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model structure for Cartesian fibrations

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

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general

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for (,1)(\infty,1)-categories

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for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Category theory

Contents

Idea

The model category structure on the category SSet +/SSSet^+/S of marked simplicial sets over a given simplicial set SS is a presentation for the (∞,1)-category of Cartesian fibrations over SS. Every object is cofibrant and the fibrant objects of SSet +/SSSet^+/S are precisely the Cartesian fibrations over SS.

Notably for S=*S = {*} this is a presentation of the (∞,1)-category of (∞,1)-categories: as a plain model category this is Quillen equivalent to the model structure for quasi-categories, but it is indeed an sSet QuillensSet_{Quillen}-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets that models ∞-groupoids).

The (,1)(\infty,1)-categorical Grothendieck construction that exhibits the correspondence between Cartesian fibrations and (∞,1)-presheaves is in turn modeled by a Quillen equivalence between the model structure on marked simplicial over-sets and the projective global model structure on simplicial presheaves.

Marked simplicial sets

Marked simplicial sets are simplicial sets with a little bit of extra structure: a marking that remembers which edges are supposed to be Cartesian morphisms.

Definition

In components

Definition

A marked simplicial set is

  • a pair (S,E)(S,E) consisting of

    • a simplicial set SS

    • and a subset ES 1E \subset S_1 of edges of SS, called the marked edges,

  • such that

    • all degenerate edges are marked edges.

A morphism (S,E)(S,E)(S,E) \to (S',E') of marked simplicial sets is a morphism f:SSf : S \to S' of simplicial sets that carries marked edges to marked edges in that f(E)Ef(E) \subset E'.

Notation
  • The category of marked simplicial sets is denoted sSet +sSet^+.

  • for SS a simplicial set let

    • S S^\flat or S minS^{min} be the minimally marked simplicial set: only the degenerate edges are marked;

    • S S^\sharp or S maxS^{max} be the maximally marked simplicial set: every edge is marked.

  • for p:XSp : X \to S a Cartesian fibration of simplicial sets let

    • X X^\natural or X cartX^{cart} be the cartesian marked simplicial set: precisely the pp-cartesian morphisms are marked

As a quasi-topos

Simple as the above definition is, for seeing some of its properties it is useful to think of sSet +sSet^+ in a more abstract way.

Definition

Let Δ +\Delta^+ be the category defined as the simplex category Δ\Delta, but with one more object [1 +][1^+] that factors the unique morphism [1][0][1] \to [0] in Δ\Delta

[0] [1] = p [0] [1 +]. \array{ [0] &\stackrel{\to}{\to}& [1] \\ {}^{\mathllap{=}}\downarrow &\swarrow& \downarrow^{\mathrlap{p}} \\ [0] &\leftarrow& [1^+] } \,.

Equip this category with a coverage whose only non-trivial covering family is {p:[1][1] +}\{p : [1] \to [1]^+\}.

Observation

The category sSet +sSet^+ is the quasi-topos of separated presheaves on Δ +\Delta^+:

sSet +SepPSh(Δ +). sSet^+ \simeq SepPSh(\Delta^+) \,.
Proof

A presheaf X:(Δ +) opSetX : (\Delta^+)^{op} \to Set is separated precisely if the morphism

X(p):X 1 +X 1 X(p) : X_{1^+} \to X_1

is a monomorphism, hence if X 1 +X_{1^+} is a subset of X 1X_1. By functoriality this subset contains all the degenerate 1-cells

X 0 σ X 1 + X 1. \array{ X_0 \\ \downarrow & \searrow^{\mathrlap{\sigma}} \\ X_{1^+} &\hookrightarrow& X_1 } \,.

Therefore we may naturally identify XX as a simplicial set equipped with a subset of X 1X_1 that contains all degenerate 1-cells.

Moreover, a morphism of separated preseheaves on Δ +\Delta^+ is by definition just a natural transformation between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.

Notice that sSet +sSet^+ is a genuine quasi-topos:

Observation

sSet +sSet^+ is not a topos.

Proof

The canonical morphisms X X X^\flat \to X^\sharp are monomorphisms and epimorphisms, but not isomorphisms. Therefore sSet +sSet^+ is not a balanced category, hence cannot be a topos.

Cartesian closure

Lemma

The category sSet +sSet^+ is a cartesian closed category.

Proof

This is an immediate consequence of the above observation that sSet +sSet^+ is a quasitopos. But it is useful to spell out the Cartesian closure in detail.

By the general logic of the closed monoidal structure on presheaves we have that PSh(Δ +)PSh(\Delta^+) is cartesian closed. It remains to check that if X,YPSh(Δ +)X,Y \in PSh(\Delta^+) are marked simplicial sets in that X 1 +X 1X_{1^+} \to X_1 is a monomorphism and similarly for YY, that then also Y XY^X has this property.

We find that the marked edges of Y XY^X are

(Y X) 1 +Hom PSh(Δ +)([1 +],Y X)Hom PSh(Δ +)([1 +]×X,Y) (Y^X)_{1^+} \simeq Hom_{PSh(\Delta^+)}([1^+], Y^X) \simeq Hom_{PSh(\Delta^+)}([1^+] \times X, Y)

and the morphism (Y X) 1 +(Y X) 1(Y^X)_{1^+} \to (Y^X)_1 sends X×[1 +]ηYX \times [1^+] \stackrel{\eta}{\to} Y to

X×[1](Id,p)X×[1 +]ηY. X \times [1] \stackrel{(Id,p)}{\to} X \times [1^+] \stackrel{\eta}{\to} Y \,.

Now, by construction, every non-identity morphism U[1 +]U \to [1^+] in Δ +\Delta^+ factors through U[1]U \to [1], which implies that if the components of p *η 1p^* \eta_1 and p *η 2p^* \eta_2 coincide on U[1 +]U \neq [1^+], then already the components of η 1\eta_1 and η 2\eta_2 on UU coincided. By assumption on XX the values of η 1\eta_1 and η 2\eta_2 on U=[1 +]U = [1^+] are already fixed, due to the inclusion X 1 +×[1 +] 1 +X 1×[1 +] 1X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}. Hence p *p^* is injective, and so Y XY^X formed in PSh(Δ +)PSh(\Delta^+) is itself a marked simplicial set.

Definition
  • For XX and YY marked simplicial sets let

    • Map (X,Y)Map^\flat(X,Y) be the simplicial set underlying the cartesian internal hom Y XsSet +Y^X \in sSet^+

    • Map (X,Y)Map^\sharp(X,Y) the simplicial set consisting of all simplices σMap (X,Y) \sigma \in Map^\flat(X,Y) such that every edge of σ\sigma is a marked edge of Y XY^X.

Corollary

These mapping complexes are characterized by the fact that we have natural bijections

Hom sSet(K,Map (X,Y))Hom sSet +(K ,Y X)Hom sSet +(K ×X,Y) Hom_{sSet}(K, Map^\flat(X,Y)) \simeq Hom_{sSet^+}(K^\flat, Y^X) \simeq Hom_{sSet^+}(K^\flat \times X, Y)

and

Hom sSet(K,Map (X,Y))Hom sSet +(K ,Map (X,Y))Hom sSet +(K ×X,Y) Hom_{sSet}(K, Map^\sharp(X,Y)) \simeq Hom_{sSet^+}(K^\sharp, Map^\sharp(X,Y)) \simeq Hom_{sSet^+}(K^\sharp \times X, Y)

for KsSetK \in sSet and X,YsSet +X,Y \in sSet^+. In particular

Map (X,Y) n=Hom sSet +(X×Δ[n] ,Y) Map^\flat(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\flat, Y)

and

Map (X,Y) n=Hom sSet +(X×Δ[n] ,Y). Map^\sharp(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\sharp, Y) \,.

In words we have

  • The nn-simplices of the internal hom Y XY^X are simplicial maps X×Δ[n]YX \times \Delta[n] \rightarrow Y such that when you restrict X 1×Δ[n] 1Y 1X_1 \times \Delta[n]_1 \rightarrow Y_1 to E×Δ[n] 0E \times \Delta[n]_0 (where EE is the set of marked edges of XX), this morphism factors through the marked edges of YY.

  • The marked edges of Y XY^X are those simplicial maps X×Δ[1]YX \times \Delta[1] \rightarrow Y such that the restriction of X 1×Δ[1] 1Y 1X_1 \times \Delta[1]_1 \rightarrow Y_1 to E×Δ[1] 1E \times \Delta[1]_1 factors though the marked edges of YY. In the presence of the previous condition, this says that when you apply the homotopy X×Δ[1]YX \times \Delta[1] \rightarrow Y to a marked edge of XX paired with the identity at [1][1], the result should be marked.

Definition

We generalize all this notation from sSet +sSet^+ to the overcategory sSet +/S:=sSet +/(S )sSet^+/S := sSet^+/(S^\sharp) for any given (plain) simplicial set SS, by declaring

Map S (X,Y)Map (X,Y) Map_S^\flat(X,Y) \subset Map^\flat(X,Y)

and

Map S (X,Y)Map (X,Y) Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)

to be the subcomplexes spanned by the cells that respect that map to the base SS.

Observation

Let YSY \to S be a Cartesian fibration of simplicial sets, and X X^\natural as above the marked simplicial set with precisely the Cartesian morphisms marked.

Then

  • Map S (X,Y )Map_S^\flat(X,Y^\natural) is an quasi-category;

  • Map S (X,Y )Map_S^\sharp(X, Y^\natural) is its core, the maximal Kan complex inside it.

This is HTT, remark 3.1.3.1.

Proof

The nn-cells of Map S (X,Y )Map_S^\flat(X,Y^\natural) are morphisms X×Δ[n] Y X \times \Delta[n]^\flat \to Y^\natural over SS. This means that for fixed xX 0x \in X_0, Δ[n]\Delta[n] maps into a fiber of YSY\to S. But fibers of Cartesian fibrations are fibers of inner fibrations, hence are quasi-categories.

Similarly, the nn-cells of Map S (X,Y )Map_S^\sharp(X,Y^\natural) are morphisms X×Δ[n] Y X \times \Delta[n]^\sharp \to Y^\natural over SS. Again for fixed xX 0x \in X_0, Δ[n]\Delta[n] maps into a fiber of YSY\to S, but now only hitting Cartesian edges there. But (as discussed at Cartesian morphism), an edge over a point is Cartesian precisely if it is an equivalence.

Proposition

We have a sequence of adjoint functors

() () () () : () () sSet () sSet + () (-)^{\flat} \dashv (-)^{\flat} \dashv (-)^{\natural} \dashv (-)^{\natural} : \array{ & \stackrel{(-)^{\flat}}{\to} & \\ & \stackrel{(-)^{\flat}}{\leftarrow} & \\ sSet & \stackrel{(-)^{\natural}}{\to} & sSet^+ \\ & \stackrel{(-)^{\natural}}{\leftarrow} & }

Model structure on marked simplicial sets

Cartesian weak equivalences

Observe that weak equivalences in the model structure for quasi-categories may be characterized as follows.

Lemma

A morphism f:CDf : C \to D between simplicial sets that are quasi-categories is a weak equivalence in the model structure for quasi-categories precisely if the following equivalent coditions hold:

  • For every simplicial set KK, the morphism sSet(K,f):sSet(K,D)sSet(K,C)sSet(K,f) : sSet(K,D) \to sSet(K,C) is a weak equivalence in the model structure for quasi-categories.

  • For every simplicial set KK, the morphism Core(sSet(K,f)):Core(sSet(K,D))Core(sSet(K,C))Core(sSet(K,f)) : Core(sSet(K,D)) \to Core(sSet(K,C)) on the cores, the maximal Kan complexes inside, is a weak equivalence in the standard model structure on simplicial sets, hence a homotopy equivalence.

Proof

This is HTT, lemma 3.1.3.2.

This may be taken as motivation for the following definition.

Definition/Proposition

For every Cartesian fibration ZSZ \to S, we have that

Map (X,Z ) Map^\flat(X, Z^{\natural})

is a quasi-category and

Map (X,Z )=Core(Map (X,Z )) Map^\sharp(X,Z^\natural) = Core(Map^\flat(X,Z^\natural))

is the maximal Kan complex inside it.

A morphism p:XYp : X \to Y in sSet +/SsSet^+/S is a Cartesian equivalence if for every Cartesian fibration ZZ we have

  • The induced morphism Map S (Y,Z )Map S (X,Z )Map_S^\flat(Y,Z^{\natural}) \to Map_S^\flat(X,Z^{\natural}) is an equivalence of quasi-categories;

Or equivalently:

This is HTT, prop. 3.1.3.3 with HTT, remark 3.1.3.1.

Proposition

Let

X p Y S \array{ X &&\stackrel{p}{\to}&& Y \\ & \searrow && \swarrow \\ && S }

be a morphism in sSet/SsSet/S such that both vertical maps to SS are Cartesian fibrations. Then the following are equivalent:

Proof

This is HTT, lemma 3.1.3.5.

The model structure

The model structure on marked simplicial over-sets Set +/SSet^+/S over SSSetS \in SSet – also called the Cartesian model structure since it models Cartesian fibrations – is defined as follows.

Definition/Proposition

(Cartesian model structure on sSet +/SsSet^+/S)

The category SSet +/SSSet^+/S of marked simplicial sets over a marked simplicial set SS carries a structure of a proper combinatorial simplicial model category defined as follows.

The SSet-enrichment is given by

sSet +/S(X,Y):=Map S (X,Y). sSet^+/S(X,Y) := Map_S^\sharp(X,Y) \,.

A morphism f:XXf : X \to X' in SSet +/SSSet^+/S of marked simplicial sets is

Proof

The model structure is proposition 3.1.3.7 in HTT. The simplicial enrichment is corollary 3.1.4.4.

Remark

Using Map S (X,Y)Map_S^\flat(X,Y) for the mapping objects makes sSet +/SsSet^+/S a sSet JoyalsSet_{Joyal}-enriched model category (i.e. enriched in the model structure for quasi-categories). This is HTT, remark 3.1.4.5.

Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the (,1)(\infty,1)-category CartFib(S)CartFib(S) of Cartesian fibrations.

Proposition

An object p:XSp : X \to S in sSet +/SsSet^+/S is fibrant with respect to the above model structure precisely if it is isomorphic to an object of the form Y Y^\natural, for YSY \to S a Cartesian fibration in sSet.

Proof

This is HTT, prop. 3.1.4.1.

In particular, the fibrant objects of sSet +sSet +/*sSet^+ \cong sSet^+/* are precisely the quasicategories in which the marked edges are precisely the equivalences. Note that the Cartesian model structure on sSet +/SsSet^+/S is not the model structure on an over category induced on sSet +/SsSet^+/S from the Cartesian model structure on sSet +sSet^+!

Definition/Proposition

(coCartesian model structure on sSet +/SsSet^+/S)

There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.

Marked anodyne morphisms

A class of morphisms with left lifting property again some class of fibrations is usually called anodyne . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all left/right fibrations or inner fibrations, respectively.

The class of marked anodyne morphisms in sSet +sSet^+ as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.

Definition (HTT, Def 3.1.1.1)

The collection of marked anodyne morphisms in SSet +/SSSet^+/S is the class of morphisms An +=LLP(RLP(An 0 +))An^+ = LLP(RLP(An^+_0)) where the generating set An 0 +An^+_0 consists of

  • for 0<i<n0 \lt i \lt n the minimally marked horn inclusions

    (Λ[n] i) Δ[n] (\Lambda[n]_i)^\flat \to \Delta[n]^\flat
  • for i=ni = n the horn inclusion with the last edge markeed:

    (Λ[n] i,(Λ[n] i) 1)(Δ[n],), (\Lambda[n]_i, \mathcal{E} \cap (\Lambda[n]_i)_1) \to (\Delta[n], \mathcal{E} ) \,,

    where \mathcal{E} is the union of all degenete edges in Δ[n]\Delta[n] together with the edge Δ {n1,n}Δ[n]\Delta^{\{n-1,n\}} \to \Delta[n].

  • the inclusion

    (Λ[2] 1) (Λ[2] 1) (Δ[2]) (Δ[2]) . (\Lambda[2]_1)^\natural \coprod_{(\Lambda[2]_1)^\flat} (\Delta[2])^\flat \to (\Delta[2])^\natural \,.
  • for every Kan complex KK the morphism K K K^\flat \to K^\sharp.

The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.

Proposition

A morphism p:XSp : X \to S in SSet +SSet^+ has the right lifting property with respect to the class An +An^+ of marked anodyne maps precisely if

  1. pp is an inner fibration

  2. an edge ee of XX is marked precisely if it is a pp-Cartesian morphism and p(e)p(e) is marked in SS

  3. for every object yy of XX and every marked edge e¯:x¯p(y)\bar e : \bar x \to p(y) in SS there exists a marked edge e:xye : x \to y of XX with p(e)=e¯p(e) = \bar e.

Proof

This is HTT, prop. 3.1.1.6

Remark

Thus, if (X,E X)(S,E S)(X, E_X) \to (S,E_S) is a morphism in sSet +sSet^+ with RLP against marked anodyne morphisms, then its underlying morphism XSX\to S in sSetsSet is almost a Cartesian fibration: it may fail to be such only due to missing markings in E SE_S.

However, if all morphisms in SS are marked, then (X,E X)S (X,E_X) \to S^\sharp has the RLP against marked anodyne morphisms precisely when the underlying morphism XSX\to S is a Cartesian fibration and exactly the Cartesian morphisms are marked in XX, (X,E X)=X (X,E_X) = X^\natural — in other words, precisely if it is a fibrant object in the model structure on sSet +/SsSet^+/S.

See also HTT, remark. 3.1.1.11.

The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in sSet +sSet^+ is a morphism whose underlying morphism in sSet is a monomorphism.

Proposition

(stability under smash product with cofibrations)

Marked anodyne morphisms are stable under “smash product” with cofibrations:

for f:XXf : X \to X' marked anodyne, and g:YYg : Y \to Y' a cofibration, the induced morphism

(X×Y) X×Y(X×Y)X×Y (X \times Y') \coprod_{X \times Y} (X' \times Y) \to X' \times Y'

out of the pushout in sSet +sSet^+ is marked anodyne.

Proof

This is HTT, prop. 3.1.2.3.

As a model for the (,1)(\infty,1)-category of (,1)(\infty,1)-categories

The Joyal model structure for quasi-categories sSet JoyalsSet_{Joyal} is an enriched category enriched over itself. So it is not a simplicial model category in the standard sense, which means sSet QuillensSet_{Quillen}-enriched.

Indeed, the full sSet-enriched subcategory (sSet Joyal) (sSet_{Joyal})^\circ on fibrant-cofibrant objects is a model for the (∞,2)-category (∞,1)Cat of (∞,1)-categories. For many applications it is more convenient to work just with the (∞,1)-category of (∞,1)-categories inside that, obtained by taking in each hom-object quasi-category the maximal Kan complex.

The resulting (∞,1)-category should have a presentation by a simplicial model category. And the model structure on marked simplicial sets does accomplish this.

References

Marked simplicial sets are introduced in section 3.1 of

The model structure on marked simplicial oversets is described in section 3.1.3

Revised on January 13, 2013 00:10:28 by Stephan Alexander Spahn (192.87.226.73)