# Schreiber Day on Higher Category Theory - July 2010

A workshop on higher category theory.

# Contents

## Where and when

If you intend to participate, we would appreciate it if you drop us a note.

## Schedule

We will have the following talks.

All talks are supposed to be about 50 minutes, with 10 minutes discussion.

### Algebraic fibrant objects

Abstract: We establish a model category structure on algebraic Kan complexes. In fact, we introduce the notion of an algebraic fibrant object in a general model category (obeying certain technical conditions). Based on this construction we propose algebraic Kan complexes as an algebraic model for ∞-groupoids and algebraic quasi-categories as an algebraic model for (∞,1)-categories.

### Euler characteristics of categories and homotopy colimits

• 11:30 - 12:20 – Thomas Fiore (University of Michigan-Dearborn, USA)

Abstract: The Euler characteristic is among the earliest and most elementary homotopy invariants. For a finite simplicial complex, it is the alternating sum of the numbers of simplices in each dimension. This combinatorially defined invariant has remarkable connections to geometric notions, such as genus, curvature, and area.

Euler characteristics are not only defined for simplicial complexes or manifolds, but for many other objects as well, such as posets and, more generally, categories. We propose in this talk a topological approach to Euler characteristics of categories. The idea, phrased in homological algebra, is the following. Given a category $\Gamma$ and a ring $R$, we take a finite projective $R\Gamma$-module resolution ${P}_{*}$ of the constant module $\underline{R}$ (assuming such a resolution exists). The alternating sum of the modules ${P}_{i}$ is the finiteness obstruction $o\left(\Gamma ,R\right)$. It is a class in the projective class group ${K}_{0}\left(R\Gamma \right)$, which is the free abelian group on isomorphism classes of finitely generated projective $R\Gamma$-modules modulo short exact sequences. From the finiteness obstruction we obtain the Euler characteristic respectively ${L}^{2}$-Euler characteristic , by adding the entries of the $R\Gamma$-rank respectively the ${L}^{2}$-rank of the finiteness obstruction.

This topological approach has many advantages, several of which now follow. First of all, this approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. It works equally well for infinite categores and finite categories. There are many examples. Classical constructions are special cases, for example, under appropriate hypotheses the functorial ${L}^{2}$-Euler characteristic of the proper orbit category for a group $G$ is the equivariant Euler characteristic of the classifying space for proper $G$-actions. The K-theoretic Möbius inversion has Möbius-Rota inversion and Leinster’s Möbius inversion as special cases. We also obtain the classical Burnside ring congruences.

This talk will focus on our Homotopy Colimit Formula for Euler characteristics.

In certain cases, the ${L}^{2}$-Euler characteristic agrees with the groupoid cardinality of Baez-Dolan and the Euler characteristic of Leinster, and comparisons will be made.

This is joint work with Wolfgang Lück and Roman Sauer. Our preprints are available online: Finiteness obstructions and Euler characteristics of categories and Euler characteristics of categories and homotopy colimits.

### Dendroidal weak $n$-categories

• 14:15 - 15:05 – Andor Lukacs (University of Utrecht, Netherlands)

Abstract: Dendroidal sets were introduced by I. Moerdijk and I. Weiss as a generalization of simplicial sets, suitable to study operads in the context of homotopy theory in MoerWeiss07a. The idea behind the notion of dendroidal sets is that in the same way as simplicial sets help us understanding categories via the nerve functor, there should be an analogous notion for studying coloured operads as generalization of categories. Much of the fundamentals of simplicial sets that relate to category theory extend to dendroidal sets. To name a few,

The purpose of this talk is to show that dendroidal sets can also be used to give a new definition of weak n-categories, and to compare the result with the corresponding classical notions in low degrees: bicategories and tricategories.

### The nerve theorem and Grothendieck’s hypothesis on homotopy types

Abstract: The nerve theorem gives sufficient conditions for a monad so that the category of its algebras embeds fully faithfully into a presheaf topos. According to Grothendieck the homotopy type of an n-truncated space should be representable by a suitably defined fundamental n-groupoid. In this expositary talk, I will first give some details entering into the nerve theorem, then explain in which way the nerve theorem offers a strategy to prove Grothendieck's hypothesis, and finally discuss the present state-of-art of the subject.

### Diaconescu’s theorem for 2-topoi

Abstract: (joint work with Branislav Jurčo)

The investigation of sheaf-theoretical and cohomological structures associated to higher categories led Ross Street to define the notion of a Grothendieck 2-topos in the case of strict 2-categories, which he later generalized to the case of bicategories. Street defined a bisite as a bicategory supplied with a suitable notion of a Grothendieck topology and he defined Grothendieck 2-topoi as those bicategories which are biequivalent to bicategories of stacks over the bisite. The main result of his work was a bicategorical version of Giraud's theorem providing a characterization of Grothendieck 2-topoi in terms of bilimits, bicolimits, exactness and size conditions.

We discuss different (non)equivalent notions of regularity and exactness for 2-categories and bicategories by reviewing basic examples: the 2-category Cat of small categories, then the 2-category ${\mathrm{Cat}}^{{B}^{\mathrm{coop}}}$ of $B$-indexed categories for a small bicategory $ℬ$, and the 2-category $\mathrm{St}\left(X\right)$ of stacks over a topological space $X$. All these examples are crucial in two-dimensional topos theory: the 2-topos Cat plays a role of the topos Set in ordinary topos theory and a point of any 2-topos $ℰ$ is defined as a 2-geometric morphism $p:\mathrm{Cat}\to ℰ$. The second example ${\mathrm{Cat}}^{{B}^{\mathrm{coop}}}$ is a 2-topos which we denote by $Bℬ$, and call the classifying 2-topos of a small bicategory $ℬ$, in an analogy with a classifying topos of small bicategory $ℬ$ introduced in BaJu09. We use a third example of a 2-topos $\mathrm{St}\left(X\right)$ in order to introduce a notion of a 2-torsor over $X$ under a bicategory $ℬ$. These are homomorphisms $𝒫:ℬ\to \mathrm{St}\left(X\right)$ such that stalks over $X$ of the canonical fibration of bicategories ${\pi }_{𝒫}:{\int }_{ℬ}𝒫\to ℬ$ given by a bicategorical version of the Grothendieck construction, are 2-filtered in the sense of Dubuc and Street. Our main result is a bicategorical version of Diaconescu's theorem which says that there exists a natural biequivalence

$2\mathrm{Top}\left(\mathrm{St}\left(X\right),Bℬ\right){\simeq }_{\mathrm{bi}}2-\mathrm{Tors}\left(X;ℬ\right)$2Top(St(X),B\mathcal{B}) \simeq_{bi} 2-Tors(X;\mathcal{B})

where the left side is a 2-category of geometric 2-morphisms from the Grothendieck 2-topos $\mathrm{St}\left(X\right)$ to the classifying 2-topos $Bℬ$ of $ℬ$ and the right side is the 2-category of left $ℬ$-2-torsors over $X$. We discuss connections of this result with a joint work of Bunge and Hermida.

Where and when

Revised on May 29, 2012 22:04:00 by Andrew Stacey (129.241.15.200)