nLab simplicial model category

Idea

The term simplicial model category is short for sSet${}_{Quillen}$-enriched model category.

A simplicial model category is a model or presentation for an (∞,1)-category that is half way in between a bare model category and a Kan complex-enriched category.

Specifically, a simplicial model category is an sSet-enriched category $C$ together with the structure of a model category on its underlying category $C_0$ such that both structures are compatible in a reasonable way.

One important use of simplicial model categories comes from the fact that the full sSet-subcategory $C^\circ \hookrightarrow C$ on the fibrant-cofibrant objects – which is not just sSet-enriched but actually Kan complex-enriched – is the (∞,1)-category-enhancement of the homotopy category of the model category $C_0$.

For generalizations of this construction with sSet replaced by another monoidal model category see enriched homotopical category.

Remark

The term simplicial model category for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of simplicially enriched category, which is a proper special case of a simplicial object in Cat (that for which the simplicial set of objects is discrete).

The other caveat is that there are different model category structures on sSet and hence even the term $sSet$-enriched model category is ambiguous.

For instance the model structure for quasi-categories is an $sSet$-enriched model category, but not for the standard Quillen model structure on the enriching category: since $sSet_{Joyal}$ is a closed monoidal model category it is enriched over itself, hence is a $sSet_{Joyal}$-enriched model category, not an $sSet_{Quillen}$-enriched one. So in the standard terminology, $sSet_{Joyal}$ is not a “simplicial model category”.

Definition

Definition

A simplicial model category is an enriched model category which is enriched over $sSet_{Quillen}$: the category sSet equipped with its standard model structure on simplicial sets.

Spelled out, this means that a simplicial model category is

• an sSet-enriched category

  • which is powered and tensored over sSet

• with the structure of a model category on the underlying category $C_0$

• such that for every cofibration $i : A \to B$ and every fibration $p : X \to Y$ in $C_0$ the pullback powering of simplicial sets $C(B,X) \stackrel{i^* \times p_*}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)$ is a Kan fibration;

  • and such that this fibration is an acyclic fibration whenever either $i$ or $p$ are acyclic.

Properties

Enrichment, tensoring, and cotensoring

Let $\mathcal{C}$ be a category equipped with the structure of a model category and with that of an sSet-enriched category, which is tensored and cotensored over sSet.

Proposition

The following conditions – that each make $\mathcal{C}$ into a simplicial model category – are equivalent:

1. the tensoring $\otimes : \mathcal{C} \times sSet \to \mathcal{C}$ is a left Quillen bifunctor;

2. pullback-power axiom:

   for any cofibration $X \to Y$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism

   $$\mathcal{C}(Y, A) \to \mathcal{C}(X, A) \times_{\mathcal{C}(X,B)} \mathcal{C}(Y,B)$$

   is a fibration, and is in addition a weak equivalence if either of the two morphisms is;

3. for any cofibration $X \to Y$ in $sSet$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism

   $$A^Y \to A^X \times_{B^X} B^Y$$

   is a fibration, and is in addition a weak equivalence if either of the two morphisms is.

This follows directly, by Joyal-Tierney calculus, from the defining properties of tensoring and cotensoring (as in this Prop.).

We list in the following some implications of these equivalent conditions.

Let $\mathcal{C}$ now be a simplicial model category.

Corollary

If $A \in \mathcal{C}$ is fibrant, and $X \hookrightarrow Y$ is a cofibration in sSet, then

$$A^{Y} \to A^{X}$$

is a fibration in $\mathcal{C}$.

Proof

Apply prop. to the case of the cofibration $X \to Y$ and the fibration $A \to *$, where “$*$” denotes the terminal object. This yields that

$$A^Y \to A^X \times_{{*}^X} {*}^Y$$

is a fibration. But ${*}^Y = {*}^X = {*}$ and hence the claim follows.

Similarly we have

Corollary

If $X \in \mathcal{C}$ is cofibrant and $A \in \mathcal{C}$ is fibrant, then $\mathcal{C}(X,A)$ is fibrant in sSet, hence is a Kan complex.

Proof

Apply prop. to the cofibration $\emptyset \to X$, where “$\emptyset$” denotes the initial object, and to the fibration $A \to *$ to find that

$$\mathcal{C}(X, A) \to \mathcal{C}(\emptyset, A) \times_{\mathcal{C}(\emptyset,*)} \mathcal{C}(X,*)$$

is a fibration. But since $\emptyset$ is initial and $*$ is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration

$$\mathcal{C}(X,A) \to * \,.$$

Derived hom-spaces

Proposition

For $X$ and $A$ any two objects and $Q X$ and $P A$ a cofibrant and fibrant replacement, respectively, $\mathcal{C}(Q X, P A)$ is the correct derived hom-space between $X$ and $A$ (see the discussion there). In particular the full $sSet$-enriched subcategory on cofibrant fibrant objects is therefore an sSet-enriched category which is fibrant in the model structure on simplicially enriched categories. Its homotopy coherent nerve is a quasi-category. All this are intrinsic incarnatons of the (∞,1)-category that is presented by $C$.

Although $\mathcal{C}(X,A)$ need not have the correct homotopy type for the mapping space if $X$ is not cofibrant or $A$ is not fibrant, simplicial notions of homotopy are still sufficient to detect the model-categorical ones.

Proposition

If $f,g:X\to A$ are simplicially homotopic (i.e. in the same connected component of the simplicial set $\mathcal{C}(X,A)$), then they represent the same map in the homotopy category of $\mathcal{C}$. Therefore, any simplicial homotopy equivalence in $\mathcal{C}$ is a weak equivalence.

Proof

This is Lemma 9.5.15 and Proposition 9.5.16 of Hirschhorn.

Examples

General

• The classical model structure on simplicial sets $sSet_{Quillen}$ is a closed monoidal model category and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.

• The classical model structure on topological spaces for compactly generated topological spaces (here) is similarly enriched over itself. Under geometric realization this also makes it a simplicial model category.

  (See also for instance Goerss-Schemmerhorn 06, p. 26)

• For $C$ any small sSet-enriched category and $A$ simplicial combinatorial model category, the global model structure on functors $[C^{op}, A]_{proj}$ and $[C^{op},A]_{inj}$ are themselved simplicial combinatorial model categories. See model structure on simplicial presheaves.

• The left Bousfield localization of a combinatorial simplicial model category at any set of morphisms is again a combinatorial simplicial model category. Large classes of examples arise this way.

Simplicial Quillen equivalent models

While many model categories do not admit an $sSet_{Qu}$-enrichment, for large classes of model categories one can find a Quillen equivalence to a model category that does carry an $sSet_{Qu}$-enrichment.

These are constructed as Bousfield localization of Reedy model structures on the category of simplicial objects in the given model category.

Definition

Let $C$ be a

• left proper

• combinatorial model category.

By the discussion at cofibrantly generated model category in the section Presentation and generation, there exists a small set $E \subset Obj(C)$ of objects that detect weak equivalences. For some such choice of $E$, let

$$S \;\coloneqq\; \Big\{ \; e \cdot \big( \Delta[k] \longrightarrow \Delta[l] \big) \; \Big\}_{ {e \in E,} \atop {([k] \to [l]) \in \Delta} } \;\subset\; Mor\big( [\Delta^{op}, C] \big)$$

where $\cdot$ denotes the tensoring of $C$ over Set, hence $e \cdot \Delta[n]$ denotes the simplicial object consisting of coproducts of $e$ as

$$e \cdot \Delta[k] \;\;\colon\;\; [n] \;\mapsto\; \underset {\Delta([n],[k])} {\amalg} e \,.$$

Write

$$[\Delta^{op}, C]_{proj, S} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\;\;\bot\;\;} [\Delta^{op}, C]_{proj}$$

for the left Bousfield localization of the projective model structure on functors at this set $S$ of morphisms.

Similarly, write

$$[\Delta^{op}, C]_{Reedy, S} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\;\;\bot\;\;} [\Delta^{op}, C]_{Reedy}$$

for the left Bousfield localization of the Reedy model structure at $S$.

Lemma

Let $C$ be a cofibrantly generated model category.

If $X \in [\Delta^{op}, C]$ is degreewise cofibrant and has all structure maps being weak equivalences, then all coprojections $X_i \to hocolim X$ into the homotopy colimit are weak equivalences and hence $X \to const\,hocolim X$ is a weak equivalence.

This appears as Dugger, prop. 5.4 corollary 5.5.

Theorem

The model structures from def. have the following properties.

1. The weak equivalences in both are precisely those morphisms which become weak equivalences under homotopy colimit over $\Delta^{op}$.

2. The fibrant objects in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the face and degeneracy maps are weak equivalences in $C$.

3. The colimit/constant adjoint functors

   $$(\lim_{\to} \dashv const) \colon C \underoverset {\underset{\;const\;}{\longrightarrow}} {\overset{\underset{\to}{\lim}}{\longleftarrow}} {\;\;\bot\;\;} [\Delta^{op}, C]_{proj, S}$$

   constitute a Quillen equivalence, the identity functors constitute a Quillen equivalence

   $$[\Delta^{op}, C]_{Reedy, S} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot} [\Delta^{op}, C]_{proj, S} \,,$$

   and the constant/limit adjoint functors (notice that the limit is evaluation in degree 0, since $\Delta^{op}$ has an initial object) constitute a Quillen equivalence

   $$(const \dashv ev_0) \;\colon\; [\Delta^{op}, C]_{Reedy,S} \underoverset {\underset{\;ev_0\;}{\longrightarrow}} {\overset{const}{\longleftarrow}} {\bot} C \,;$$

4. The canonical sSet-enrichment/tensoring/powering of the category of simplicial objects $[\Delta^{op}, C]$ makes $[\Delta^{op}, C]_{Reedy,S}$ (but not in general $[\Delta^{op}, C]_{proj,S}$) into a simplicial model category.

This is Dugger, theorem 5.2, theorem 5.7, theorem 6.1.

So in particular every left proper combinatorial model category is Quillen equivalent to a simplicial model category.

Proof

We first show that the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the objectwise fibrant objects all whose structure maps are weak equivalences in $C$. The argument for the fibrant objects in $[\Delta^{op}, C]_{Reedy,S}$ is directly analogous.

By general properties of left Bousfield localization, the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the projective fibrant objects $X$ for which all induced morphisms on derived hom spaces

$$\mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}(s \cdot(\Delta[k] \to \Delta[l]), X)$$

are weak equivalences. Since $s$ is cofibrant in $C$ by definition, also $s \cdot \Delta[k]$ is cofibrant in $[\Delta^{op}, C]_{proj}$.

So for $X \in [\Delta^{op}, C]_{proj}$ fibrant, let $X_\bullet \in [\Delta^{op}, [\Delta^{op}, C]]$ be a simplicial framing for it. Notice that this means that for all $n \in \mathbb{N}$ also $X_\bullet([n])$ is a simplicial framing for $X([n])$. This is because

1. $const X \to X_\bullet$ being a weak equivalence means that for all $n$ the morphism $X \to X_n$ is a weak equivalence, which means that for all $k$ the morphism $X([k]) \to X_n([k])$ is a weak equivalence.

2. $X_\bullet$ being fibrant in $[\Delta^{op}, [\Delta^{op}, C]_{proj}]_{Reedy}$ means that for all $n\in \mathbb{N}$ the morphism $X_{\Delta[n]} \to X_{\partial \Delta[n]}$ is a fibration in $[\Delta^{op}, C]_{proj}$, hence that for all $k \in \mathbb{N}$ the morphism $X_{\Delta[n]}([k]) \to X_{\partial \Delta[n]}([k])$ is a fibration in $C$, hence that $X([k])$ is Reedy fibrant.

Then we find

$$\begin{aligned} \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}( s \cdot(\Delta[k] \to \Delta[l]), X) & \simeq Hom_{[\Delta^{op}, C]}(s \cdot(\Delta[k] \to \Delta[l]), X_\bullet) \\ & \simeq Hom_{C}(s , X_\bullet([l]) \to X_\bullet([k])) \\ & \simeq \mathbb{R}Hom_C(s, X([l]) \to X([k])) \,. \end{aligned}$$

By assumption on the set $S$, this implies the claim.

Now we show that the weak equivalences in $[\Delta^{op}, C]_{proj,S}$ are precisely those morphisms that become weak equivalences under the homotopy colimit.

By functorial cofibrant resolution and two-out-of-three, it is sufficient to show that this holds for morphisms between cofibrant objects.

By lemma , we have weak equivalences

$$\mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) \stackrel{\simeq}{\to} \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) \stackrel{\simeq}{\to} \mathbb{R}Hom_{C}(\lim_\to A , Z)$$

seen by computing the derived homs by simplicial framings.

Now, by properties of left Bousfield localization, $A \to B$ is a weak equivalence if for all $S$-local objects $X$ the morphism of derived hom-spaces $\mathbb{R}Hom(A \to B, X)$ is a weak equivalence. Looking at the diagram

$$\array{ \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to A , Z) \\ \uparrow && \uparrow && \uparrow \\ \mathbb{R}Hom_{[\Delta^{op}, C]}(B,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(B, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to B , Z) }$$

we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant $Z \in C$, which is the case if $\lim_\to A \to \lim_\to B$ is a weak equivalence. Since $A$ and $B$ here are cofibrant in $[\Delta^{op}, C]_{proj}$, the colimits here are indeed homotopy colimits (as discussed there).

Now we discuss that $(\lim_\to \dashv const) \colon C \to [\Delta^{op}, C]_{proj,S}$ is a Quillen equivalence. First observe that on the global model structure $const \colon C \to [\Delta^{op}, C]_{proj}$ is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the derived functor of the left adjoint $\lim_\to$, namely the homotopy colimit, takes the localizing set $S$ to weak equivalences in $C$. Therefore the assumptions of the discussion at Quillen equivalence - Behaviour under localization are met, and hence it follows that $(\lim_\to \dashv const)$ descends as a Quillen adjunction also to the localization.

To see that this is a Quillen equivalence, it is sufficient to show that for $A \in [\Delta^{op}, C]_{proj}$ cofibrant and $Z \in C$ fibrant, a morphism $\lim_\to A \to Z$ is a weak equivalence in $C$ precisely if the adjunct $A \to const Z$ becomes a weak equivalence under the homotopy colimit.

For this notice that we have a commuting diagram

$$\array{ hocolim A &\longrightarrow& hocolim(const Z) \\ \big\downarrow && \big\downarrow \\ colim A &\longrightarrow& colim (const Z) & \simeq Z }$$

and so our statement follows (by 2-out-of-3) once we know that the vertical morphisms here are weak equivalences. The left one is because $A$ is cofibrant, by assumption, as before. To see that the right one is, too, consider the factorization

$$Z \to (const Z)_i \to hocolim (const Z) \to Z$$

of the identity morphism on $Z$, for any $i \in \mathbb{N}$. By lemma the first morphism is a weak equivalence, and hence so is the morphism in question.

Now we show that the weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$ are the hocolim-equivalences.

By a general result on functoriality of localization, we have that the $(id \dashv id ) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj,S}$ is at least a Quillen adjunction.

Let then $A \to B$ be a morphism in $[\Delta^{op}, C]$ and consider two fibrant replacements

$$\array{ A &\to& \bar A &\to & \hat A \\ \downarrow && \downarrow && \downarrow \\ B &\to& \bar B &\to& \hat B } \,,$$

where the first one ($\bar A \to \bar B$) is taken in $[\Delta^{op}, C]_{proj,S}$ and the second ($\hat A \to \hat B$) in $[\Delta^{op}, C]_{Reedy}$.

Assume first that $A \to B$ is a hocolim-equivalence. Then so is $\hat A \to \hat B$, because the horizontal morphisms are all objectwise weak equivalences. But $\hat A$ and $\hat B$ are fibrant in $[\Delta^{op}, C]_{Reedy}$, hence in $[\Delta^{op}, C]_{proj}$ by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in $[\Delta^{op}, C]_{proj,S}$. By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure – so $\hat A \to \hat B$ is indeed even an objectwise weak equivalence. It follows then that so is $\bar A \to \bar B$, which is therefore in partiular a weak equivalence in $[\Delta^{op}, C]_{Reedy, S}$. Finally the left horizontal morphisms are also weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$, by the above Quillen adjunction. So finally by 2-out-of-3 in $[\Delta^{op}, C]_{Reedy,S}$ it follows that also $A \to B$ is a weak equivalence there.

By an analogous diagram chase, one shows the converse implication holds, that $A \to B$ being a weak equivalence in $[\Delta^{op}, C]_{Reedy,S}$ implies that it is a hocolim-equivalence.

With this now it is clear that the identity adjunction above is in fact a Quillen equivalence.

Finally we show that $(const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C$ is a Quillen equivalence.

First, it is immediate to check that $const \colon C \to [\Delta^{op}, C]_{Reedy}$ is left Quillen, and since $id : [\Delta^{op}, C]_{Reedy} \to [\Delta^{op}, C]_{Reedy,S}$ is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction.

To see that it is a Quillen equivalence, let $A \in C$ be cofibrant and $X \in [\Delta^{op}, C]_{Reedy,S}$ be fibrant – which by the above means that it is a simplicial resolution – and consider a morphism $const A \to X$. We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if $A \to X_0$ is a weak equivalence in $C$.

To that end, find a cofibrant resolution $const \tilde A \to \tilde X$ of $const A \to X$ in $[\Delta^{op}, C]_{proj}$ and consider the diagram

$$\array{ A &\stackrel{\simeq}{\leftarrow}& \tilde A &\stackrel{\simeq}{\to}& colim(const \tilde A) \\ \downarrow && \downarrow && \downarrow \\ X_0 &\stackrel{\simeq}{\leftarrow}& \tilde X_0 &\stackrel{\simeq}{\to}& colim \tilde X } \,.$$

The colimits on the right compute the homotopy colimit. By 2-out-of-3 it follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is.

Finally it remains to show that $[\Delta^{op}, C]_{Reedy,S}$ is a simplicially enriched model category. (…)

Remark

(uniqueness)

Let $C$ be a model category. Then there is a unique model category structure on $s C \coloneqq [\Delta^{op}, C]$ such that

• every morphism that is degreewise a weak equivalence in $C$ is a weak equivalence in $s C$;

• the cofibrations are those of the Reedy model structure;

• the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in $C$.

This is Rezk, Schwede & Shipley, theorem 3.1.

Proof

By theorem at least one such model structure exists. By the discussion at model category – Redundancy of the axioms, the classes of cofibrations and fibrant objects already determine a model category structure.

Corollary

For $C$ any left proper combinatorial model category, the derived hom-space between two objects $X, A$ may be computed by

• choosing a cofibrant replacement $\hat X$ of $X$ in $C$;

• choosing a Reedy fibrant replacement $\hat A$ of $const A$ in $[\Delta^{op}, C]$ such that all face and degeneracy maps are weak equivalences,

setting

$$Maps(X,A) \;\colon\; [n] \mapsto Hom_C(\hat X, \hat A_n) \,.$$

Proof

By theorem we may compute the derived hom space in $[\Delta^{op}, C]_{Reedy,S}$ after the inclusion $const \colon C \to [\Delta^{op}, C]$. Since by that theorem $[\Delta^{op}, C]_{Reedy,S}$ is a simplicial model category, by prop. the derived hom space is given by the simplicial function complex between a cofibrant replacement of $const X$ and a fibrant replacement of $const A$. If $\hat X$ is cofibrant, then $const \hat X$ is already Reedy cofibrant, and by the theorem $\hat A$ as stated is a a fibrant resolution of $const A$. Finally, the theorem says that the simplicial function complex is given by

$$\begin{aligned} [\Delta^{op}, C](const \hat X, \hat A)_n & = Hom_{[\Delta^{op}, C]}((const \hat X) \cdot \Delta[n], \hat A) \\ & \simeq Hom_{[\Delta^{op}, C]}((const \hat X) , \hat A^{\Delta[n]}) \\ & \simeq Hom_C(\hat X, \hat A_n) \end{aligned} \,.$$

There is also a version for stable model categories:

Theorem

Every proper cofibrantly generated stable model category is Quillen equivalent to a simplicial model category

This is Rezk, Schwede & Shipley, prop 1.3.

Combinatorial simplicial model categories

A particularly important type of simplicial model categories are those that are also combinatorial model categories.

A combinatorial simplicial model category is precisely a presentation for a locally presentable (∞,1)-category. See there for more details.

Related concepts

• Dugger's theorem

• combinatorial model category

• Ho(CombModCat)

• simplicial monoidel model category?

References

The definition originates in

• Daniel Quillen, chapter II, section 2 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

Textbook accounts:

• Philip Hirschhorn, Chapter 9 in: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

• Jacob Lurie, section A.3 in Higher Topos Theory (2009)

• Garth Waner?: Categorical Homotopy Theory, EPrint Collection, University of Washington (2012) [hdl:1773/19589, pdf, pdf]

• Emily Riehl, §3.8 and §11.4 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

Further review:

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods, Notes from lectures given at the University of Chicago in August 2004, published as: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436 AMS (2007) [arXiv:math.AT/0609537, doi:10.1090/conm/436]

On Quillen-equivalent simplicial enhancements of model categories:

• Charles Rezk, Stefan Schwede, Brooke Shipley, Simplicial structures on model categories and functors, American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575 (arXiv:0101162, jstor)

• Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 12 (2001) 5003-5027 [ams:S0002-9947-01-02661-7, pdf]

• Daniel Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 1 177-201 [arXiv:math/0007068, doi:10.1006/aima.2001.2015]

  (cf. Dugger's theorem)