### Algebraic Models for Higher Categories

#### Posted by Urs Schreiber

[ *guest post by Thomas Nikolaus* ]

A few months ago we had here a discussion about $\infty$-groupoids and the homotopy hypothesis. In this discussion I had the idea to use algebraic Kan complexes as a model for $\infty$-groupoids and to endow the category of algebraic Kan complexes with a model stucture. After a fair amount of work I finally could prove that these things really do work out.

Notes on Algebraic Kan Complexes(pdf)

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 quasicategories as an algebraic model for (∞,1)-categories.

I would be grateful for comments and criticism. Please let me know what you think about this.

Simplicial sets have been introduced as a combinatorial model for topological spaces. It has been known for a long time that topological spaces and certain simplicial sets, called Kan complexes, are the same from the viewpoint of homotopy theory. To make this statement precise Quillen introduced the concept of model categories and equivalence of model categories as an abstract framework for homotopy theory. He endowed the category $\Top$ of topological spaces and the category $\sSet$ of simplicial sets with model category structures and showed that $\Top$ and $\sSet$ are equivalent in his sense. He could identify Kan complexes as fibrant objects in the model structure on $\sSet$.

Later higher category theory came up. A 2-category has not only objects and morphisms, like an ordinary (1-)category, but also 2-morphisms, which are morphisms between morphisms. A 3-category has also 3-morphisms between 2-morphisms and so on. Finally an $\infty$-category has $n$-morphisms for all $n \geq 1$. Unfortunately it is very hard to give a tractable definition of $\infty$-categories.

An interesting subclass of all $\infty$-categories are those $\infty$-categories for which all $n$-morphisms are invertible. They are called ∞-groupoids. A standard construction from algebraic topology is the fundamental groupoid construction $\Pi_1(X)$ of a topological space $X$. Allowing higher paths in $X$ (i.e. homotopies) extends this construction to a fundamental ∞-groupoid $\Pi_\infty(X)$. It is widely believed that every $\infty$-groupoid is, up to equivalence, of this form. This believe is called the homotopy hypothesis.

There is another important subclass of $\infty$-categories, called (∞,1)-categories. These are $\infty$-categories where all $n$-morphisms for $n \geq 2$ are invertible. Thus the only difference to $\infty$-groupoids is that there may be non-invertible $1$-morphisms. In particular the collection of all $\infty$-groupoids forms a $(\infty,1)$-category. Another example of a $(\infty,1)$-category is the category of topological spaces where the $n$-morphisms are given by $n$-homotopies. In the language of $(\infty,1)$-categories a more refined version of the homotopy hypothesis is the assertion that the fundamental $\infty$-groupoid construction provides an equivalence of the respective $(\infty,1)$-categories.

From the perspective of higher category theory Quillen model structures are really presentations of $(\infty,1)$-categories, see e.g. HTT appendix A.2 and A.3. Hence we think about a model category structure as a generators and relations description of a $(\infty,1)$-category. A Quillen equivalence then becomes an adjoint equivalence of the presented $(\infty,1)$-categories. Thus the classical Quillen equivalence between topological spaces and simplicial sets really encodes an equivalence of $(\infty,1)$-categories.

Keeping this statement in mind, it is reasonable to think of a simplicial set $S$ as a model for an $\infty$-groupoid. The $n$-morphisms are then the $n$ simplices $S_n$. And in fact there has been much progress in higher category theory using simplicial sets as a model for $\infty$-groupoids. This model has certain disadvantages. First of all a simplicial set does not encode how to compose $n$-morphisms. But such a composition is inevitable for higher categories. This problem is usually adressed as follows:

The model structure axioms on $\sSet$ imply that in the corresponding $(\infty,1)$ category each simplicial set is equivalent to a fibrant object i.e. a Kan complex. It is possible to interpret the lifting properties of a Kan complex $S$ as the existence of compositions in the $\infty$-groupoid.. Although the lifting conditions ensure the existence of compositions for $S$, these compositions are only unique up to homotopy. This makes it sometimes hard to work with Kan complexes as a model for $\infty$-groupoids. Another disadvantage is that the subcategory of Kan complexes is not very well behaved, for example it does not have colimits.

The idea of this paper to solve these problems is to consider a more algebraic version of Kan complexes as model for $\infty$-groupoids. More precisely we will consider Kan complexes endowed with the additional structure of distinguished fillers. We call them algebraic Kan complexes. We show that the category of algebraic Kan complexes has all colimits and limits. Furthermore we endow it with a model structure and show that it is Quillen equivalent to simplicial sets. The name algebraic will be justified by identifing algebraic Kan complexes as algebras for a certain monad on simplicial sets. The fact that algebraic Kan complexes really model $\infty$-groupoids will be justified by a proof of the appropriate version of the homotopy hypothesis.

We will generalize this notion of algebraic Kan complex to algebraic fibrant objects in a general model category $\mathcal{C}$, which satisfies some technical conditions. In particular we show that $\mathcal{C}$ is Quillen equivalent to the model category $Alg\mathcal{C}$ of algebraic fibrant objects in $\mathcal{C}$. We show that $Alg\mathcal{C}$ is monadic over $\mathcal{C}$ and that all objects are fibrant. In addition we give a formula how to compute (co)limits in $Alg\mathcal{C}$ from (co)limits in $\mathcal{C}$.

Finally we apply the general construction to the Joyal model structure on $\sSet$. This is a simplicial model for $(\infty,1)$-categories. The fibrant objects are called quasicategories. We propose the category $Alg\mathcal{Q}$ of algebraic quasicategories as our model for $(\infty,1)$-categories. One of the major advantages is that the model structure on algebraic quasicategories can be described very explicitly, in particular we will give sets of generating cofibrations and trivial cofibrations. Such a generating set is not known for the Joyal structure.

## Re: Algebraic Models for Higher Categories

Around proposition 2.11 I think there are some references to AlgKan and sSet which are misplaced.

Awesome paper by the way!