nLab
model structure for left fibrations

Idea

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

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

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

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

$$((\mathrm{SSet}/S{)}_{\mathrm{lfib}}{)}^{\circ }\simeq \mathrm{LFib}(S).$$((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 $\mathrm{Fun}(S,\mathrm{\infty }\mathrm{Grpd})$ (for left fibrations) and $\mathrm{Fun}(S^{\mathrm{op}},\mathrm{\infty }\mathrm{Grpd})$ (for right fibrations).

Motivation

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 $p:X\to S$ of quasi-categories, the (∞,1)-functor $S^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Grpd}$ that the left adjoint to the Grothendieck construction extracts from it is all encoded in the pushout $X^{◃}{\coprod }_{X}S$ in

$$\begin{array}{ccc}X& \stackrel{p}{\to }& S\\ \downarrow & & \downarrow \\ X^{◃}& \to & X^{◃}\coprod _{X}S\end{array},$$\array{ X &\stackrel{p}{\to}& S \\ \downarrow && \downarrow \\ X^{\triangleleft} &\to& X^{\triangleleft} \coprod_X S } \,,

where $X^{◃}=(*)\star X$ is the join of $X$ with the point, i.e. $X$ 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 $\mathrm{sSet}/S$ 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 $(\mathrm{\infty },1)$-Grothendieck construction it induces a weak equivalences of the $(\mathrm{\infty },1)$-functors that classify these fibrations.

Definition

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

$$\begin{array}{ccc}X& \hookrightarrow & X^{◃}\\ \downarrow & & \downarrow \\ S& \to & S\coprod _X{X}^{◃}& =:C^{◃}(f)\end{array}$$\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 $f$.

The colimits in sSet are computed componentwise, so that the set of vertices $C^{◃}(f{)}_0$ is the disjoint union of the vertices of $S$ and one extra vertex $v$, the cone point.

Definition

(HTT, def. 2.1.4.5)

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

A morphism $f:X\to Y$ 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

  $$X^{◃}\coprod _{X}S\to Y^{◃}\coprod _{Y}S$$X^{\triangleleft} \coprod_X S \to Y^{\triangleleft} \coprod_Y S

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

Proposition

This is a

• left proper

• simplicial

• combinatorial model category.

(HTT, prop 2.1.4.7, 2.1.4.8)

We have

• The fibrant objects are precisely the left fibrations (HTT, corollary 2.2.3.12)

• Every left anodyne morphism is an acyclic cofibration. (HTT, prop 2.1.4.9)

Properties

Weak equivalences

Proposition

Let $S$ be any simplicial set. Every morphism

$$\begin{array}{ccccc}X& & \to & & Y\\ & \searrow & & \swarrow \\ & & S\end{array}$$\array{ X &&\to&& Y \\ & \searrow && \swarrow \\ && S }

in ${\mathrm{sSet}}_S$ for which $X\to Y$ is left anodyne is a weak equivalence in the model structure for left fibrations.

This is HTT, prop. 2.1.4.6.

Proof

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

$$(\Lambda [n{]}_i{)}^{◃}\coprod _{\Lambda [n{]}_i}S\to (\Delta [n]{)}^{◃}\coprod _{\Delta [n{]}_i}S$$(\Lambda[n]_i)^{\triangleleft} \coprod_{\Lambda[n]_i} S \to (\Delta[n])^{\triangleleft} \coprod_{\Delta[n]_i} S

is a weak equivalence in ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

Observe that this is a pushout

$$\begin{array}{ccc}\Lambda [n+1{]}_{i+1}& \to & (\Lambda [n{]}_i{)}^{◃}\coprod _{{\Lambda }_i^n}S\\ \downarrow & & \downarrow \\ \Delta [n+1]& \to & (\Delta [n]{)}^{◃}\coprod _{{\Delta }^n}S\end{array}$$\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 $\Lambda [n+1{]}_{i+1}\to \Delta [n+1]$ and therefore a weak equivalence.

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

The simplicial set $({\Lambda }_1^2{)}^{◃}{\coprod }_{{\Lambda }_1^2}S$ is accordingly the simplicial set $S$ with only two of the three faces of this tetrahedron over $\sigma$ erected.

The map $({\Lambda }_2^3)\to ({\Delta }^2{)}^{◃}{\coprod }_{{\Delta }^2}S$ identifies the horn of this tetrahedron given by these two new faces and the original face $\sigma$.

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

Change of base

For every morphism $j:S\to S\prime$ we have the corresponding adjunction on overcategories

$$(j_{!}⊣j^{*}):\mathrm{sSet}/S\stackrel{\stackrel{f_{!}}{\to }}{\underset{j^{*}}{\leftarrow }}\mathrm{sSet}/S\prime ,$$(j_! \dashv j^*) : sSet/S \stackrel{\overset{f_!}{\to}}{\underset{j^*}{\leftarrow}} sSet/{S'} \,,

where

• $j_{!}$ is given by postcomposition of $j$;

• $j^{*}$ is given by pullback along $j$.

Proposition

(change of base)

This is a Quillen adjunction with respect to the model structures for left fibrations over $S$ and $S\prime$, respectively. (HTT, prop. 2.1.4.10)

If $j$ is a weak equivalence in ${\mathrm{sSet}}_{\mathrm{Joyal}}$, then this is a Quillen equivalence. (HTT, remark. 2.1.4.11)

Grothendieck construction

Proposition

($(\mathrm{\infty },1)$-Grothendieck construction)

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

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

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

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

Related concepts

The operadic generalization is the

• model structure for dendroidal left fibrations.

References

This is the content of section 2.1.4 of

• Jacob Lurie, Higher Topos Theory

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