nLab Dwyer map

Idea

A Dwyer map is a type of cofibration in the category of small categories designed for the purpose of constructing homotopy invariant colimits with respect to weak homotopy equivalences, i.e. weak equivalences created from weak homotopy equivalences of simplicial sets by the nerve functor. It plays an essential role in the construction of the Thomason model structure. Its definition is inspired by neighborhood deformation retracts.

A suitable generalization serves a similar purpose in the homotopy theory of relative categories, see Barwick, Kan.

Definition

A functor of small categories $i \colon C \to D$ is a Dwyer map if it is a sieve and factors as a composite of $f \colon C \to C'$ and $j \colon C' \to D$ such that

• $f$ admits a deformation retraction, i.e. a functor $r \colon C' \to C$ such that $r f = \id_C$ toghether with a natural transformation $h \colon f r \to \id_{C'}$ such that $h f = \id_f$,
• $j$ is a cosieve.

In the original definition $r$ was assumed to be a right adjoint of $f$. The definition above is due to Cisinski who called this more general notion a pseudo-Dwyer map. Cisinski’s definition is now considered more useful and hence usually called by the simpler name “Dwyer map”.

Properties

Proposition

Dwyer maps are closed under:

• coproducts,
• pushouts along arbitrary functors,
• transfinite composition,
• retracts.

Moreover, the nerve functor preserves pushouts along Dwyer maps and transfinite composites of Dwyer maps up to weak homotopy equivalence.

The proofs can be found in Thomason, Proposition 4.3, Lemma 4.7 and Cisinski, Lemme 4.

Proposition

Let $c$ denote the left adjoint of the nerve functor and let $Sd$ denote the barycentric subdivision functor. The composite $c Sd^2$ carries injective simplicial maps to Dwyer maps.

This is Thomason, Proposition 4.2.

So among the Dwyer maps are all cofibrations in the Thomason model structure. But not all Dwyer maps are Thomason cofibrations – for example, the unique functor $\emptyset \to C$ is always a Dwyer map, but not every category is Thomason-cofibrant.

References

• R. W. Thomason, Cat as a closed model category,

  Cahiers Topologie Géom. Différentielle 21, no. 3 (1980), pp. 305–324. MR0591388 (82b:18005) numdam scan

• 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)
• Clark Barwick, Daniel Kan, Relative categories: Another model for the homotopy theory of homotopy theories. Indagationes Mathematicae 23 (2012) pp. 42–68.