Contents

category theory

for ∞-groupoids

# Contents

## Idea

By regarding a simplicial set as an object in the standard model structure on simplicial sets, one effectively identifies it (up to weak equivalence) with that ∞-groupoid that it presents under Kan fibrant replacement.

If the original simplicial set is the nerve of a category, the corresponding Kan fibrant replacement is something like the $\infty$-groupoidification of that category: see geometric realization of categories.

This way each ordinary category models an ∞-groupoid. The Thomason model category structure on Cat exhibits this: in this model category a morphism between two categories is a weak equivalence, precisely if it induces a weak equivalence of the corresponding $\infty$-groupoids.

It turns out the Thomason model structure on Cat is Quillen equivalent to the standard model structure on simplicial sets.

This is remarkable, in that it says that every homotopy type, i. e. every weak equivalence class of $\infty$-groupoids, is obtained by $\infty$-groupoidifying just categories.

In fact, every cofibrant object in this structure is a poset. Since every object in a model category is weakly equivalent to a cofibrant one, this means that even the nerves of just posets are sufficient to model all homotopy types.

This is a rather curious aspect of the Thomason model on Cat: it does not really have anything intrinsically to do with categories, but rather uses these as a way to present ∞-groupoids. In particular, the class of weak equivalences is much larger than just the equivalences of categories. There is a different model structure on Cat in which the weak equivalences are precisely the “true” weak equivalences of categories (not of anything constructed from them). This is called the canonical model structure on Cat.

## Definition

Recall the subdivision functor $Sd$ and its right adjoint $Ex$. The Thomason model structure on $Cat$ is given by

• A functor is a Thomason cofibration iff it has the left lifting property against the Thomason trivial fibrations
• A functor $f : C \to D$ is a Thomason weak equivalence iff $N(f)$ is a weak homotopy equivalence
• A functor $f : C \to D$ is a Thomason fibration iff $Ex^2 N(f)$ is a Kan fibration

## Properties

###### Proposition

$Cat$ is a proper model category.

This is corollary 5.5 of Thomason.

### Equivalence with the classical model structure on simplicial sets

The Thomason model structure on Cat is Quillen equivalent to the classical model structure on simplicial sets by the adjunction

$h Sd^2 : sSet \rightleftarrows Cat : Ex^2 N$

We can assert more: this is also an adjoint weak equivalence. These are relative functors between the relative categories $(sSet, Kan)$ and $(Cat, Thomason)$, and both the adjunction unit and counit are natural weak equivalences.

### Relation to ∞Gpd

The localization $L : Cat \to \infty Gpd$ sending a category to its homotopy type has a direct interpretation in terms of ∞-category theory:

###### Proposition

$L$ is the ∞-groupoidification functor functor $C \mapsto C[C^{-1}]$.

### Homotopy colimits

Let $f : C \to Cat$ be a functor. Its homotopy colimit in the Thomason model structure can be computed using the Grothendieck construction:

### The nerve is a homotopy colimit of simplicial sets

By Hirschhorn proposition 18.1.6, the nerve of any category is the homotopy colimit of the constant point-valued diagram:

###### Proposition

For any category $C$, $N(C) \simeq hocolim_{c \in C} \Delta^0$

In fact, when Hirschhorn’s definition of the homotopy colimit for functors valued in simplicial model categories:

$hocolim(f) = N(- \downarrow C)^{op} \otimes_{C} f$

we get an isomorphism $N(C)^{op} \cong hocolim(1)$.

The original reference is

• Denis-Charles Cisinski, Les morphisme de Dwyer ne sont pas stables par rétractes, Cahiers Topologie et Géom. Différentielle Catégoriques, 40 no. 3 (1999), pp. 227–231. (Numdam)

There it was clarified that every cofibrant object in the Thomason model structure is a poset (although this is already explicitly mentioned in Thomason’s paper – see the beginning of section 5).

A useful review of the Thomason model structure and a generalization of the model structure to n-fold categories is given in

• Thomas M. Fiore, Simona Paoli, A Thomason model structure on the category of small n-fold categories (arXiv)

The following article gives a large class of categories which are fibrant in the Thomason model structure.

Some posets that are cofibrant in the Thomason model structure are studied in

• Roman Bruckner, Christoph Pegel, Cofibrant objects in the Thomason Model Structure, arXiv

The analog of the Thomason model structure for the stable case – an equivalence between connective stable homotopy theory and nerves of symmetric monoidal categories – is discussed in

• Robert Thomason, Symmetric monoidal categories model all connective spectra, Theory and Applications of Categories, Vol. 1, No. 5, (1995) pp. 78-118

Some other references

• Hirschhorn Model categories and their localizations.