model structure for left fibrations


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The model structure for right fibrations (SSet/S) rfib(SSet/S)_{rfib} is a model category structure on the overcategory SSet/SSSet/S of simplicial sets over a given quasi-category SS, that presents the (∞,1)-category of right Kan fibrations of quasi-categories over SS

((SSet/S) rfib) RFib(S). ((SSet/S)_{rfib})^\circ \simeq RFib(S) \,.

This is the (,1)(\infty,1)-analog of the (2,1)(2,1)-category Fib grpd(S)Fib_{grpd}(S) of categories fibered in groupoids over a category SS.

Similarly there is an analogous model structure for left fibrations that models left Kan fibrations, i.e. op-fibrations in ∞-groupoids

((SSet/S) lfib) LFib(S). ((SSet/S)_{lfib})^\circ \simeq LFib(S) \,.

The extension of this from right fibrations to Cartesian fibrations and from left fibrations to coCartesian fibrations is the model structure for coCartesian fibrations.

The (∞,1)-Grothendieck construction relates this to the global model structure on functors that presents the (∞,1)-category of (∞,1)-functors Fun(S,Grpd)Fun(S,\infty Grpd) (for left fibrations) and Fun(S op,Grpd)Fun(S^{op},\infty Grpd) (for right fibrations).


The following model category structure is best understood with the (∞,1)-Grothendieck construction in mind, which it serves to model.

Recall from the discussion there that given a morphism p:XSp : X \to S of quasi-categories, the (∞,1)-functor S opGrpdS^{op} \to \infty Grpd that the left adjoint to the Grothendieck construction extracts from it is all encoded in the pushout X XSX^{\triangleleft} \coprod_X S in

X p S X X XS, \array{ X &\stackrel{p}{\to}& S \\ \downarrow && \downarrow \\ X^{\triangleleft} &\to& X^{\triangleleft} \coprod_X S } \,,

where X =(*)XX^\triangleleft = (*) \star X is the join of XX with the point, i.e. XX with an initial object freely adjoined to it.

More discussion of why this is the case is at Adjoints to the Grothendieck construction.

The model category structure described below declares that a morphism in the overcategory sSet/SsSet/S is a weak equivalence if it induces a weak equivalence of the quasi-categories given by these pushouts. So this is effectively saying that we regard a morphism of right fibration of quasi-categories as a weak equivalences, if under the left adjoint to the (,1)(\infty,1)-Grothendieck construction it induces a weak equivalences of the (,1)(\infty,1)-functors that classify these fibrations.


For f:XSf : X \to S a morphism of simplicial sets, write C (f)C^{\triangleleft}(f) for the pushout

X X S S XX =:C (f) \array{ X &\hookrightarrow& X^{\triangleleft} \\ \downarrow && \downarrow \\ S &\to& S \coprod_{X} X^{\triangleleft} & =: C^{\triangleleft}(f) }

in the category sSet of simplicial sets. Call this the left cone over ff.

The colimits in sSet are computed componentwise, so that the set of vertices C (f) 0C^{\triangleleft}(f)_0 is the disjoint union of the vertices of SS and one extra vertex vv, the cone point.


(HTT, def.

The model structure for left fibrations or covariant model structure (sSet/S) lfib(sSet/S)_{lfib} on SSet/SSSet/S is given by

A morphism f:XYf : X \to Y is

  • a cofibration if the underlying morphism of simplicial sets is a cofibration in the standard model structure on simplicial sets, i.e. a monomorphism;

  • a weak equivalence if the induced morphism of cones

    X XSY YS X^{\triangleleft} \coprod_X S \to Y^{\triangleleft} \coprod_Y S

    is a weak equivalence in the Joyal model structure for quasi-categories, where X X^{\triangleleft} is the join X :=*XX^{\triangleleft} := {*} \star X.


This is a

(HTT, prop,

We have

An alternative characterization of this model structure is:


The model structure for left fibrations is the left Bousfield localization of the overcategory model structure on SSet/XSSet/X induced by the model structure for quasicategories on SSetSSet at the set of maps {Δ 0 nΔ n|n0,Δ nX}\{ \Delta^n_0 \hookrightarrow \Delta^n | n\ge 0, \Delta^n \to X \} indexed by all the simplices of XX.

This is mentioned in Heuts-Moerdijk, p.5; see also this discussion.


Weak equivalences


Let SS be any simplicial set. Every morphism

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

in sSet SsSet_S for which XYX\to Y is left anodyne is a weak equivalence in the model structure for left fibrations.

This is HTT, prop.


Recall from here that left anodyne morphisms are the weakly saturated class generated by the horn inclusions (i.e. under transfinite composition of retracts of pushouts). Therefore it is sufficient to check the statement for these generating morphisms.

By the definition of weak equivalences above, this means that we need to check that

(Λ[n] i) Λ[n] iS(Δ[n]) Δ[n] iS (\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda[n]_i} S \to (\Delta[n])^{\triangleleft} \coprod_{\Delta[n]_i} S

is a weak equivalence in sSet QuillensSet_{Quillen}.

Observe that this is a pushout

Λ[n+1] i+1 (Λ[n] i) Λ i nS Δ[n+1] (Δ[n]) Δ nS \array{ \Lambda[n+1]_{i+1} &\to& (\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda^n_i} S \\ \downarrow && \downarrow \\ \Delta[n+1] &\to& (\Delta[n])^{\triangleleft} \coprod_{\Delta^n} S }

of the inner anodyne morphism Λ[n+1] i+1Δ[n+1]\Lambda[n+1]_{i+1} \to \Delta[n+1] and therefore a weak equivalence.

To illustrate the above pushout property set n=2n = 2 for example. Start with a 2-simplex σ\sigma in SS. Then (Δ 2) Δ 2S(\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S is the original simplicial set SS together with a tetrahedron Δ 3\Delta^3 built over σ\sigma. One face of the tetrahedron is the original 2-simplex σ\sigma in SS, the three others “stick out” of SS:

The simplicial set (Λ 1 2) Λ 1 2S(\Lambda^2_1)^{\triangleleft} \coprod_{\Lambda^2_1} S is accordingly the simplicial set SS with only two of the three faces of this tetrahedron over σ\sigma erected.

The map (Λ 2 3)(Δ 2) Δ 2S(\Lambda^3_2) \to (\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S identifies the horn of this tetrahedron given by these two new faces and the original face σ\sigma.

The pushout therefore glues in the remaining face of the tetrahedron and fills it with a 3-cell.

Change of base

For every morphism j:SSj : S \to S' we have the corresponding adjunction on overcategories

(j !j *):sSet/Sj *f !sSet/S, (j_! \dashv j^*) : sSet/S \stackrel{\overset{f_!}{\to}}{\underset{j^*}{\leftarrow}} sSet/{S'} \,,


  • j !j_! is given by postcomposition of jj;

  • j *j^* is given by pullback along jj.


(change of base)

This is a Quillen adjunction with respect to the model structures for left fibrations over SS and SS', respectively. (HTT, prop.

If jj is a weak equivalence in sSet JoyalsSet_{Joyal}, then this is a Quillen equivalence. (HTT, remark.

Grothendieck construction


((,1)(\infty,1)-Grothendieck construction)

For CC a simplicially enriched category and S=N(C)S = N(C) its homotopy coherent nerve, there is a Quillen equivalence

(sSet/S) lfib[C,sSet Quillen] (sSet/S)_{lfib} \stackrel{\leftarrow}{\to} [C, sSet_{Quillen}]

between the model structure for left fibrations over SS and the global model structure on sSet-functors on CC with values in sSet equipped with the standard model structure on simplicial sets.

For more on this see (∞,1)-Grothendieck construction.

The operadic generalization is the


This is the content of section 2.1.4 of

There the model structure (sSet/S) lfib(sSet/S)_{lfib} is called the covariant model structure and the model structure (sSet/S) rfib(sSet/S)_{rfib} the contravariant model structure.

The alternative construction as a localization is mentioned in

Further discussion is in

Revised on December 16, 2015 04:11:17 by Urs Schreiber (