom Leinster, Basic category theory, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, 2014, arxiv/1612.09375

and say the denser but shorter van Oosten notes pdf. Nice choice of important examples is in the introduction to categories and functors in the chapter II of Gelfand-Manin Methods of homological algbera listed below. Russian-speaking students can also look at the youtube videos Категории и главные расслоения 1, 2 (Лекции Федорова Р.М., Летняя школа «Современная математика», Дубна, 2013).

Discrete groupoids are just categories in which all arrows are invertible. There is a generalization which we viewed implicitly, the internal groupoid in some “ambient” category $C$, see $n$Lab internal category. The examples are smooth and topological groupoids.

A short algebraic treatment of discrete groupoids is on the wikipedia page groupoid. A nice geometric overview is

• Alan Weinstein, Groupoids: unifying internal and external symmetry, Notices Amer. Math. Soc. July 1996, pdf

For usages of groupoids in homotopy theory see materials linked at R. Brown’s page.

A short overview of simplicial sets is at the page homotopija lekcija9 and a longer one are sections I.1, I.2, I.3 and part of II.4 (from the definition of the nerve of a category II.4.20) of the book

• S. Gelfand, Yu. Manin, Methods of homological algebra, Springer translated from Russian original С. И. Гельфанд, Ю. И. Манин, Методы гомологической алгебры, Т. 1: Введение в теорию когомологий и производные категории. - М. : Наука, 1988. 22 см. - 415 с.

In I.5 (chapter I, section 5) basic notions from sheaf theory (on topological spaces) are presented and chapter II is an introduction to the category theory (II.2 has equivalences and udalities of categories including examples like duality between spaces and algebras, II.3 has representable functors and categorical limits and colimits) including the ringed spaces, Grothendieck topologies and sheaves on them in II.4).

A local homeomorphism is a continuous map $p:E\to B$ of topological spaces such that for every point $e\in E$ in the domain there is an open neighborhood $x\in U\subset B$ such that $p|_{U}:U\to p(U)\subset E$ is a homeomorphism on the image $p(U)$ (with the induced/subspace topology), see local homeomorphism and etale space. Etale spaces are a viewpoint on sheaves over topological spaces. Another is via presheaves. A very basic information on sheaves can be found already on wikipedia

A topologically oriented classical introduction to etale spaces/sheaf theory is in the very readable book of Godement

• Roger Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris 1958 (translated to Russian: Годеман Р. Алгебраическая топология и теория пучков, М., Издательство иностранной литературы 1961.)

At higher level, a short introduction to many categorical notions and a category oriented introduction into sheaves on topological spaces and on Grothendieck pretopology is in the early chapters of the more advanced book by Moerdijk and MacLane, Sheaves in geometry and logic.

A short introduction to the basic notions on group actions and principal bundles is in my lectures hom11lec1 English which are still under translation from Croatian to English (soon to be translated, now just small part translated). See also hom11lec2.

The approach to orbifolds nowdays standard as otutlined in the course is in the early part of the book

• Alejandro Adem, Johann Leida, Yongbin Ruan, Orbifolds and stringy topology, Cambridge University Press (2007)

which draws partly on research article

• Ieke Moerdijk, Dorette A. Pronk, Orbifolds, sheaves and groupoids, K-theory 12: 3–21, 1997.pdf

and lecture notes

• Ieke Moerdijk, Orbifolds as groupoids: an introduction, in: Orbifolds in mathematics and physics (Madison, WI, 2001), 205–222, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 math.DG/0203100

Basic definitions of groupoid, smooth groupoid, orbifold, refinements and so on are also given in a slightly different fashion in chapter 2 of the article

• Joel W. Robbin, Dietmar A. Salamon, A construction of the Deligne–Mumford orbifold, JEMS 8 (2006), 611-699, arxiv:math.SG/0407090

Many classical examples of effective orbifolds can be found in the chapter 13 of the book

• William P. Thurston, The Geometry and Topology of Three-Manifolds

and this chapter can be found at this link.

For the principal bundles (Rus. главные расслоения) and their classification by the first nonabelian Čech cohomology some good references are

• Постников М.М. Лекции по геометрии. Семестр IV. Дифференциальная геометрия, Москва, Наука 1988
• Миша Вербицкий, Векторные расслоения 8: главные расслоения, 2013, pdf (see http://verbit.ru/MATH/BUNDLES-2013 for other chapters)
• Dale Husemoller, Fibre bundles, Springer

Smooth principal bundles and their associated fibre bundles often carry connection, an additional structure of central importance in differential geometry, algebraic topology (Chern-Weil theory of characteristic classes, see J. L. Dupont, Fibre bundles and Chern-Weil theory, Univ. of Aarhus 2003, 115 pp. pdf) and physics (gauge fields in classical gauge theories are connections). See wikipedia connection (principal_bundle), Ehresmann connection, Postnikov’s Semester IV lectures listed above and for short reference also hom11lec9.

To get the idea of descent and Grothendieck stacks starting with the example of gluing of vector bundles as an example see

• Barbara Fantechi, Stacks for everybody, European Congress of Mathematics Volume I, Progr. Math., 201, Basel: Birkhäuser, pp. 349–359, 2001 pdf

For the Grothendieck stacks in terms of pseudofunctors, a reference is

• Ieke Moerdijk, Introduction to the language of stacks and gerbes, math.AT/0212266

and a far more detailed treatment in terms of fibered categories one can look into

• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory, math.AG/0412512

which is however oriented toward algebraic geometers rather than differential geometers. At the beginning you can find the treatment of representable functors and Yoneda lemma, as well as Grothendieck pretopologies (systems of covers on objects of a category). People with background in algebraic geometry can also look for an inspiration in the short article

• Dan Edidin, What is… a stack, Notices Amer. Math. Soc. April 2003 pdf

If we add various representability conditions to Grothendieck stacks we get stacks in a more geometric sense, e.g. the topological stacks as exposed in the work of Noohi:

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