Zoran Skoda seminar Nerves and Grothendieck's homotopy hypothesis

Here I will sketch the outline and literature from my expositional seminar

• Nerves and Grothendieck’s homotopy hypothesis, Seminar for topology, Department of Mathematics, University of Zagreb, 31 Jan 2011, 11:15-12:45

General ideas on higher groupoids and homotopy types

Introduction about Grothendieck’s Pursuing stacks (1983) and the idea of sheaves of higher groupoids; generalizing nonabelian cohomology to higher coefficients.

A sketch of the definition of weak 2-categories (bicategories).

Fundamental groupoid, bigroupoid,…infinity groupoid. Insufficency of strict infinity-groupoids. Toward weak infinity categories in order to model homotopy types and present coefficients for higher nonabelian cohomology. Homotopy conjecture: weak infinity groupoids model weak homotopy types. It has many variants, some almost trivial, soem very hard, depending how precisely one models homotopy types and which model for infinity groupodis one takes. Now people consider models involving simplicial techniques, what makes it easier, but Grothendieck wanted to consider the true algebaric generalizations of groupoids and bigroupoids.

Nerves

General nonsense on nerve and realization (see the link). Main examples: Grothendieck’s nerve $Cat\to sSet$, Duskin’s nerve $Bicat\to sSet$, homotopy coherent nerve $sSet-Cat\to sSet$, $\omega$-nerve $\omega Cat\to sSet$, Moore’s normalization functor (in Dold-Kan correspondence), cyclic nerve of cyclic sets… The corresponding geometric realization is in all these situations defined as certain left Kan extension. In all these situations, which lead to pairs of a nerve functor and a geometric realization, the geometric realization is a left adjoint to the nerve functor. Characterization of the essential image of the Grothendieck’s nerve by Grothendieck via the unique horn filler conditions. Kan complexes are characterized by horn filler conditions without uniqueness. The subcategory $Kan \subset sSet$ of Kan complexes should be the image of a nerve functor from $\infty$-groupoids (of any kind) to $sSet$. In particular the Street’s $\omega$-nerve is such; Duskin’s “hypergroupoids” are such by the definition.

Simplicial homotopy theory

Simplicial homotopy theory. Kan fibrations, inner Kan fibrations, Kan complexes, inner Kan complexes aka quasicategories. Higher fundamental groupoids and the homotopy conjecture via Kan complexes.

Other models of $(\infty,1)$-categories.

Categories enriched over monoidal categories. Closed monoidal categories are enriched over itself. Usually it is useful to enrich over close monoidal categories. Topological categories - categories enriched over $Top$ (in this case, $Top$ is a convenient category of topological spaces). Simplicially enriched categories – categories enriched and over $sSet$ ($Top$ and $sSet$ are closed symmetric monoidal categories with respect to categorical product). $Top-Cat$ and $sSet-Cat$ can be treated as the models of the $(\infty,1)$-categories of $(\infinity,1)$-categories. Many $\infty$-categorical notions in this model differ from the default notions of enriched category theory (e.g. limits are not quite the same as enriched limits…). It is difficult to define the infinity category of infinity functors in that approach, even among (co)fibrant objects; this is an important point where it is easier to work with quasicategories.

Quillen model categories as presentations

Quillen model categories: axioms, via the notion of weak factorization systems. Outline of general tools and rough passage through basic terminology (e.g. fibrant objects, left homotopy, homotopy category…). Main examples: Quillen model structure on $sSet$. Strøm model structure on the category of all topological spaces. Model categories on categories of simplicial presheaves. Quillen adjunctions, Quillen equivalences.

Idea of model structures as presentations of bicomplete locally presentable $(\infinity,1)$-categories.

Topos $\mathbb{U}$ and strong shape

Coherent nerve $\mathbb{U}$ of the large category of Kan complexes is important in many situations; in particular it is the terminal $(\infty,1)$-topos.

$(\infty,1)$-topoi are special $(\infty,1)$-categories which are analogues of usual Grothendieck topoi in ordinary category theory. The usual Grothendieck topos is a category which is equivalent to a category of sheaves on some Grothendieck site (which is a generalization of the category of open subsets of a topological space); a Grotehndieck topos can be internally characterized by abstract properties which are called Giraud’s axioms. A Grothendieck $(\infty,1)$-topos is an $(\infty,1)$-category satisfying the infinite/categorical analogues of Giraud’s axioms. In the quasicategory language this is expanded in Lurie’s book Higher Topos Theory; in an another model (“Segal topoi”) it appeared earlier in a work of Toen and Vezzosi; there is also some earlier treatment in the works of Rezk, Joyal and others.

For a locally presentable infinity category $D$, the $(\infinity,1)$-category $Pro(D)= Ind(D^{op})^{op}$ is the full $(\infty,1)$-subcategory of $Fun(D,D)^{op}$ whose objects are pro objects which are defined as left exact accessible $(\infty,1)$-functors $D\to \mathbb{U}$. In particular, for $D = \mathbb{U}$, the $(\infty,1)$-topos $Pro(\mathbb{U})$ is the $(\infinity,1)$-category of shapes.

A geometric morphism $p_* : C\to \mathbb{U} : p^*$ for $C$ a $(\infty,1)$-topos, canonical up to homotopy. Strong shape of $(\infty,1)$-topos (after Lurie’s book) as $p_* p^*\in Pro(\mathbb{U})$. If $X$ is a paracompact space then the strong shape of $X$ is induced by the geometric morphism $p_*:Sh_\infty(X)\to Sh_\infty(Y):p^*$ induced by the map $p : X\to \star$ of spaces. The strong shape of $C$ is defined even for the infinity topoi which are not necessarily of the form $Sh_\infty(X)$ for a paracompact $X$; examples not strictly in classical topology abound.

Questions asked

I was asked if it is worthy to learn so many abstract infinite-categorical tools and other tools presented; of course it depends on wanted applications and personal inclinations, but the machinery actually gives cleaner and simpler perspective, it is easier to think in wide range of applications and (in the point of view of our nLab), it is the most natural way of doing things involving homotopy, structured (geometric) spaces and so on. Now there are very many results and fruitful new fields of mathematics, including the proof of cobordism hypothesis, derived algebraic geometry, “new brave worlds of homotopy theory” including motivic homotopy theory, higher gauge theory in physics etc. Lurie’s book Higher Topos Theory in chapter 7 gives some applications of $(\infty,1)$-topoi to the classical topology, for example to the dimension theory and proper base change theorems.

Literature

• nLab: nerve and realization, nerve, homotopy coherent nerve, homotopy hypothesis, model category, model structure on simplicial sets, model structure on simplicial presheaves, fundamental groupoid, Quillen adjunction, Quillen equivalence, quasicategory
• nLab:zoranskoda: course pages on abstract homotopy theory, homotopija lekcija9 (intro to simplicial sets)
• Jacob Lurie, What is an infinity category, Notices AMS, pdf (a very short and accessible introduction for a newcomer)
• Joyal's Catlab
• Jacob Lurie, Higher Topos Theory, Princeton Univ. Press
• Bill Dwyer, Homotopy theory and classifying spaces (Copenhagen notes), pdf
• Julia Bergner, A survey of $(\infinity,1)$-categories, arxiv/0610.5239
• André Joyal, Quasi-categories and Kan complexes, (in Special volume celebrating the 70th birthday of Prof. Max Kelly) J. Pure Appl. Algebra 175 (2002), no. 1-3, 207–222 (doi)
• A. Joyal, M. Tierney, Simplicial homotopy theory (preparatory lectures for an advanced course on simplicial methods, Barcelona 2008)
• A. Joyal, Notes on quasicategories, (draft)
• (applications, using mainly strict higher categories) Urs Schreiber, Z. Škoda, Categorified symmetries, SFIN, XXII A1, (2009), 397-424. Extended 55 page version: arXiv/1004.2472
• John C. Baez, Michael Shulman, Lectures on $n$-categories and cohomology, math/0608420
• Tom Leinster, Higher categories, higher operads, London Math. Soc. Lec. Note Series 298, math.CT/0305049
• Ross Street, Combinatorial aspects of descent theory, math.CT/0303175, doi