nLab Thomason model structure

Idea

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

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

In fact, every cofibrant object in the Thomason model structure 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-functor.

Definition

Let $C$ and $D$ be small categories. A functor $f \colon C \to D$ is called :

• a Thomason fibration iff $Ex^2 N(f)$ is a Kan fibration,

• Thomason weak equivalence iff its nerve $N(f)$ is a weak homotopy equivalence,

• Thomason-cofibration iff it has the left lifting property against the morphisms that are both Thomason-fibrations as well as Thomason-equivalences.

Proposition

Equipped with these classes of morphisms, the 1-category of small categories is a proper model category.

This is Thomason (1980), corollary 5.5.

Properties

(Co)Fibrant objects

Proposition

Every cofibrant object in the Thomas model structure (Def. ) is a poset (regarded as a small category).

This is Thomason (1980), proposition 5.7.

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 $N(C)^{op} \cong hocolim(1)$.

Related concepts

• basic localizer

• Dwyer map

• homotopy hypothesis

• model structure on simplicial sets, model structure on topological spaces

• Strøm model structure

References

The original reference:

• Robert W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21 3 (1980) 305-324 [numdam:CTGDC_1980__21_3_305_0, MR0591388 (82b:18005)]

A correction to this article was made in

• Denis-Charles Cisinski, La classe des morphismes de Dwyer n’est pas stable par rétractes, Cahiers Topologie et Géom. Différentielle Catégoriques, 40 no. 3 (1999), pp. 227–231. (Numdam)

Review and generalization to $n$-fold categories:

• Thomas M. Fiore, Simona Paoli, A Thomason model structure on the category of small $n$-fold categories, Algebr. Geom. Topol. 10 4 (2010) 1933-2008 [arXiv:0808.4108, doi:10.2140/agt.2010.10.1933]

Restriction of the Thomason model structure to the category of just posets:

• George Raptis, Homotopy theory of posets, Homology Homotopy Appl. 12 2 (2010) 211-230 [euclid:hha/1296223882]

and to the category of just (small) acyclic categories:

• Roman Bruckner, A Model Structure On The Category Of Small Acyclic Categories [arXiv:1508.00992]

Discussion of fibrant objects in the Thomason model structure:

• Lennart Meier, Viktoriya Ozornova, Fibrancy of partial model categories, Homology, Homotopy and Applications, Volume 17 (2015) Number 2. (arXiv:1408.2743, doi:10.4310/HHA.2015.v17.n2.a5)

• Lennart Meier, Fibrancy of (Relative) Categories, talk at Young Topologists Meeting 2014 (slides pdf)

Discussion of cofibrant objects (Thomason-cofibrant posets):

• Roman Bruckner, Christoph Pegel, Cofibrant objects in the Thomason Model Structure, [arXiv:1603.05448]

• Roman Bruckner, Abstract Homotopy Theory and the Thomason Model Structure, PhD thesis, Bremen 2016 [gbv:46-00105527-15, pdf]

On the analog of the Thomason model structure for the stable case – an equivalence between connective stable homotopy theory and nerves of symmetric monoidal categories:

• Robert W. Thomason, Symmetric monoidal categories model all connective spectra, Theory and Applications of Categories 1 5 (1995) 78-118

Some other references

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf, pdf]